Hello my friends, my name is Michał Śmigiel, but you can call me Michael. In this video I will show you the solutions for 2017 Math Kangaroo competition for levels 11 and 12. I hope you will enjoy it, I hope it will help you better understand these questions and math in general. If you have any questions about those solutions you can call me on an email that you can see on the bottom of this video. You can also check my YouTube channel Let's Solve It Together and well hope you will enjoy those solutions. So question number one 20 times 17 divided by 2 plus 0 plus 1 plus 7. Okay so this is 20 times 17 divided by and here we have 10. And now before multiplying let's simplify this fraction and we get 2 times 17 divided by 1 which means just 2 times 17 and this is 34. So the first answer for the question number one is C. Let's check in answer key. So here we have number one C for level 11 and 12. So one tip here if you have in denominator or numerator only multiplication then do not multiply it in the beginning just make everything you need in the other part of fraction and then check if you can simplify. Because you get you get a simpler multiplication, smaller numbers it lowers the probability of making a simple mistake in calculations which unfortunately in kangaroo is well in kangaroo is very often problem. Okay so question number two Ben likes to play with his H0 model railroad. He modeled some things in the H0 ratio of 1 to 87 even a two centimeter high model of his brother. How tall is Ben's brother? Okay so this ratio 1 to 87 means that everything is lowered times 87. So if you have a height of his brother the real height it was just divided by the 87 and you got two centimeters. So if you know that the the lowered height of his brother is two centimeters then if you want to have the real height you need to multiply this by 87. You need to reverse the thing you did. So two times 87 160 plus 14 it's 174 centimeters which is 1.74 meters and that's answer A. Okay let's check in this yeah this is answer A. Okay so we can go to question number three. In the figure we see 10 islands that are connected by 15 bridges. What is the smallest number of bridges that need to be eliminated in order to make it impossible to get from A to B by using the bridges? Okay so maybe let's check it closer. So here we have this this picture. We want to get from A to B. Well simply we can see that if we cut these four bridges that go from A then we are cut off. Okay so it maybe it's four it's really not five. But can we make it smaller? Well by cutting this for example let's take a look at B. We could also cut those five bridges and we cannot get there. Okay so that's also the possibility but it's it's not the answer because we need to cut even more bridges than here. But those two bridges well they come from this line. If we cut this one bridge here in A we it is it's like cutting those two bridges close to B. So let's cut this one bridge and we we don't have this southern path from A to B. Now let's check into north and central path. And well here we have two bridges but from this island we get to this island and then from this island we only have this one way. So if we cut this one this one bridge it's like cutting those two bridges. And that's that lowers our possibilities. And the only thing here well we need to cut this because if we don't cut this then we have a possibility to get to B. So we need to cut this one this one and this one and we get three bridges. And we will not find the smaller amount because well we cannot cut one bridge instead of those two bridges. This is impossible. So the lowest answer is C. 3C. So let's check the answer key. So 3C. Yeah. So it's like just watching on a picture and finding the easiest the smallest amount. You need to just try a little a few things. Okay. And the fourth one. Two positive numbers A and B are each are such that 75% of A equals 40% of B. This means that okay so 75% of A it's 0.75 times A. And this equals to 40% of B. Which is 0.4 B. So we have a question. Well but this equation is not in the answers. So we need to do something. Well watching into these answers we can see that we need those two numbers to be integers. Okay. So let's first make this number an integer. Well we need simply to multiply it by 4. I want it to disappear. Okay. We multiply it by 4. And here 0.75. It's 3 fourth of A. Which multiplied by 4 will give us 3 A. Here well 0.4 times 4 is 1.6 B. Okay. And now we also need to to make this integer. So here we need just to multiply by 5. And here we get 15 A equals 1.6 times 5. It's 8 B. 15 A equals 8 B. This is answer A. So 4 A. Once again let's check in the answer key. 4 A. Yes. So a fun way to multiply by 5. First divide by this number by 2. 1.6 divided by 2 is 0.8. And now multiply it by 10. Which is very simple because you just need to move your comma or point or dot. Here we are using dot on the right by one place. So 0.8 times 10. It's 8. And the same simple well not maybe exactly the same but very similar way you can multiply by 5 in a easy way. So number 5. Four of the following five figures are part of the graph of the same quadratic function. Which one is not a part of the graph? Okay. So let's try to draw this in a one graph. So here we have x and y axis. So let's say that two those parts will be 1. So here is 1. Here is 2. And so on. And the same on y axis. Y 1 2 and so on. So here we see the part between 1 and let's say 2.5. So it starts bottom from 1. And it has a root just before 2. And it goes and it's increasing. Here we see something that is close to the y axis. So we see that in it crosses the y axis just between minus 1.5 and minus 2. So here is minus 1. Here is minus 2. And it will go through this point. And here it's increasing. And here also increasing. Okay. Then we see C. 3 between 3 and 4. Let's say 2.5 and 4. So something like here and here. And it goes down. And we see it's not right. It doesn't look like a quadratic function. The graph of a quadratic function is a parabola. So it cannot look like this. But let's draw it more. So here we see more on the left side of the y axis. So here it looks like it connects with this part. And somewhere here it starts to decrease. Okay. And here we see between minus 4 and minus 3. So here is minus 1, minus 2, minus 3 and minus 4. And we see that it also has a root just above minus 4. And here this function is decreasing. So what can we say? Well, what doesn't fit here? This part doesn't fit here. The graph of a quadratic function is a parabola. And no matter which parabola, if you take this interior here as a figure, it has to be convex. So convex, which means that if you take two points that are in the interior and you connect them with a line, this line doesn't go outside this figure. Here, this part breaks the convexity. When you take this point, two points, and you draw a line, you see it goes outside our figure. So it's not convex, it's concave. So this part doesn't fit. Which means that the right answer is answer C. Let's check this. And 5C. Yes, so that's the right answer. So now question 6. Given a circle with center O and diameters A, B and C, X, such that OB equals BC, what portion of the area of the circle is shaded? Okay, so here we have this graph, this picture. Here is O. These are diameters. So this is AB, this is C, X. If they are diameters, they have to move through this point O. And what we also know is that OB, so this segment, has the same length as BC. And it also has the same length as a segment OC, because they are both radius of the circle. So they need to be the same. So we get equilateral triangle here. And this is 60 degrees. And also this is 60 degrees, because if you have two angles made by two intersecting lines, then they have the same amount of degrees, the same measure. So there also needs to be 60. And these figures here, these parts of circle, how do they call? I know how they call in English, in Polish, but... Section of a circle, slice of circle. Okay, so they are slice of circle. Okay, so how do we measure area of them? Well, if we want to measure an area of whole circle, then we just have a formula PR², where R is a radius of our circle. Here we don't know what is the radius, and we don't need it. What we need that the area of all circle is PR². And we also want to calculate the sums of the areas of those slices of circle using R. R will be unknown number, which will simplify. Because how do we measure area of slice of a circle? Well, we take this number of degrees, this angle, measure of this angle, 60 degrees, divided by 360 degrees, which is whole the circle. And we multiply it by area of a circle. So this angle, the measure of this angle, tells us what part of all the circle is this slice of a circle. If it would be 40 degrees, the area of the circle would be 40 times divided by 330 part of the area of circle. So here we have 60 divided by 360. Degrees simplify, zeros simplify, 6 and 36 also simplify, and we get 1 divided by 6. So it's one-sixth of a circle. And this part here is exactly the same. So the area of a shaded part of a circle is 2 times one-sixth PR², which we can also simplify. Here we get 1, here we get 3, and we