Hello, and welcome to the Math Kangaroo Media Library. You are about to view interactive solutions to levels 9 and 10 of the 2011 competition. You will likely notice that some of the solutions are slightly different from the suggested solutions you may have already reviewed. As you follow along, compare your own solutions and the suggested solutions with the current presentation and make sure you understand the differences. If you have any questions or comments, feel free to contact me at the address provided. My name is Luke, and I'm a past Math Kangaroo participant, and I hope you will find this presentation useful in preparing for your next Math Kangaroo competition. Solution number 1. A pedestrian crosswalk consists of black and white stripes, each half a meter wide. The crosswalk begins and ends with a white stripe, and there are 10 white stripes in all. How wide is the crosswalk? I have here a typical diagram representing a white stripe as it would appear somewhere in the middle of the crosswalk, and we notice here a pattern of white followed by black that would have to be repeated as we draw our crosswalk, so this is our pattern A. We would have in our crosswalk, we would have 10 times A, and that is a little bit too much. We would have to subtract from this a black stripe, so we see that repeating this pattern over and over would actually produce the larger pattern here displayed where it ends with a black stripe. So this has length 10 times the length of A minus the length of the black stripe. So 10 times 1 meter minus half a meter, that gives us 10 meters minus half a meter, and 9.5 meters is the answer we're looking for that's deep. Question number 2. Points A and B are the midpoints of the sides of a trapezoid, and the area of the shaded rectangle is equal to 13 centimeters squared. What is the area of the trapezoid in square centimeters? I have copied the provided picture here and labeled some additional sides. Capital A and capital B are still the midpoints of the two non-parallel sides, and the parallel sides I have labeled with lowercase a and lowercase b. H is the distance between them, that's the height of the trapezoid. I also recall that the area of the trapezoid is the product of the height with the average of the two parallel sides. So with that, we can compute the area of the blue rectangle. I'm going to say that the area is equal to, as usual, length times width, where the length is exactly the distance between capital A and capital B, or the average of the lengths of the two parallel sides, divided by 2, and the width would be the height, divided by 2, as we can see from the picture. So we can factor out one half out of this expression, and the remaining expression is exactly the height times the median of the trapezoid, that's another name for its area. So this is equal to one half the area of the trapezoid. Since we know that the area of the blue rectangle is exactly 13 centimeters squared, doubling that we obtain the area of the trapezoid, which is therefore 26 centimeters squared, and our answer is C. Question number three. If X, Y, and Z are defined as follows, then which of the following inequalities holds? Let's begin by simplifying. The expression for Y is the simplest. We have Y is equal to 2 squared plus 3 squared plus 4 squared. We can compute that as 4 plus 9 plus 16, which gives us 29. Then X is 2 times 3 plus 3 times 4 plus 4 times 5, which we can also compute, 6 plus 12 plus 20, which gives us 38. So we already know that X is bigger than Y. And finally, Z is 1 times 2 plus 2 times 3 plus 3 times 4, which simplifies to 2 plus 6 plus 12, and that altogether gives us 20, which makes Z the smallest number, followed by Y, and finally followed by X. So that is the relationship here in D. Question number four. Each point in the picture is labeled with a number in such a way that the sum of the numbers at the endpoints of any edge is equal to the sum of the numbers at the endpoints of any other edge. What is the value of X? So in my bigger diagram here, if the value of X is some number, which I'll just let represent the color red, then the points on the adjacent sides of X, so at the endpoints of the two edges X is connected to, here and here, have to be the same value, so they are represented by the same color. Now in order to keep the rule that any edge sums to the same number as any other edge, the two remaining points here adjacent to the green point have to be in red, and likewise on the other side. So we have to keep this pattern, and we can repeat it like so until we see what color is assigned to the value of X. So either it's going to be 1 or 4, and if we keep going like that, we see that it is in fact going to be 1, because already 4 came out to be green, not the value of X, and 1 happens to be red in our pattern. So X is exactly 1, and the answer is A. Question number 5. How many natural numbers N have the property that when dividing 31 by N, the remainder is 7? Let's set up the division. We have the number 31, the dividend, the divisor is N, that's equal to the quotient plus the remainder over N, that's by the division law, and here Q is the quotient, and N cannot divide 7. In fact, it has to be greater than 7. So working with this equation, we can multiply both sides by N. Equivalently, we have 31 is the quotient times the divisor plus 7, or subtracting from both sides, we have 24 is Q times N, where now we know that N divides 24. So the divisors of 24 greater than 7 is what we have to count, and we have 3 of them, 8, 12, and finally the number itself, 24. So D is the answer, 3. Question number 6. When Arnold adds the length of three sides of a rectangle, he gets 20 centimeters. When Betty adds the length of three sides of the same rectangle, she gets 22 centimeters. What is the perimeter of this rectangle? So imagine there is a rectangle here with sides labeled A and B, and let's add up the sides in two different ways. So let's say that Arnold chose side A twice and side B once, and his answer came out to 20 centimeters. To obtain a different answer from the same rectangle, Betty must have chosen side B twice and side A once, otherwise she would have also obtained 20 centimeters. So now we have two equations. We have 2A plus B is equal to 