This is the Math Kangaroo Solutions video library, presenting solution suggestions for levels 9 and 10 for the year 2018. These solutions are presented by Lucas Nalaskowski. The purpose of the Math Kangaroo Solutions video library is to help you learn how to solve math problems, such as those presented in the Math Kangaroo competition. It is important that you make sure to read the question first. If you're reading the question, listen as I read the question, pause the video, and try to solve the problem on your own. Question 1. In my family, each child has at least two brothers and at least one sister. What is the smallest possible number of children in my family? Since in the problem, we do not know if the person is a boy or a girl, we have to assume for both scenarios. If the person is a boy and they have at least two brothers, then the family will have three boys. However, if the person is a girl, then the family will have two girls. With this, if we add in 3 and 2 together, we get the answer 5. Question 2. Some of the rings in the picture form a chain that includes the ring indicated by the arrow. How many rings are there in the longest possible chain? If we look at this more closely, we can see that the two rings to the right are not connected to the rest of the rings. Therefore, if we go from the topmost ring, we can follow that we have 1, 2, 3, 4, 5 rings connected. Therefore, the answer is C, 5. Question 3. The lengths of the two sides of a triangle are 5 and 2. The length of the third side is an odd integer number. Find the length of the third side. To help us imagine this, we can first draw a drawing. So we have sides with the length of 2, length of 5, and then for the unknown length, we can have T. Now, knowing what we know about a triangle, we know that both sides, 5 plus 2, are greater than the third side, T. 5 plus T is greater than 2. 2 plus T is greater than 5. So we know that 7 is greater than T, which is greater than 3. So the side has to be between the values of 3 and 7. Since we know that it is an odd integer, there is only one number that satisfies this. That is 5. The correct answer is C, 5. Question 4. The distance from the top of the cat sleeping on the floor to the top of the cat sitting on the table is 150 cm. The distance from the top of the cat sitting on the floor to the top of the cat sleeping on the table is 110 cm. Both sitting cats are the same size and both sleeping cats are the same size. What is the height of the table? First we must define our values. Capital H will be the height of the sitting cat, lowercase h is the height of the sleeping cat, and T is the height of the table. We know that T is equal to the height of the sleeping cat plus 150 cm minus the height of the sitting cat. We also know that the height of the sitting cat plus 110 minus the height of the sleeping cat is also equal to the height of the table. We can set both equal to each other. After simplifying this solution, we get h minus h equals 20. So using this, we can put it back into the formula and find out that the correct answer is C, 130 cm. Question 5. The sum of five consecutive integers is 10 to the power of 2018. What is the middle integer? We line up five consecutive numbers. We can have them integer values as m minus 2, then m minus 1, m, m plus 1, and m plus 2. We can do this since we know they are consecutive. And we try to find out m, the middle integer. Adding together all of these integers, we get 5m, and we know that the sum is equal to 10 to the power of 2018. This can also be displayed as 10 times 10 to 2017. If we divide both sides by 5, we get the value of m equals 2 times 10 to the power of 2017. Therefore, the correct answer is E, 2 times 10 to the power of 2017. Question 6. Three congruent rectangular hexagons A, B, and C are given. X is the shaded area of A, Y is the shaded area of B, and Z is the shaded area of C, as shown below. Which of the following statements is true? If we take a closer look at any one of the hexagons, for example, hexagon A, we notice that it can be divided into three equal parts, and we see that the shaded area is equal to the not shaded area. And this is true if we divide hexagon B as well as hexagon C. Each hexagon's shaded area is exactly half of the full area of the hexagon. Therefore, the correct answer is A, X equals Y equals Z. Question 7. Mary collected 42 apples, 60 apricots, and 90 cherries. She wants to divide them into identical piles using all of the fruit. What is the largest number of piles she can make? Since we have 42, 60, and 90 fruits, to find the amount of piles we must find the greatest common factor. We can do this by dividing. The greatest common factor is 6, since 42 can be divided into 6 and 7, 60 into 6 and 10, and 90 into 6 and 15. Therefore, the greatest common factor is 6. Therefore, Mary can