get one-third PR². And what the part of all the circle is? Well, we can already say that this is one-third, or we can divide these areas, one-third PR² divided by PR². And they simplify, and we get one-third of a circle. So this is answer B. Let's check in answer key. 6B. Yeah, so I will tell it once again. Always check the answers if you have them, because that's how you get confident that you are right. And that's very important to be confident when you are learning something new. Okay, so now question 7. A bar consists of two white and two gray cubes glued together such that the result is 4 times 1 times 1 bar with two white cubes on one end and two gray cubes on the other end. So something looking like this. Which figure can be built with 4 bars? Okay, which can be built? So four of them cannot be built. Well, A looks okay. But B? B is bad, because when you look on the top back of this figure, you see four white cubes. That's impossible, because how can you build such a figure from these bars? Well, you just need to put them like this. One here, second on the top of this, and also the same in the second line. So here you put another bar, and here one more bar. And that's how you get this figure. And this figure means that you would need a bar consisting only of from four white cubes. And you don't have it. You have a cube that you have a bar that has two white and two black. So B is impossible. The same C. You also would need a bar that has only four white cubes in the in back on top. So once again, that's not possible. D is also not possible, because back on top you have four black cubes. You don't have these bars. So it's also impossible. And E is also impossible, because here on front, on the bottom, you have four black cubes. You would need a bar that has four black cubes. No, you don't have. So the only possible is A. 7A is possible. So let's check the answer. 7A, yeah. So that's how you can solve it fast and efficient. Okay, and now the last question in this video, number eight. Which quadrant contains no points of the graph of the linear function f of x equals minus 3.5 x plus 7. Okay, so that's, well, in Poland, I think in many countries, it's common knowledge that, first of all, what this formula tells us? This formula tells us that this linear function goes down. Let's check. That it's decreasing. Okay, very, very hard, hard word. Okay, so maybe it looks like this. Maybe like this. Maybe like this. So we see that it surely has to go through two and four. And the question is, which quadrant contains no points? So this is one or third. So this is what we get from this number minus 3.5. But what we get from seven? Well, seven, if you want to calculate the value of this function, when x equals zero, then you get seven. Well, this cancels out and you get 7, so this number here is always the value of this linear function when x equals 0, which means that for 0 you have 7, so it goes through this point and it's decreasing, so it has to look like this. So what we see? It goes through 1, 2 and 4, but it doesn't go through 3. So the answer is C. 8C. Let's check the answer. 8C, yeah, 8C. Each of the following five boxes is filled with red and blue marbles as labeled. Ben wants to take one marble out of the boxes without looking. From which box should he take the marble to have the highest probability of getting a blue marble? Well, we just need to count this. So in A, he has 18 marbles, 10 of them are blue. So what is the probability of blue? Well, 10 divided by 18, which we can simplify to 5 divided by 9, 5th 9, or however you call it. In B, he has 10 marbles at all, and 6 of them are blue, 6 10th, which is 3 divided by 5. In C, we have 14 marbles, and 8 of them are blue, which we can simplify to 4 divided by 7. In D, we have 7 blue and 7 red, which will give us 7 divided by 14, and it's half. And this number is the smallest from all those that we already have, so we can cancel this. And in E, we have at all 21 marbles, and 12 of them are blue. We can simplify this to 4 divided by 7. And this is the same as in C. And there cannot be two right answers in Kangaroo, which tells us that they cannot be right. So now we have a question. What is bigger, 5 divided by 9, or 6 divided by 10? Well, 5 divided by 9, we can divide this like this, I don't know how you call it. And we will get 0.45 in a bracket, which means that these numbers continue for infinity. So you have something like this, 4, 5, 4, 4, 5, 4, 5, and so on. And this will be infinite numbers of them. No, sorry, I've made a mistake. 5 divided by 9, it will be 0, or 0.54. OK, so not for 5, 5, 4, 5, 4, 5, 4, and so on. On the other hand, 6 divided by 10, it's 0.6. Which one is bigger? Well, 0.6. So the right answer is answer B, 9B. Of course, you can also check this 4 divided by 7. What you will get? Well, 1 divided by 7, it would be 0.142857, and so on, the same numbers. And 4 divided by 7, it will be 4 times more, so it will be 0.571428, and so on, the same numbers all the time. As you can see, it's also smaller than 0.6. So still, the answer is B, 9B. Let's check this answer in answer key, 9B, yeah, 9B. OK, so number 10, the last question for 3 points, and then we get to 4 points. The graph of which of the following functions has the most points in common with the graph of the function f of x equals 6? Well, f of x equals 6, it's a very simple graph. It goes just through this point, 0, 0, and it goes 45 degrees from the axis. So it goes like this one, and this here, and also the same this way, OK? So this is f of x equals 6. So now they give us 5 possibilities. x squared, how does x squared go? Well, it goes like this. It's easy to tell that it has 2 intersections with this function. It has 2 common points with this function. OK, so maybe that's it, but actually not, because x to the power of 3, well, in 0, it will be 0. So this is first intersection. Second will be in 1, because 1 to the power of 3, it is also 1, and it goes this. And minus 1 to the power of 3 is minus 1, and we go this way. So this has 3 intersections, so x squared is already losing. And this will be the answer, because x to the power of 4, well, it looks like x squared just higher. So here, from minus 1 to minus 1, it goes closer to 0, but when it reaches 1, then after that it goes even higher than x squared. This is more steep function than x squared. And actually, exactly the same as x squared, it has 2 intersections with function f of x equals 6 in 0 and in 1, and no other. So it's also losing. Minus x squared looks exactly the same, but it goes down, and it also has 2 intersections in minus 1 and in 0 with function f of x equals 6, so it's also not that. And the last part, well, if that would be g5 of x equals x, that it would be winning, because it would have infinite number of common points with this function. But this is minus x. Minus x goes almost the same with this 45 angle to the axis, but it's decreasing, which means that it has only one common point with f of x, which also means that the answer b wins, because it has 3 intersections with this function f of x. So let's check. 10b? 10b? Yeah. 10b. Okay. So now, question 11. Three mutually externally tangent circles with centers a, b, and c have the radii of 3, 2, and 1, respectively. What is the area of the triangle ABC? Okay, so it looks like this. I will draw these circles as good as I can, and here let's do this one. So we have something like this, and this too is the worst. Well, let's say it looks like this. I will not do it right. And it has a center here. So they ask us about what is the area of the triangle ABC. So this triangle. Well, these circles are mutually externally tangent. What this means? Well, this means that they have one common point here. For example, this biggest and the smallest circle has one common point here. And this point is the point where their radiuses go, and they make one line. Why? Because this is one segment. Why? If we would write a tangent line here to those circles, well, it will be tangent to both of those circles, and it would go through this point. And both radiuses of those circles are perpendicular to this line. And if they are perpendicular, then they make an angle, which is 180 degrees, which means that this is one line. And the same goes here, and the same goes here. So the segments of this triangle consist of radiuses of those circles. And well, the radius of this biggest one is 3. Here the smallest one is 1, and this average is 2. So we get a triangle that has 5 here, 4 here, 3 here. Okay, so we have the lengths of the sides of a triangle. How we can calculate the area of a circle? Well, one way is to use a Heron formula. What is Heron formula? It's a formula that tells us that area, I'm using P because in Polish area is pole, and we always use P as to write, to calculate the area, so sorry for that. This equals to square root of small p times p minus a times p minus b times p minus c. Where p is a half of a perimeter of this circle. So it is half of a plus b plus c, while a, b, and c are the lengths of the sides of this circle. So we can calculate it like this, and we end up with such numbers. So what is this? What is p? This is half times 3 plus 4 plus 5, and this is half times, well, this is 12, and this is 6. So p is 6. So we have 6 times 6 minus 3 times 6 minus 4 times 6 minus 5. And we can now calculate this. So this is square root of 6 times 3 times 2 times 1. And we here see 6 times 6. And square root of this is 6. So this is the area of the circle. And that's the one way that we can calculate this area. And if you have a typical triangle, and you have the lengths of the sides of this triangle, then the Heron formula is probably what you want to use. But here, we can make it easier, because this is right triangle, yeah? 3, 4, 5, well, I think every mathematician should know these numbers. If you see 3, 4, 5, the sides of the circle, the sides of the triangle, you already know this is right triangle, okay? So if you don't know this, then you should know it, okay? Remember this. 3 squared plus 4 squared equals 5 squared. 