20, and A plus 2B is equal to 22. So we can add these up, and 3A plus 3B is equal to 42 as a result. We can divide both sides by 3, and A plus B is 14. Now the perimeter here is equal to 2A plus 2B, or twice A plus B, which we know is 14, so the perimeter comes out to 28 centimeters. So in centimeters, the answer is C, 28. Question number 7. Beatrice wrote, in descending order, all 4-digit positive numbers whose digits add up to 4. In what position did she place the number 2011? We note that when she did this, she followed the convention that a 4-digit number cannot begin with a zero. So let's think of the largest number we can write, and that has to be the number 4000. Its digits add up to 4. We use 4 as the largest digit, and so we have to have three zeros. Then after that, in descending order, if we replace the 4 with a 3, we have to use a 1 in the remaining spots, and 3100 would be the next biggest number we can make like that. Then after that, 3110 and 3001, so we can put the 1 in three different positions. Now after that, if we replace 3 with 2, the remaining 3 digits have to add up to 2 also, so we can simply use 2 twice in a row, and 2200 is going to be the fifth number in the list. So after 2200, we can move the 2 around or use two 1s in the remaining 3 digits, 2110 would be the next number in our list, and then after that, we can use 2101 followed by 2020 followed by 2011. So here we have found 2011, and that is the ninth number in our list. So the answer is D. Question number 8. We have the triangle ABC shown on the right, and also here a larger version of it. We know that side AC is equal to side BC, which makes this an isosceles triangle. The measure of the angle ACB, the vertex angle here, is 36 degrees, and we also know that the segment AD is an angle bisector. What is the measure of the angle ADC? Let's label some sides here and some angle measures. We know that the measure of angle ACD here is 36 degrees, this being an isosceles triangle makes these two angles equal, and finally, because AD is an angle bisector, we also know that the angles here in blue are equal. What we have to determine for this question is the measure of this angle over here marked with a question mark. So first, let's find the measure of angle at vertex A and B. We know that 180 degrees is the sum of the angle measures, so that would be the measure of angle C plus twice the measure of angle A, and that gives us 36 degrees here plus twice the measure of angle A, and when we solve this, we have that 180 minus 36 is twice the measure of angle A, so 144 degrees is twice the measure of angle A, therefore angle A has to measure 72 degrees, and still half of that is equal to 36 degrees, and that is, again, the measure of one of the blue angles, and our triangle ADC is, again, an isosceles triangle. So the measure of angle marked in green here at D is computed in the similar method. We have twice 36 degrees plus, I will call our angle here angle ADC, that's equal to 180 degrees, so we can perform this calculation, measure of angle ADC is 180 degrees minus 72 degrees, that comes out to 108 degrees, so that is our final answer, E. Number nine. The figure in the illustration consists of a regular hexagon, that's the hexagon here in the center of the figure, with a side length of one centimeter, six squares that are positioned along the perimeter of the regular hexagon, and six triangles to fill in the spaces in between. What is the perimeter of this figure? What I have done is copied a part of it, enlarged over here to label the lengths of some edges, we know, part of the hexagon is displayed here, being a regular hexagon with length equal to one, we know that the square here is a unit square, each of these four edges has length one, and along another adjacent edge of the hexagon we have a similar square, so we are looking here at an isosceles triangle. We know furthermore that all of the edges here that we can see in this enlarged diagram, aside from these two, all have the same length, mainly one centimeter, so if we could just find the length of these unknown edges of the triangle, we would find the perimeter. So let's look at some of the angles that we know about. In a hexagon, the angle here, interior angle, measures 120 degrees, and the full angle about this point has to measure 360 degrees, so we have 120 degrees that we know about, plus the two 90 degree angles here, that's 90 degrees, that's 90 degrees, and an unknown angle here, I will call theta, so 120 plus theta, plus 90 degrees, plus 90 degrees, must equal to 360. So solving this, we have 120 plus theta, plus 180, is equal to 360, or 300 degrees on the left hand side, plus theta, equal to 360 degrees, and we see that the angle theta here, the vertex angle of one of these isosceles triangles, is 60 degrees, and that automatically makes this isosceles triangle equilateral. If this is 60 degrees, and it's an isosceles triangle, the other angles must also be 60 degrees. Well, if it's an equilateral triangles, all sides are equal, so the missing side here is also one centimeter. In fact, all the edges have length one in this figure, and the six triangles and the six squares that make up the perimeter have 12 edges that make up the perimeter, each of length one centimeter, so in centimeters, the answer here is 12. Number 10. A rectangular mosaic with area equal to 360 centimeters squared is made of square tiles, each of which is the same size. The mosaic is five tiles wide and has a length of 24 centimeters. What is the area of each tile in square centimeters? So we can make a diagram with five squares representing the width of the mosaic and a 24 centimeter long length here, so the area together has to be 360 centimeters squared, and then we can express 360 centimeters squared as the product of 24 centimeters and five tiles of, let's say, length x, so each tile would be a square of unknown length x, and then we can divide 360 by 24. That gives us 15. We can cancel the centimeters, and we have the following equation. x comes out to three, so each tile is three centimeters by three centimeters, and so it's a tile that in centimeters squared has area equal to nine, and that's our answer here, c.

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