make 6 different piles of apples, apricots, and cherries that are all identical and use all the fruit. So, the correct answer is B, 6. Question 8. Some of the digits in the following correct addition problem have been replaced by the letters P, Q, R, and S, as shown. How much is P plus Q plus R plus S? To solve this, you must notice that 5 plus a number will give us a two-digit number ending in 4, since we know S cannot be a negative. The only number that satisfies this is 9, so S must equal 9. With this, we know that 5 plus 9 gives us 14, so the 1 will be carried over and be added onto the 4. So, 4 plus 1 plus R will give us 5, so R must equal 0. With this, we know that P plus Q will equal 6, since no digit will carry over from the 1 plus 4 plus R. Now we have all our numbers. We have S, R, P, and Q, since we do not need to find out the individual values of P and Q. We add them together, P plus Q plus R plus S, we get 6 plus 0 plus 9, so the correct answer is B, 15. Question 9. What is the sum of 25% of 2018 and 2018% of 25? So, what the question is asking is 25% times 2018 plus 2018% of 25. This can be rewritten in such a way like this, 0.1 times 25 times 2018 plus 0.1 times 2018 times 25. That can be simplified to just 2 times 0.1 times 2018 times 25. Also known as, after we simplify, 0.5 times 2018. If we divide this by half, we get 1009, so the correct answer is A, 1009. Question 10. In the picture below, you should go from A to B following the arrows. In how many different routes are there from A to B along the lines following the directions of the arrows? We look at our sideways hourglass shape. We have to start out from A and reach B in as many ways as possible. If we count them out, from A there are 4 ways to reach the midpoint, as shown, and from the midpoint to reach B there are also 4 different distinct ways. Knowing this, all we must do is just multiply 4 by 4 to get 16, so the correct answer is B, 16. Question 11. Two buildings are located on one street at a distance of 250 meters from each other. 100 students live in the first building and 150 students live in the second building. Where should a bus stop be built so that the total walking distance for all residents from their buildings to this bus stop would be the shortest? If we let F be the distance from the first bus stop and S be the distance from the bus stop to the second building, then F plus S is always equal to 250 meters or more. The total walking distance for all residents is as such 100F plus 150S, which is also 100F plus S plus 50F, is greater than or equal to 100 times 250 plus 50 times 0, which will give us 2,500 meters. The total walking distance is minimal when S is equal to 0, so the bus stop should be located in front of the second building. The correct answer is A, in front of the first building. Question 11. Two buildings are located on one street at a distance of 250 meters from each other. 100 students live in the first building and 150 students live in the second building. Where should a bus stop be built so that the total walking distance for all residents from their buildings to this bus stop would be the shortest? If we let F be the distance from the bus stop to the first building and S be the distance from the bus stop to the second building, then F plus S is always 250 meters or more. The total walking distance for all residents is 100F plus 150S. We look at this equation. 100 times the sum of F and S plus 50F is greater than or equal to 100 times 250 plus 50 times 0, which gives us a total of 25,000 meters. The total walking distance is minimal when S is 0, so the bus stop should be located in front of the second building. Therefore, the correct answer is D, in front of the second building. Question 12. There are 105 numbers written in a row. 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, and so on. Each number n is written exactly n times. How many of these numbers are divisible by 3? We look at our formula. 1 plus 2 plus 3, so on, plus n minus 2 plus n minus 1 plus n, and simplify that into n times the product of n plus 1 divided by 2. In that formula, we get from 1 to 14, we get 14 plus 15 divided by 2 gives us 105, so we know that the numbers go up to 14. The only numbers that are divisible by 3 are 3, 6, 9, and 12. If we add these together, you get a total of 30, so the correct answer is D, 30. Question 13. Eight congruent semicircles are drawn inside a square with a single side of 4. What is the area of the non-shaded part of the square? If we look at the square closer, we notice we can divide it into four equal parts. If we look at it even closer, we notice that a diagonal can be drawn, and it is the same for all other three parts. If we look at the semicircles, we notice that the overlaps can be put into the white petals resulted in the overlaps, so therefore half of