9 plus 16 equals 25, yeah, it works. So it satisfies the Pythagorean formula. Actually, there's this story that Pythagoras, or whoever actually, this person, or people where found his Pythagorean formula in ancient Egypt, where people were using 3, 4, and 5 lengths of triangle, because they knew that this is right triangle. And he was thinking about this, and get to the Pythagorean formula that every triangle which satisfies a squared plus b squared equals c squared has to be right triangle. So that's the way he or they found it. So 3, 4, 5, you always see that this is right triangle. 3 and 4 are the low, the, catetus? I don't think so. 5 is hypotenuse of this triangle. So 3 and 4 are these, these sides, yeah? So close to the right angle here. And if we want to calculate the area, well, we can simply calculate 3 times 4 divided by 2. And this is 6 once again, yes? So here, we don't need to use Heron formula. We can simply see a right triangle and calculate typical, the easiest way to calculate the area of a triangle. Okay? So, the last question in this video, number 12. The positive number p is less than 1, positive, so bigger than 0. And the number q is greater than 1. Okay. Which of the following numbers is the largest? Okay. So first of all, we see that p is smaller than q. Now, p times q, well, because p is smaller than 1, we can write that this is smaller than 1 times q, which is equal to q. So p times q, answer a cannot be right because e is bigger. D cannot be also right because e is also bigger. So we can cut the answer a and d, and we get p plus q and p divided by q. So clearly, q is smaller than p plus q because p is positive. So it makes this number bigger. So e also is not the answer. The last question is p plus q or p divided by q. And well, it's in this combination, it's easy because p divided by q, well, p is smaller than 1 and this is, so this is smaller than 1 divided by q and q is bigger than 1, which means that this is smaller than 1 divided by 1, which is 1, which is smaller than q. So surely not p divided by q. So the answer is b, but if we would have answer q divided by p, then, well, there would be no right answer with this information because q divided by p can be bigger than p plus q, but can sometimes also be smaller than p plus q. So that would be a problem. We couldn't solve this. We couldn't give the answer with this information. So let's check 12b, 12b, yeah, 12b, yeah, so, and in 11, this was really a, 11a, yes. 13. Two right cylinders, a and b, have the same volume. The radius of the base of b is 10% larger than that of a. How much larger is the height of a than that of b? Well, so right cylinders, I think you mean that the basis of this cylinder is perpendicular to the side of the cylinder. That's what I think you mean. Okay, so what we know? We know that the volumes of these two cylinders are the same, and how we actually calculate the volume of cylinder? Well, it's area of basis, which in Poland we always write as PP, pole podstawy, which means area of basis, times height of the cylinder. And we know that they are the same, but the area of basis is not the same in both of them, and so thus the height cannot also be the same. In first, we know that radius of basis is 10% larger than that of a. So let's say that a has a radius r. This means that b has a radius 1.1r. So the area of basis in a equals pr², because the basis is a circle. And area of basis in b equals p times 1.1r², because the radius here is 1.1r, which give us 1.21pr². Yes, 1.1² is 1.21. Now we get that volume a equals 1.21pr, no, it equals pr² times hi, and this is the same as 1.21pr²hb, so we have, which is volume of b. So we have an equation that can give us the ratio ha to hb. Actually we want to know how much larger is the height of a than that of b. So we want to calculate from this aj. So we have pr²h, ha equals 1.21pr²hb. So if we want to calculate ha, then we have to divide by all the things that are close to aj, so by pr², and this cancels out. So we get ha equals 1.21hb, which means that aj is 21% bigger than hb, which gives us answer e. Now let's check it, 13e, 13e, yes. So that's the fast and efficient way to solve this. Okay, now 14. Okay, the faces of the polyhedron shown are either triangles or squares. Each square is surrounded by four triangles and each triangle is surrounded by three squares. If there are six squares how many triangles are there? Okay, so let's start here with this square and we know that there are three triangles, four triangles connected with it and we know also that these triangles are connected to the squares. I will draw them like this. It maybe doesn't look like on a typical net of a polyhedron where we would like to have connected some sides but I think it will make this draw easier to understand. So this side here is the same as this. They connect. The same this side connects with this side, this side this with this, that with that and that with this. Okay, so we have already five squares. We need one more but what we also know that this, for example this square, has to connect with four triangles. There are two of them, yes, and the other two, well, we can draw them here. And now let's understand what are these sides of this triangle. So this side connects with this one, this side connects with this one. Also this square has to be connected with two more triangles and now this side connects with this side and this side connects with this side. And we can ask this about these four sides. What do they connect with? So I have to admit that my personal RAM is too small to see this polyhedron, how it really looks like. But if it has to have any sense at all, then the last sixth square has to be here. And now this side connects with this side, this side connects with this side. I think that's simple. But what's not really simple, this side connects with... and yeah, well, my personal RAM is too small, so maybe with this side, or maybe with this side. All at all, if with this side, then this side connects with this side. One of those sides have to connect with one of those, both those sides have to connect with those both sides. And that's how it has to look if it has to work. So I don't see how this polyhedron really looks, but it has to be something like this. And now we can calculate, we want to know how many triangles are here. Well, 1, 2, 3, 4, 5, 6, 7, 8. So they are 8. So the answer is D. So 14D, 14D, yeah, 14D. So I don't know how it really looks, this polyhedron, but I am able to find the answer just by illogically drawing it and thinking what has to connect with what. OK, so now question 15. We have four tetrahedral dice, perfectly balanced, like all things should be, with their faces numbered 2, 0, 1, and 7. If we roll all four of these dice, what is the probability that we can compose the number 2017 using exactly one of the three visible numbers from each die? So, well, let's think that we can order them. So we have first dice, second, third, and fourth dice. And in all of them, because what, how does it look like? If we have this tetrahedral dice, then we see those three numbers when it's on the ground, but we don't see this number because, well, it's exactly on the ground, yes? So we don't see the face of this number. We know what is there because we know those four numbers, those three numbers, and there has to be this fourth one. But we don't see it. We want to compose the number 2017 using those that we can see. So, this one we cannot see. So, here we have four possibilities for a number that is on the ground, yes, so that we don't see. The same here, the same here, and the same here. So, all the numbers, all the possibilities are four to the power of four. And now the question is, how many is possibilities to compose the number 2017? So, I think most of them will be, because we have three numbers out of four from which we can choose our number. And this gives us many, many, many possibilities. So, I would rather think, how many are there possibilities that we don't, that we cannot compose this number? So, when we cannot? Well, we cannot if all the four dices have the same number on the ground, so we don't see it. For example, all the four dices are standing on number two. Then we don't have two. We only have zeros, ones, and sevens, and it's impossible to create 2017 from them. So, that's one possibility. The other is that they are, they all have four zeros. They have zeros on the ground, ones on the ground, and seven on the ground. So, it actually gives us four possibilities where we surely cannot compose number 2017. Okay, is that all? Well, let's think about a little bit less extreme situation. For example, let's say that on three, three of those dices stand on the same number, and the fourth stands on the other number. For example, three of them stands on two, and the next one stands on seven. Doesn't matter. So, can we compose 2017 from those in this situation? Yes, we can, because on this one we have to choose two, and here we have three dices where we can see zero, one, and seven, and we need to choose from them zero, one, and seven. So, zero, one, seven, and we have 2017. So, if there is at least one dice that we can see some number, that we are able to create 2017. What if there would be like two? Two, two, and seven, seven. So, two dices stand on two, and two dices stand on seven. Well, here we choose two, here we choose seven, and here doesn't matter. Doesn't matter. Zero, one, yeah, we have. What if we would have two twos, and let's say one and zero? Well, we can choose zero here, one here, two here, seven here, with no problem. So, the only possibilities that we cannot compose 2017 are those that we have all the dices standing on the same number, and they are four of them. So, what's the probability of this situation? Well, four divided by four to the power of four, which we can simplify to one divided by four to the power of three, which is one divided by 64. So, that's the probability that we cannot compose the number 2017. Now, the question is, what is the probability that we can compose this number? Well, that's easy. One divided by the other probability. One minus one divided by 64, which is 63 divided by 64, and this is answer B. So, let's check. 15B, 15B, 15B, yeah, 15B, that's the answer. So, that's how we can solve it. Okay, and the last question, 16. The polynomial 5x cubed, or to the power of three, plus ax squared, plus bx, plus 24, has integer coefficients a and b. Which of the following is certainly not a root of the polynomial? So, a few videos ago, I was talking about some zero places of a quadratic function. Well, what I meant, it was a root of this quadratic function. So, okay, so this question is if you know some theorem. I don't know if it has any name, but it states something like this. If you have a polynomial with all the coefficients that are integers, then all the integer roots of this polynomial has to be divisors of this last number, of this number without x, a free number, or however you call it in English. I don't know. So, all the roots of this polynomial have to be divisors of 24. And answer D, 5, is not the root of 24, is not a divisor of 24, which means it cannot be a root of this polynomial. So, simple answer D, 16D. Let's check it. 16D, yeah, 16D. So, that's the answer. So, you see, sometimes it's good to have some math knowledge in Kangaroo, because with this knowledge, I was able to solve this question like in seconds. Describing took much more, but solving a few seconds. Without this knowledge, oh, I think you can think, think, think, think, and don't find the answer. Julia has 2017 chips. 1009 of them are black and the rest are white. She places them in a square pattern as shown, beginning with a black chip in the upper left hand corner, alternating colors in each row and each column. How many chips of each color are left after she has completed the largest possible square? So, first of all, when I see this, I think of a chessboard. It also has black and white squares, one close to the other, and it's also a square. So, the first thing, if we have 1009 black and the rest are white, then we have 1009 black and 1008 white. Now, the largest possible square, we have 2017 chips. How is the biggest square that we can get? Well, that's simple. We just need to find the answer that if we calculate the square of it, it will be smaller than 2017, and for the next number, the square of it will be bigger than 2017. So, we can easily start with 40 squared equals 1600. On the other hand, 50 squared, it will equal 2500. So, this number has to be between 40 and 50. Okay, let's try 45, and that's a pretty good number, because 45 squared equals 2025, and we have 2017, and we clearly see 45 is the smallest number that is too big. Let's calculate 44 squared, and 44 squared, well, let's calculate it. 4 times 4, 16. So, here's 17, 16, once again, 17. We add them, 6, 13, 9, 1. So, it is 1936. So, that's how many chips we will use. So, how much will be left? So, 2017 minus 1936. Here we get 1, 9, 11. 11 minus 3 equals 8, and here it cancels out. So, 81 chips are left. So, first of all, answer A is not right, not known. 81 is not known. B is also not correct. 40 of each would be 80. We have 81. D is also not correct, because 41 each would be 82. We have left 81. So, C or E, 40 black and 41 whites, or 40 whites and 41 black ones. So, now let's look at this. Well, if we have like 2 on 2 square, this is black, this is white, this is white, and this is black. And if we would get 4 times 4, we will have pretty much the same situation in the way that there will be the same number of black and white. So, if our side has a length of even number, then we use exactly the same number of blacks as white. If it's odd number, then we use one more black than white. But it doesn't matter for us, because we have 44 sides, which is even. So, we used exactly the same number of whites and blacks. So, from all these numbers, we need to subtract the same numbers, which means that if there were one more black in the beginning, there will be also one more black in the end. So, 40 white and 41 black ones. So, answer E. If it would be an odd number, then we would use one more black and they will get to the point that there will be the same number of blacks and white. But in this situation, there has to be one more black than white, because there was one more black in the beginning than white. So, answer E. 17E. And yeah, 17E. Okay, so now let's go to the question 18. Two consecutive numbers are such that the sum of the digits of each of them are multiples of 7. At least how many digits does the smaller number have? Okay, so let's think about this. If we would like to have 27, two consecutive numbers, so the next has to be 28, the bigger one. So, well, here is some sum, here is also some sum. And well, in this situation, this sum is always bigger than 1 than here. So, they cannot be both multiples of 7. So, it cannot be that such a straightforward situation, 27 and 28, because it will be impossible that they are both, that the sums of the digits will be both multiples of 7. So, where can it break? Well, it can break when we go from 19 to 20, for example, because here we have sum 10, and here we have sum of 2. And it changed not by 1, it's not bigger by 1, but actually smaller than 8. So, let's think about this. If we would like have 129 and 130, what changes? Well, this number goes down by 9, this number goes up by 1. And this number, for example, in this situation doesn't change, which means that the difference between the sums of their digits is 8. This has 8, the sum of digits bigger by 8 than this one, which is not something that we want, because if the difference between them is 8, it's impossible that they are both multiples of 7. So, it's also not such a situation. So, let's think about other situation, 199 and 200. So, now this one goes down by 9, this also goes down by 9, but this one goes up by 1. So, in total, the difference is minus 17. So, the sum of these numbers, the sum of these digits is higher than the sum of these digits by 17. So, we see that it has changed. It's still not okay, because 17 is not a multiple of 7, which means that if one of these sums is multiple of 7, then the other cannot be. So, we want to find such a situation that this difference is on its own a multiple of 7. So, we see that two numbers were not okay, because the difference was 8. Three numbers were not okay, because the difference was 17. Actually, well, we had here three numbers, three digits, but only two of them was changing. That's what I'm saying that for two, we have the difference of 8. If you would have like 5, 6 here and 5, 6 here, it wouldn't change anything. The only important thing is that the two numbers are changing, and then this difference is 8. If three numbers are changing, then the difference is 17. If four numbers are changing, 1, 9, 9, 9 and 2000, well, this one goes down by 9, the same as this, the same as this, but this one goes up by 1. And so, what's the difference? Well, the difference is minus 26. And it's still not what we want, because this is not divisible by 7. But now we can see, here was 8, 17, 26. Do you see the pattern? Oh, here was minus 17, minus 8, minus 26. And in a simple situation, when only one number changes, the difference is 1. So, these numbers create arithmetical sequence with the difference minus 9. So, the next one for five numbers, it will be minus 35. And that's what we like. You