the square is not shaded. With an area of 16, we know that half of 16 is 8, so the correct answer is B, 8. Question 14. On a certain day, each of 40 trains traveled between two of the towns, M, N, O, P, and Q. 10 trains traveled either from M or to M. 10 trains traveled either to or from N. 10 trains traveled either to or from O. 10 trains traveled either to or from P. How many trains traveled to or from Q? We make a chart like so. We can start filling it out. We know that the row of M will be equal to 10, since that is the amount of trains that have gone to or from M. Same is true for this row, the next row, and the next row. Now for the rest of this to be true, the columns of M, N, O, and P must be equal to 10. So the row of Q will look like this, and knowing that all other totals are 10, the only conclusion we can come to is that 40 trains traveled to or from Q. So the correct answer is E, 40. Question 15. At the University of Humanities, you can study languages, history, or philosophy. 35% of students that study languages study English. 13% of the university students study a language other than English. No student studies more than one language. What percentage of the university students study languages? So if we take F and E, having the students who study English as well or a foreign language, those are the language studiers. The foreign language, English, and non-language studiers are all students. So if we do F plus E divided by the sum of F plus E plus N, that will give us our answer. To do this, we must do 0.35 of the total times the total of F plus E equals E, as well as 0.13 times the sum of F, E, and N equals F. This we know that 0.3F is equal to 0.65E, and E is equal to 35 65ths, as well as F is equal to 7 13ths. F plus E is equal to 20 F 13ths. So F plus E plus N is equal to 100 divided by 13F. And therefore, we get the solution of 0.2, or B, 20%. Question 16. Peter wanted to buy a book, but he didn't have any money. He bought it with the help of his father and his two brothers. His father gave him half the amount given by his brothers. His older brother gave him one third of what the others gave. The younger brother gave him $10. What was the price of the book? You have the father plus the brother plus the younger brother, which is 10. That will be the price of the book. From what the question told us, we know that the father is equal to the sum of the brother plus 10 divided by 2, while the brother is equal to the sum of the father plus 10 divided by 3. Simplifying this, we get 2F equals B plus 10. 6B equals B plus 10 plus 20, or plus 30. If we take that, we get 5B equals 30, and B equals 6. So we know that the other brother gave $6. This, we can plug this into the B, and 2F equals 6 plus 10. We get the father equals 3 plus 5, so the father gives $8. If we add these together, 6 for the big brother, 8 for the father, and 10 for the younger brother, we get a total of 24. So the correct answer is A, $24. Question 17. How many three-digit numbers are there with the property that the two-digit number obtained by deleting the middle digit is equal to one-ninth of the original three-digit number? We can write a formula for any given three-digit number as 100A plus 10B plus C, and then the resulting two-digit number that is created from the deletion of the middle digit will be 10A plus C. With this, we can have the formula of 9 times the product of 10A plus C equals 100A plus 10B plus C, meaning that they are one-ninth of the original three-digit number. We simplify this, we get 8C is equal to 10A plus 10B, or 4C is equal to 5 times the sum of A and B. This, you get the resulting answer, D, 4. Question 18. In the calculation shown, how many times does the term 2018 to the power of 2 appear inside the square root to make the calculation correct? We are given the calculation 2018 squared plus 2018 squared plus, and so on, square rooted is equal to 2018 to the power of 10. We write this out, the square root of the product of n times 2018 to the power of 2 is equal to 2018 to the power of 10. We get this. We square both sides, then we can have n times 2018 squared equals 2018 to the power of 10 squared, which will also result in 2018 to the power of 20. So we can have this equation of 2018 to the power of 20 minus 2, or 2018 to the power of 18, and the correct answer is E, 2018 to the power of 18. Question 19. How many digits does the number resulting from the calculation 1 9th times 10 to the power of 2018 times the difference of 10 to the power of 2018 minus 1 have? To do this, we must examine the third part of the problem and look for the pattern. 