know why? Because 35 is a multiple of 7, which means that they can both be multiples of 7. So, at least how many digits does the smaller number have? Okay, let's think about this, because it can be tricky. Because, okay, if we get this number and this number. so yeah the difference is 35 well they are not both multiples but doesn't matter but what with this number 9 9 9 9 and 10,000 well it's not something that we like because the sum of the digits of this number is one as and it's not multiple of 7 so it has to be at least 5 but let's find some number that actually satisfies it here we have the sum of these numbers is 37 well that's not what we want and we need these four nines in the end so what we would have what we would need here ah but what we would need here so that the sum of the digits is divisible by 7 well 36 what is divisible by 7 42 so 6 9 9 9 9 69,999 the sum of the digits is 42 and is divisible by 7 and we've if we add up one we have 70,000 and the sum is 7 and that's okay so it has a five numbers and that's the lowest possible number of digits that we can get so the answer is C 18 C let's check 18 C yes so that's the answer okay so now question number 19 in a convex quadrilateral ABCD the diagonals are perpendicular the sides have lengths AB equals 2017 BC equals 2018 CD equals 2019 figure not to scale but with the right numbers what is the length of AD okay so here this information that those those how do you call it Diagonals I was wondering yeah diagonals they are perpendicular and when you see information that they are perpendicular well what do you think of first well I mostly think about Pythagorean theorem and that's the way here 2019 2018 2017 we want to know this length so let's call it a B C D yeah this parts of those diagonals so we know that a square plus b square equals let's call 2017 X then this will be X plus 2 this will be X plus 1 okay it will be shorter so this is X plus 2 squared okay now b square plus c square equals X plus 1 squared c squared plus d squared equals X squared and we want to know what is a squared plus d squared okay so how can we find this well this from the first equation we got that b square equals X plus 2 squared minus a squared okay from the third we get that c squared equals X squared minus d squared and now we can substitute them into the second we get X plus 2 squared minus a squared equals nope nope plus c squared which means X squared minus d squared equals X plus 1 squared okay and here we have X squared and d squared and we can put them in the into the right side and we will get a squared plus d squared so something that we are looking for and here we will get X plus 2 squared minus a squared and the minus X plus 1 squared and the length this this is actually the square of the length that we are looking for we just need to simplify this first of all so let's simplify this well this is a square I know there shouldn't be a square because we put this on the right side there should be plus X squared okay so first of all X plus 2 squared minus X plus 1 squared and plus X squared this is our number so here I can see the formula because if you have a squared minus b squared this is exactly the same as a minus b times a plus b if we call a X plus 2 and b equals X plus 1 then we can get X plus 2 minus X plus 1 times X plus 2 plus X plus 1 and this is the same as this squares this this difference of squares plus X squared so now we can simplify them and actually in the first first bracket we get just one in the second bracket we get 2 X plus 3 and plus X squared so this is X squared plus 2 X plus 3 okay so this is this is the length that we are looking for this is the square of the length that we are looking for now we can well what we can do here more so did I do it right so X squared is 17 ah I see X squared plus 2 X plus 3 we can rewrite this as X squared plus 2 X plus 1 plus 2 why would I do this 1 plus 2 equals 3 okay simple but why would I do this because X squared plus 2 X plus 1 is the same as X plus 1 squared plus 2 so the square of our length that we are looking equals X plus 1 squared plus 2 now X was 2017 which get gives us 2018 square plus 2 this is the square of the length that we are looking for so the length that we are looking for is D the square root of this number and this is D so let's check 19 D yeah 19 D so that's that's a I think a nice way to find the the answer here and okay the last question 20 TT tries to be a good little kangaroo but lying is too much fun therefore every third thing that she says is a lie and the rest is true sometimes she starts with a lie and sometimes with one or two true statements TT is thinking about of a two-digit number and is telling her friend about it she says one of the digits is two it is larger than 50 it is an even number I will write this like this it is less than 30 so something like this it is divisible by 3 one of the digits is 7 okay so these are the six statements what is the sum of the digits of the number TT is thinking of okay so as I understand every third thing she says is a lie so well if that was a lie then this means that this is true this is true this is a lie this is true and this is also true okay and so on if this is true false true ah sorry true starts with true but in Polish Pravda starts with P sorry sorry okay so once again we have false so every third of these statements is false so let's think about so we have three possibilities false true true false these are the three possibilities so think about I would start with the last one why because the last one tells us that this number is bigger than 50 it is true the second is true but also the fourth is true so the truth that you would be that it's smaller than 30 so this number would have to be bigger than 50 and smaller than 30 it's impossible so we can cancel this it's not possible okay let's go to the second so the first one is right so this has seven this has two and the last one is right which means that this has also seven so this can be 27 or 72 because this has to be the two-digit number okay so let's think about the other questions it's false that this is bigger than 50 so it cannot be this one it has to be 27 now it's true that this is an even number no it's not true so no so it's also impossible if I have this right okay so let's go to the first one so it doesn't have two it is bigger than 50 it is even it is it is not smaller than 30 and it's okay because it's bigger than 50 already we know it's divisible by 3 and it has 7 okay it has 7 but it is even so it has to be 70 something and it has to be bigger than 50 so 70 something is bigger than 50 so that's one does this one is okay it has to be even so there has can be 0 2 4 6 or 8 but it doesn't have 2 so it cannot be 72 which we had already in the previous one okay so we have four possibilities it is not smaller than 30 okay it is divisible by 3 70 is not divisible by 3 also 74 and 76 is not so this has to be 78 what is the sum of the digits well 15 so the answer is D so 20d 20d 20d yes this is 20d so so that's it okay so number 21 how many positive integers have the property that the number obtained by deleting the last digit is equal to 1 14th of the original number okay so let's think about some examples let's say 36 this is positive integer and if we delete the last digit well we get 3 so from 36 we get 3 well obviously it's not something that satisfies us because it's 3 is not 1 14 for 36 it's what 1 12 so okay so how can we solve this well so let's say that the number that we get after deleting is a and the number we delete is B what we can clearly say about B that this is a one digit number because we delete the last digit so it is from 0 to 9 what we can say about a well it doesn't have to be one digit number for example 3 365 well we can get 36 and that would be a so it is a positive integer that's what we know okay but the number that we get is a the number that we were starting with was B plus there was a but this a was 10 times bigger I mean this part is a but actually in this all this number this is 10 a because before deleting it was this a was multiplied by 10 it was 360 plus 5 which means 36 times 10 plus 5 so here we have B and here we have a and we multiply a by time by 10 so the original number was B plus 10 a after deleting we get a and we know that this a has to be 1 14th of the original number so we get a equation that B plus 10 a equals 14 a okay so now we can a little bit rearrange this and we can write that the B equals for a and we know that B is positive integer with one digit so a actually cannot be too big it has to be some small number for example if a would be 0 well then B would be 0 and the number that we were starting was 0 and that was not a positive integer yeah it's 0 it's not positive so no but if a would be 1 then B is 4 and the original number was 14 after deleting 4 we get 1 and yeah that's 1 14th of 14 so okay so that's one possibility the other possibility is a equals 2 then B equals 8 and the number that we were starting was 28 after deleting is 2 and yes this is the 1 14th and that's all because if a would be 3 then B would be 12 but we know that B has to be one digit 12 is not one digit so that's not okay and if a would be bigger then B would be even more even bigger so it wouldn't be also one in one digit number so there are only two possibilities 14 and 28 so the answer is C so let's check 21 C yes so okay so we can go to 22 the picture shows a regular hexagon with side lengths equal to 1 the flower was constructed with sectors of circles of radius 1 and centers at the vertices of the hexagon what is the area of the flower okay so let's look at this picture so the centers of the circle was were on the vertices with radius of 1 the same as the length of the site of this hexagon so for example if here was center of circle then the circle goes here okay and the same here with the center here here we have a center here and so on so actually this flower we can divide in 12 equal parts equal by by everything like this one so something like this two of them give us one leaf of this flower okay so if we would calculate the area of this part then we just have to multiply it by 12 and we get the area of all the flower okay so how to calculate this part well let's see this circle so this circle looked like this and this part well this all part is a slice of a circle and if we subtract from it this triangle we get this thing this part of this circle so so first of all we need to know the the measure of this angle okay so what is the measure of this angle well so this triangle this one is one this is also one and this is one this is one and how much is this well this is also one because okay here is not because this is this line here so this one is a site is a site of for example is it's a radius of this circle and we know that this circle has also radius of one so this is equilateral triangle with sides equal to one okay so what what it tells us that there is 60 degrees this angle is 60 degrees so the area of all this slice is 60 degrees divided by 360 degrees times PR square so the area of all the circle so it cancels out to one sixth one sixth times P and R equals 1 so 1 squared so it's just one sixth of P so this is the area of all this slice but we want we want to have only this part so we need to subtract from this number the area of this equilateral triangle with a side of one so the area of equilateral triangle is a squared times square root of 3 divided by 4 and a is a length of the side which in our case is 1 so it is 1 square times square root of 3 divided by 4 which is just simply square root of 3 divided by 4 so the area of this part is one sixth P minus square root of 3 divided by 4 and as I said in the beginning it's one twelfth of all the area of the flower so we need to multiply this by 12 and while we get one sixth of P times 12 it equals to P minus square root of 3 divided by 4 times 12 well 12 and 4 cancels out so simplify so from 12 we have 3 and 4 just vanishes and we have 3 square roots of 3 and that's the area of this flower and this is the answer e Okay, so let's check the answer 22 e 22 e yeah, so that's how we can solve this Okay 23 consider the sequence a n with a 1 equals 2017 and a n plus 1 equals a n Minus 1 divided by n a n. So we have a sequence that is given by a recursive formula Then a 2017 equals well finding the general formula For this sequence work is not so easy Well, there are some ways in linear algebra using matrices to find the general formula for for example Fibonacci sequence but It wouldn't work for this and well we are in high school, we don't know matrices yet So what we do? Well, we hope We hope that it will work so let's calculate a 2 Well, it is a 1 minus 1 divided by a 1. What is this? A 1 is 2017 minus 1 divided by 2017 it is 2016 16 divided by 2017. Okay Let's see What is a 3 and let's hope It will work a 3 a 2 minus 1 divided by a 2 which is 2016 divided by 2017 minus 1 divided by 2016 divided by 2017 okay, so in The numerator we get minus 1 divided by 2017 divided by 2016 divided by 2017 and they cancel this Denominators cancel out and we get 1 minus 1 divided by 2016. So this is a 3 Okay Sounds like Do we have some numbers, but what do they tell us? Okay, my mouse had the problems but okay, so let's try a 4. What is a 4? a 3 minus 1 divided by a 3 So it is minus 1 divided by 2016 minus 1 divided by minus 1 divided by 2016 So in numerator We will get minus 2017 divided by 2016. In the denominator we get minus 1 divided by 2016 and now we see a really cool thing. First of all minuses cancel out but also 2016s cancel out and we get 2017 which is exactly the same as a 1 which says that a 5 Will be exactly the same as a 2 because we will calculate this using exactly the same numbers a 6 will be like a 3, a 7 like a 4 and like a 1. So this sequence just looks like this 2017 2016 divided by 2017 and then minus 1 divided by 2017 and then we go back to the beginning and go down and go back and go down and go there Yeah, so now it's very easy A2017 we just need to know Which number of those three it is? How can we find it? So we need to know what is the rest from the dividing 2017 by 3 Okay, let's calculate this. Actually, we don't need it because you know the rule of division by 3. You have to add all those numbers so we get 10 and if it would be Dividable by 3 then the whole number was also dividable by 3, but it's better 10 is not dividable by 3 but we know that it will give 3 and the rest will be equal to 1 and This rest is exactly the same as here. If this rest would be 0 so it would be dividable it would also have the rest 0 from dividing by 3, which would mean that this is dividable by 3, but if this has a rest of 1 this also has a rest of 1. If it would have a rest of 2 this one would also have a rest of 2. So we don't have to divide this by 3. We can just use the rule for division by 3 and just find the rest of this and this is 1 So this tells us that 2017 is on the first place in this sequence. So it is 2017. So the answer is E, 23E 23E. Yes, so that's how we can do it fast and efficient Okay, and the last question in this movie 24 Consider a regular tetrahedron Its four corners are cut off by four planes each passing through the midpoints of three adjacent edges. See figure What part of the volume of the original tetrahedron is the volume of the resulting solid? Okay, so let's look So we just cut those four edges Yes, and these are the midpoints of these segments, the points that we are cutting through So this line is half of this line This line is half of this line. This line is a half of this line and This line here It also has to be half of this line because here we have triangle and This is half of this. This is half of this which means that we get Thales formula, which says that this line is Parallel to this line and it's also a half of this line and the same with this line the same with this line so this small tetrahedron here is also a regular tetrahedron and All its sides are half of the original sides Which means that the scale is 1 to 2 Which means that the volume is 1 to 2 to the power of 3 So it is 1 8th so this tetrahedron here has a volume of 1 of 1 8th of the original tetrahedron and Exactly the same thing we can do with all the other four tetrahedrons. The one that we don't see is on that side in back on back of this tetrahedron so we cut four tetrahedrons with all that all of them are 1 8th of The original tetrahedron So we cut half of this tetrahedron and what is left is also a half of this Tetrahedron because 1 minus half is half which means that the answer is D 24 D. Let's check the answer 24 D. Yeah, that's it. So that's a very easy way to solve this question number 25 The sum of the lengths of the three sides of a rated triangle is equal to 18 and the sum of the squares of the lengths of the three sides is equal to 128. What is the area of the triangle? Okay, so let's say that these Sides are A, B and C. So the first thing they told us That the sum of these lengths is 18. A plus B plus C equals 18 They also told us that the sum of squares is equal to 128. So A squared plus B squared plus C squared equals 128 But is it right? I said that the first thing they said is that this sum is equals A equals 18 No the first thing that they said is that this is right triangle and This is very important information so this means that let's say that the C is hypotenuse then A Squared plus B squared has to be equal to C squared and that's something that we need to Solve this and it will help us to solve it very efficient because In this second equation A squared plus B squared. Well, that's C squared So we get C squared plus C squared equals 128 2 C squared equals 128 Which means C squared equals 64, which means C equals 8 No, and now we get that A plus B plus 8 equals 18 which means A plus B equals 10 and we also know that A squared plus B squared equals 64. Okay, how we can solve this in an efficient way? I think Some substitution will be the first Oh, no I see a bit better one because if this is a right triangle and C is the hypotenuse then The area of this triangle is A B divided by 2. So of course we can Calculate A and B and I've done this and these are not very nice numbers. I'm not so bad, but Not the nicest way to solve this there has to be there is a Better way. We don't need A and B. We need A times