10 to the power of 1 minus 1 divided by 9 gives us 1. We raise the power to 2, we get 11. Raise the power to 3, we get 111. So surmising this, we can notice that whatever the power is will be the number of 1s in the resulting number. So there will be 2018 1s in this number. Now, from the 10 to the 2018, there will be a resulting 2018 zeros after the 1s. So if we add all the 1s, 2018 of them, plus all the 0s, also 2018 of them, we get a total of 4036, which gives us D, 4036. There are two diagonals drawn in a regular 2018 GAN, with its vertices numbered from 1 to 2018. One diagonal connects the vertices with the numbers 18 and 1018. The other connects the vertices with the numbers 1018 and 2000. How many vertices do the resulting three polygons have? Draw this out, and label the vertices 18, 1018, and 2000, and we get a shape looking like this. To find the total of vertices, we must simply subtract 2018 minus 1999, and add 19 for the first smaller segment, which gives us 38. Then we must do 2000 minus 1017 to get the number of vertices, 983. Finally we must do 1018 minus 17 to get the final number of vertices, 1001. Therefore the correct answer is A, 38, 983, and 1001. Several integers are written on a blackboard, including the number 2018. The sum of all of these integers is 2018. The product of these integers is also 2018. Which of the following could be the number of integers written on the blackboard? Since we know that the sum of the integers is 2018, and one of the integers is 2018, we know there will be negative numbers. Since we know that the product will also be 2018, we know that this can only be made up of 2018 and a series of 1s and negative 1s. Since 1 plus negative 1 will give us 0, so the sum will be able to stay 2018. Now if we write out a formula of 1 times negative 1, plus, and so on, we get negative 1 to the power of k, and k must be 2p, p being a substitute number, since we know that the number has to be even, otherwise we will end up with a negative number, and the product is 2018, not negative 2018. The only way to satisfy it being k equal to 2p, and also the sum of the integers being an even number, 0, then we must have 4p plus 1, which will be the total integers on the blackboard. The plus 1 representing the integer 2018. One number satisfies this, 2017. Since 2016 can be divided into 4, so the correct answer is b, 2017. Question 22. 4 positive numbers are given. You choose 3 of them. Calculate their arithmetic mean, and then add the 4th number. This can be done in 4 different ways. The results are 17, 21, 23, and 29 respectively. What is the largest of the 4 given numbers? Write the 4 different numbers as unknown a, b, c, and d, and write down the formulas, such as the sum of a plus b plus c divided by 3, which is the mean, and then add on d, the 4th number, which is 17. This can be done with the numbers being in a different order every time, to get 21, 23, and 29. Using these numbers, we can get a plus b plus c gives us 51 minus 3d. Using this, we can get to the sum of 3a plus 3b plus 3c plus 3d divided by 3 plus a plus b plus c plus d gives us 90. The two parts are equal to each other, meaning that 3a plus 3b plus 3c plus 3d, all divided by 3, is also equal to a plus b plus c plus d, meaning a plus b plus c plus d is 45. Put this back into a formula, we can divide it by 3 and get 15. Knowing that the value of 15, we can find out that the future numbers will be 3, 9, 12, and 21, being the only solution set that works for a, b, c, and d. Therefore, the correct answer is c, 21. Question 23. The points a0, a1, a2, and so on lie on a line such that a0 to a1 equals 1, and the point aN is the midpoint of the segment aN plus 1 to aN plus 2 for every non-negative integer n. What is the length of the segment aN to a11? Let's first realize that aN is the midpoint of the segment aN plus 1 to aN plus 2. So xn is equal to the sum of xn plus 2 plus xn plus 1 divided by 2. Subtract xn plus 1 from both sides, we see that we can get negative xn plus 1 plus xn. With this, we can lead on to dn is equal to xn plus 1 minus xn. Figuring this out, we lead on to dn plus 1 is equal to negative 2dn, also known as dn is equal to negative 2 to the power of n. This leads us to the solution going on from d0 all the way to d10, adding all the numbers together, which is equal to negative 2 to the power of 0 plus negative 2 to the power of 1 plus negative 2 squared, so on, until negative 2 to the power of 10, which can also be written as the difference between 1 and negative 2 to the power of 11 divided by the difference of 1 and negative 2, giving us 1 plus 2048 divided by 3, or the answer E, 683. Question 24. Two concentric circles of radii 1 and 9 make a ring. In the interior of this ring, n circles are drawn without overlapping, each being tangent to both of the circles of the ring. An example of such a shape for n equals 1 and different radii is shown in the picture. What is the largest possible value for n? Start off by drawing a picture of this. We can divide the circles into 4. And since we are