B and we can find it by Squaring the first first equation here because A plus B squared is A squared plus 2AB Plus B squared and this equals 100 But we also know that A squared plus B squared equals 64 Which means that we can substitute A squared plus B squared In the first first equation, we can write just simply 64 plus 2AB equals 100 and now we can Subtract 64 and we get 2AB Equals 36 which means AB equals 18 But we want to have AB divided by 2 so let's divide this by 2 AB divided by 2 equals 9 and this is the area of this triangle very efficient way without solving quadratic equation So the answer here is E So let's check 25E 25E. Yeah 25E Okay so now 26 You are given four four five black balls five white balls and five boxes You choose how to put the balls in the boxes. Each box has to contain at least one ball Your opponent comes and draws one ball from the from one box of his choice and He wins if he draws a white ball Otherwise you win How should you arrange the balls in the boxes to have the best chance of winning? and we have five possibilities here and Okay, so Let's think about this. For example, you put one white and one black Ball in each box. Well first thing but what will be the probability of? Of drawing a white ball well You have a one-fifth probability of taking a first first first box and in this box You have a probability of half of getting white and the same goes for all the other five boxes So we can just multiply it by five and without any surprise we get half Yeah, there are the same number of white and black. So that seems Legit that it should be half but we can be We can be forgot the word So We can be clever Yeah We can be more clever and find much better way to lower our chance of winning for example What if we would put all the white boxes all the white balls to one box and to others? Black balls. Well, then we have then the opponent has one-fifth Probability of getting this box. There are the in which are only white balls So if we will he will get this box, then he has a probability one of winning, but he also has a four-fifth Chance of getting a box in which there surely is not no white ball So his probability of winning is zero then Which give us one-fifth and this is much smaller than half So we clearly see a is not the right answer what we Be you arrange all the black balls in three boxes and all the white balls in two boxes no, it will it will be better than the first one, but it will be worse than our because Then he has two-fifth probability of winning because if we get he has two-fifth probability of getting a box in which He clearly wins and three-fifth in he in which day he loses So it will give it will look like two-fifth times one plus three-fifth times zero and it is two-fifth So our is still better and our you arrange all the black balls in four boxes and all the white balls in one box so this is what we What we Tried and that's much better But it's not the best still Because in D you put one black box in every box and add all the white balls in one box So let's calculate this probability so he has one-fifth probability of getting the box in which there is white ball and Here in this box, he has five black balls and one black so he's not still Sure that he will win the the probability of winning is high here fifth five sixth But it's not one and he also has four-fifth probability of losing at all. So If we calculate this it will give us one sixth and it is even smaller than one-fifth so D is better than C and The last one you put one white ball in every box and add all the black balls in one box. Well if The winning would be for black boxes for black balls for him Then that would be the best in this case. Well, we will probably get Five six probability of losing so no, so the best one is D That's what we want so 26 26 D. Yes, is it this answer? Yes. Yes because Okay, so So as we can see we can be much clever than a typical Typical action Okay, so 27. Nine integers are written in the cells of three times three table. The sum of the nine Numbers is equal to 500. It is known that the numbers in any two neighboring cells That is cells sharing a common side Differ by one What is the number in the center cell? Let's think about this If we put 50 in the center well 50 times 9 is 450. We would like to have 500. So we need to go up. Let's give here 51 51 51 51 and even here we can get give 52. Yes, because we can have the Number that is differs from those by one so it can be maximally 52 and what we get Well, we get 9 times 50 plus 4 ones plus 4 twos Which will give us 4 plus 8. So it's 12 12 plus 450 462 so it's way too small. So it cannot be 50 It has to be more So Actually 500 divided by 9 well 5 45 5 so this is 55 and we need 5 more So if we would put like 54 in the middle, then we will need to get higher So the highest number that we can get is like this and well, so this is 9 9 times 55, which is 495 and we so we have 456 so this is plus 4 times 1 because we have one more in every of those cells in the rows in the vertices we have also 9 455 which doesn't change the result and we have 154 which Subtracts 1 from this so we get 495 plus 4 minus 1 so this is only 498 So this is still too small. So 54 cannot be this number Well, let's try 55 We can have 55 But let's think about also this one if we have in the center the number that is Even then we have five numbers that are even and four numbers that are odd Yeah, it has to be even if we would go down here to 54 It would also be even so if we start here with even we need to have these four odd numbers and those four will be even and the result of The sum of this will be even because we have even number of odd numbers. So in In summing up it will give even number So and what we want to get we want to get 500. This is even number so we have we need to have an even number in the middle so it cannot be 55 and 57 so we need to be 56 so let's just find this sequence 56 so we need to go down i think 55 55 55 55 let's try 56 56 56 56 so here we have 4 times 55 plus 5 times 56 it will count pretty nice because it will be 220 plus 280 and wow this is 500 so this means that in the middle in the center it has to be 56 so d 27 d let's check this 27 d 27 d yeah 27 d okay and the last question in this video 28 if absolute value of x plus x plus y equals 5 and x plus absolute value of y minus y equals 10 what is the value of x plus y first of all what is absolute value well absolute value it's the distance from zero that's the general definition that is that is essential for example in in complex numbers in real numbers we can think about this kind of easier way so absolute value is a function that doesn't change positive numbers and changes negative numbers into their the positive numbers so absolute value of 5 equals 5 but absolute value of minus 8 equals 8 so it cancels the minus well of course absolute value of minus 8 plus 1 it's not 8 plus 1 we first need to calculate this this is minus 7 so this is 7 or we can just change the designs and we get 8 minus 1 and we get 7 we have to change them because we see that this number is negative but if it would be positive you didn't change so if we will get we will have absolute value of 7 minus 2 that would be just 7 minus 2 which is 5 yeah because this is positive number at all okay so how can we solve this well we can check four possibilities first of all x and y let's say that they are non-negative positive or equal to zero well absolute value of zero is just zero so it doesn't change so in this case we get that absolute value of x is x plus x plus y equals 5 and the other the the second equation is x y is non-negative so it's also plus y minus y equals 10 which tells us that x equals 10 which means that 2 x is 20 plus y equals 5 which means y equals minus 15 but we assume that y is not non-negative so it's not the case it's not the right answer so they cannot be both positive so what can we do else we can for example well let's say x is positive non-negative and y is negative then we get x plus x plus y equals 5 and here we have x so if y is negative then absolute value of y actually is minus y yes because minus y if y is negative then minus y is positive so we get x minus y minus y equals 10 so we have 2x plus y equals 5 and x minus 2y equals 10 so we can see that x equals 10 plus 2y which means that 2 times 10 plus 2y plus y equals 5 so 20 plus 4y plus y equals 5 which means that 5y equals minus 15 which means y equals minus 3 and that's what satisfies aha once again x x equals 10 plus 2 minus times minus 3 which is 10 minus 6 which is 4 and this satisfies us x is positive non-negative and y is negative that's what we wanted so that's possibility x plus y would be then 4 plus minus 3 which is 1 so this has to be an answer let's check the other possibilities what if x is negative and y is non-negative then in the first well this will be minus x plus x plus y so it will be y equals 5 and in the the second we will get x equals 10 because the absolute value of y will be y and we will it will cancel out with minus y and this is not okay because x had to be negative and not 10 and what if they are both negative then well in the first x absolute value of x and x