trying to find the number of circles that will fit, we can draw lines like this. We can make a triangle by extending out a radius of the new circle. Using math, we can figure out that the radius will be 4. Since we know that the radius of the big circle is 9, the radius of the smaller circle is 1. The diameter of this new circle is 8, or divided by 2, the radius is 4. We also know that the hypotenuse of our new triangle will be 5. To figure out how many circles will fit in, we must find the value of x. To do this, we can use sine. Sine of x equals 4 divided by 5. Since we know that root 2 divided by 2 is less than 4 divided by 5 is less than root 3 over 2, we know that sine of 45 is equal to root 2 over 2, and sine of 60 is equal to root 3 over 2. So therefore, x will be somewhere between 45 and 60 degrees. Now since this is only half the circle, we must double this to get 90 and 120 degrees. With knowing 2x, we can figure out that since a circle has 360 degrees, then we can fit three circles inside of the larger circle. Since if it was less than 90 degrees, we could fit more, or if it was more than 120 degrees, we could fit less. But since it is somewhere between 90 and 120, the correct answer is C, 3. At each vertex of the 18-gon, in the picture, a number should be written which is equal to the sum of the numbers at the two adjacent vertices. Two of the numbers are given. What should the number be written at vertex A? So looking at the pattern we are given, the fact that one number is equal to the sum of the two adjacent numbers, we can make such a pattern, A, B, B minus A, negative A, negative B, and so on. This is thanks to the fact that we know that either side added is equal to the center vertice. For example, B is equal to B minus A plus A. That gives us B. Or we can look at it in another way, that B minus A is equal to the two previous vertices before it. So the vertices that you are looking at is equal to the two previous vertices subtracted from each other. So B minus A, so B, and then minus A. If we draw this 18-gon, we know that 18 and 20 are given values. We can ignore those for now. We start off with A as the top point, can lead to B, then we subtract them, B minus A, then we can subtract those. B minus A minus B gives us negative A. If we do negative A minus B minus A, we get negative B. And if we continue on with this pattern, we can see that it circles around the 18-gon in a fashion that it loops completely, since A, B, B minus A, negative A, and so on, will end up in a continuous loop. Now if we look at our values of 18 and 20, we can see that B is equal to 18, and A minus B is 20. And this will give us D, 38, because once we do the substitution for B, we can find out that A minus 18 gives us 20. So the answer is D, 38. Question 26. Diana draws a rectangular grid of 12 squares on a squared paper. Some of the squares are painted black. In each blank square, she writes the number of black squares that share a side with it. The figure shows an example. Now she does the same in a rectangular grid with 2,018 squares. What is the maximum value that she can obtain as the result of the sum of all the numbers in the grid? If we look at the given rectangle, we notice that if a square that is white shares a side with a square that is black, it gets a value plus 1. And for every black square, it will be worth more. So the most efficient way to make a rectangle is to make it 2 squares high and then as long as possible. Since we have 2,018 squares, we make a checkerboard pattern such as this and go on for 1,009 squares on each side. Then we will get the most efficient shape. To find out the total value, you must realize that there will be 504 black squares and 503 black squares on the bottom and 504 black squares at the top. Also meaning that there will be 504 white squares on top and 503 white squares on bottom. Each of these squares will have a value of 3, since they will be surrounded by 3 different black squares. And this will give us a total of 3,021. However, we must also remember the 2 white squares at the bottom left and the top right, which are only sharing 2 black squares, so they will add 4. So the correct answer is D, 3,025. Question 27. A 3x3x3 cube is built from unit cubes. Seven of these unit cubes have been removed. The picture shows three faces, but the cube looks the same when rotated. We cut this cube by the plane passing through the center of the cube and perpendicular to one of its four space diagonals. What will the cross section look like? We know that due to it being cut through one of the four space diagonals, it will be a hexagon, meaning it will end up with a shape that is six-sided. All the possible solutions are also six-sided. Now if we imagine the cube, we know that it has missing cubes, which can be reimagined as rods, such as this. And we know that when three rods intersect from three different planes of x, y, and z, for example, they will form together, and when cut straight, they will make a triangle. We must imagine three other rods from the other side of the cube, since a cube has six sides, and this will make a triangle, such as this. When these two come together, they give you a six-pointed star, such as this. So on the hexagon, there will be a cutout like this, where all six rods meet. So the answer is A. Question 28. Each number of the set, 1, 2, 3, 4, 5, and 6, is written in exactly one cell of a 2x3 table. In how many ways can this be done so that each row and each column, the sum of the numbers is divisible by 3? To draw our 2x3 table, we have to recognize that in our list from 1 through 6, only 3 and 6 can be divided by 3 on their own, so they must be in the same column, no matter where they are placed. So there are six different ways to place 3, and only one way to place 6, or vice versa, since regardless of where we place 3, 6 will have to be in the opposite box. Same is true if we want to put 6 in, 3 will have to be in the opposite box. After this, we must realize that 1 and 4 cannot be in the same column, so no matter where they are placed, they must be diagonal of each other. So there are four ways to put the number 1, and only one way to put the number 4 in. Then we can put 2 in, and it can go in either box, and there are two different places it can be put in. So if we do this, and multiply all the options, 6 options by 1 by 4 by 2 by 5, we get the answer D, 48. Question 29. Ed made a large cube by gluing together a number of identical small cubes, and then he painted some of the faces of the large cube. His sister Nicole dropped the cube, and it broke into the original small cubes. 45 of these small cubes didn't have any painted faces. How many faces of the large cube did Ed paint? We take the number of small cubes as n cubed, for the natural number n, such that n cubed is greater than or equal to 45. We can also get n is greater than or equal to 4. Only one face is painted, and we remove the layer of cubes attached to it, and then the resulting cuboid does not have any painted faces. Cuboid contains n cubed minus n squared, which is also equal to n squared times n minus 1, for all small cubes, but in n squared times the difference of n and 1, this is always an even number, so it cannot be equal to 45. If we remove the wall, there must be another painted face of the cuboid. If it's the opposite wall, then its size is n times n, after removing the corresponding wall. We are left with n cubed minus 2n squared, which also equals n squared times n minus 2. But this does not have a solution, even if it is simplified to n squared times the difference of n minus 1 minus n times the difference of n minus 1. Now, if we wish to jump and try to paint four sides of the cube, then we can do so by doing the top side, or any side, as well as the side opposite to it being not painted. By removing the painted walls, the original cube is shrinked to a cuboid of the dimensions n times the difference of n minus 1 times the difference of n minus 1. So the number of small cubes not being painted is n times n minus 2 squared. The equation of n times n minus 2 squared equals 45 has one solution, 5. So when we have four sides painted of the larger cube, with the top and bottom sides not being painted at all, we can have a solution. So the answer is C, four sides. Question 30. Two cords, AB and AC, are drawn in a circle with a diameter AD. Angle BAC equals 60 degrees. BE is perpendicular to AC. AB is equal to 24 centimeters. And EC is equal to 3 centimeters. What is the length of the cord BD? We know that angle ABE is 30 degrees, since angle EAB is 60 degrees. And AEB is 90 degrees, since it is an angle inscribed in a circle with AD being the diameter of the circle. The same is true for ACD. Angle EBD is equal to angle ABD minus angle ABE, which can also be written as 90 degrees minus 30 degrees, which of course is 60 degrees. Now lines DC is perpendicular to line AC, since ACD is 90 degrees. So HD is equal to EC, where H is a point on BE, such that DH is perpendicular to BE. In the right triangle DHB, angle HBD equals angle EBD, which equals 60 degrees. So we know that triangle DHB is half of an equilateral triangle with a height of 3 centimeters. We look at the properties of an equilateral triangle, and we know that H height is equal to the side length times root 3 divided by 2. This can be rewritten as S is equal to H divided by root 3 divided by 2. We put H as 3, and we get that BD is equal to 2 root 3 centimeters. So the correct answer is D, 2 root 3 centimeters.

Math Kangaroo

Lucas Nalaskowski

problem-solving

geometric shapes

logical reasoning

interactive learning

competition preparation

mathematical puzzles