cancels out and we get y equals 5 and this non-negative is positive so now it cannot be so the only possibility is that x equals x 4 and y equals minus 3 and then the sum that we are looking for equals 1 and the answer is a 28a let's check in the answers so 28a 28a yeah yeah 28a yeah so that's the answer so in 29 how many three digit positive integers a b c exist such that a plus b to the power of c is a three digit integer and an integer power of d so well first of all i understand that a b c we should understand actually as a times 100 plus b times 10 plus c not as a times b times c that's what i think um if that's the case then well a plus b to the power of c has to be three digit and an integer power of two okay what does it tell us well first of all a and b with this with this understanding is they are a and a b and c are one digit numbers integers so a plus b is something between 2 and 18 okay well 1 and 18 a has to cannot be 0 but b can be 0 so from 1 to 18 and now a plus b to the power of c has to be the power of 2 this means that a plus b on its own also has to be an integer power of 2 for it cannot be 3 it cannot be 5 or something like this because then if we if our power here is also an integer then it won't be this number won't be an integer an integer power of 2 so a plus b can be 1 or 2 or 4 or 8 or 16 so that these are the possibilities okay so now let's think about them if a plus b is 1 which actually means that a would be 1 and b would be 0 then 1 to any power is 1 and 1 is an integer power of 2 it's 2 to the power of 0 but it's not three digit integer yes it's just 1 so this one no so we got to 2 okay so here we have 2 to the power of c well let's think about integer powers of 2 so they are 2, 4, 8, 16, 32, 64, 128, 256, 512 and then 1024 but it's not okay the what we want we want those three because they are three digit so 2 to the power of c would should have should be 128, 256 or 512 so this one would be here c would equal 7 here 8 and here 9 and they are all okay they they are okay so now let's think how can we get 2 well a plus b has to be equal to which means that it can be 2, 0, a can be 2, b can be 0 or 1, 1 so we have two possibilities for our basis of this of this power of however you call it and with those two possibilities we have for each of them three possible three possible powers so we have six possibilities here so for example it can be number 207, 208, 209 but also it can be 117, 118 or 190 a cannot be zero because it has to be three digit okay so that's for two what's for four so here we have six possibilities if a plus b is four then to the power of c we will not get 2 to the power of 7 because then 2 to the power of 7 is equal to 2 to the power of 2 to the power of 3, 3.5 which is 4 to the power of 3.5 and 3.5 is not an integer so we cannot this get this number 128 and also we cannot get 512 because c would have to be 4.5 we can only get this one in this in this situation this c is 4 so okay so we have only one possibility for c now how can we get this four as a plus b well it can be 4 0 or 3 1 or 2 2 or 1 3 four possibilities each one with one possibility for c this gives us four new possibilities okay so i will turn back the internet so we have four two we have four four how many possibilities now we will go to eight if the basis is eight then well if we want to power this eight to the integer power so we cannot get 128 because 8 is 2 to the power of 3 so 2 to the power of 7 would be 2 to the power of 3 and this one to the power of 2 and one third and it would be 8 to the power of 2 and one third and 2 and one third is not an integer so it cannot be c the same goes for 256 we will not get 8 but we can get 9 8 to the power of 3 equals 512 so it's okay so c can only be 3 now how many possibilities to get 8 here well we can have 8 0 7 1 and up to 1 7 so here we have 8 possibilities for each of them one possibility for c which gave us 8 possibilities if the the basis is 8 okay and the last last one 16 what if we have a basis equal to 16 we will not get 128 because 16 is 2 to the power of 4 which means that 2 to the power of 7 would be 16 to the power of 7 4th which is 1 and 3 4th and this is not an integer we cannot also get 512 but we can get 256 actually 16 squared is 256 okay so one possibility for c and how many possibilities for a and b well 9 7 8 8 or 7 9 so three possibilities here okay so now let's sum them let's sum them up here is 10 here is 11 in total we get 21 so this is answer e okay so let's check 29 e 29 e yeah that's the answer okay and the last question question number 30 well each of the 2017 people living on a certain island is either a liar and always lies or a truth teller and always tells the truth more than 1000 of them take part in a banquet all sitting together at a round table each of them says of the two people beside me one is a liar and the other one is truth teller how many truth tellers are there on the island at most okay so let's think about this well we have some person let's say that this person is a truth teller then it means that on one side has to be truth teller and on the other side has to be a liar now for this person if there is this person is also truth teller so the he already has he or she already has truth teller on one side then on the other side has to be liar here we have a liar so it's not true that it that this person have truth teller and liar on its sides on his sides so if he or she already has a truth teller then there cannot be liar because then the liar would not be lying so there has to be truth teller and it goes like this a this truth teller has a liar so there has to be truth teller then has to be liar then has to be truth teller then has to be a truth teller liar truth teller truth teller liar and so on and it could go like this infinite but it is a round table so first of all we see truth teller truth teller liar truth teller truth teller liar so two truth tellers then liar then true two truth tellers and liar and if that was a line it could go forever but this is a round table which means that in some point they actually have has to meet and what we have must have here we must have truth teller truth teller and liar truth teller truth teller and liar this has to be the sequence if this has to work so what does it mean for us well this means that we have the the number of people that are on this banquet are is divisible by three because you see here you have this this this and so on and when you go this way truth teller truth teller liar and here and those those blocks of three people there are no there are only those blocks of three people there are no any person that is not in some block of this and they are also disconnected so in every person is in exactly one of these blocks and they are of three people which means that the number of these people on this banquet is divided by three we want to know and we want to have as many truth tellers as it is possible so this pattern works only for people in banquet so every person that is not on this banquet can be truth teller and that's the maximal way because here we have two third of people being truth tellers and all the others that are not on this banquet are truth tellers so we won't have to we won't actually have to the least possible number of people on this banquet but we know that they are that more than one thousand of them take part in this banquet so we are looking for a number that is bigger than one thousand but the smallest possible and dividable by three and it's pretty easy one thousand and two one thousand and two are people on this banquet so pretty big banquet and how many of them are truth tellers well two thirds so we need to multiply this by two third so we can simplify this and here we get one two thousand one thousand and two divided by three well it will be three hundred thirty four and now we multiply this by two and we get six six eight six hundred sixty eight and all the other people who are not on this banquet are can be truth tellers and how many other people well on the island are 2017 one thousand and two of them are on this banquet so the others are 1015 and they can all be truth tellers and on this banquet are 668 truth tellers so all truth tellers well we just need to sum them up we need to add them so five plus eight equals 13 116 this is 8 0661 1683 and that's the answer this is answer a let's check this uh 30 a yeah so if you are looking for the most possible truth tellers and you get the result that is the biggest of the possible answers then you feel good because if your calculations are good then you didn't miss anything that could give you the bigger answer so that's also something that strengthens your confidence in your result so that's all with this with this uh this competition we will be doing some other so thank you for watching and have a nice day goodbye

Math Kangaroo

Michael Śmigiel

numerical expressions

ratio problems

geometric shapes

triangle area

polyhedron properties

systems of equations

probability puzzles

polynomial equations

integer coefficients

Pythagorean theorem

mathematical theories

problem-solving strategies

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