Math Kangaroo Competition Questions - Samples of Suggested Solutions.
Level 9-10, 2003
1. E) QUOTE
When adding, both denominators must be equal or like.
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Next, multiply opposites of the given equation.
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Plug in an value in equation. bn cancels out.
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2. E) 22
First calculate the area of the entire square.
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Next calculate the areas of the three smaller triangles.
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Subtract the sum of the three triangles from the area of the entire square.
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48 – (8 + 6 + 12)
22
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C)
4. E)
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5. B) 2a + b
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Level 11-12, 2010
15. B) 1
Because
p consists of only prime numbers as its digits, the only possible digits are 2, 3, 5, 7. There are potentially 64 different configurations of these numbers. Obviously, these will all be odd numbers, so p – 1 will be even, and its only prime divisor must be 2. Therefore, only one configuration satisfies p – 1 = a number with only one prime divisor. This configuration is 257.p = 257
p – 1 = 256 = 2816. A) 1
There can only be 1 blue and 1 green marble, otherwise there is a possibility that drawing 5 marbles at random will produce the wrong result. If 2 red marbles have to be drawn every time and 3 marbles have to be the same color, we can assume there are 3 red marbles. This leaves no more than 2 marbles left in the bag. Because there are 2 colors left, we can assume 1 is blue and 1 is green.
17. C) 84
Inscribe (or circumscribe) the 14-gon in (or about) a circle and use the Theorem of Thales, which states that a triangle with one edge being the diameter of that circle and the other edges intersecting on the circle is necessarily a right triangle. There are
142 ways to select the vertices which will lie on the hypotenuse, but by symmetry half the choices will be redundant. Thus there are 142 – 7 = 91 – 7 = 84 choices.
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Level 7-8, 2005
19. C) 2002
499 + 500 + 501 + 502 = 2002
20. D) 19
The segments connecting the vertices of the cube will be the edges (12), two diagonals on each of the faces of the cube (12), and segments connecting the opposite vertices of the cube through the inside of the cube (4). The segments along the edges give 12 distinct midpoints, the two segments on each face share a point at their midpoints, thus giving 6 distinct points, and the 4 segments going through the inside of the cube all intersect at the same point at their midpoints, thus giving only 1 more distinct point. 12 + 6 + 1 =19
21. C) 5
We need to find numbers which have exactly 3 prime factors, and the smallest of which must be at least 3, because the number is not divisible by 2.
3 ´ 3 ´ 3 = 27 3 ´ 3 ´ 5 = 45 3 ´ 3 ´ 7 = 63 3 ´ 3 ´ 11 = 99
If we tried using 13 as the third factor at this point, the number would be too large, so we will start with the next set of factors.
3 ´ 5 ´ 5 = 75 (because 3 ´ 5 ´ 3 is the same as 3 ´ 3 ´ 5)
3 ´ 5 ´ 7 is not a two-digit number
5 ´ 5 ´ 5 is also not a two-digit number
So we see that no other attempts will result in a two-digit number.
Thus, the numbers are 27, 45, 63, 75, 99, and there are 5 of them.
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Level 3-4, 2004
7. E) Green
The red building is only next to the blue building, which means that the red building is either building 1 or 5, and the blue building is either building 2 or 4. The blue building is between the red building and the green building. If the blue building was number 2, then the red building is number 1 and the green building is number 3. If the blue building is number 4, then the red building is number 5 and the green building is number 3. Either way, the green building is number 3.
8. B) 3
The figure is made of 24 squares. For the number of shaded squares to equal half of the number of white squares, the number of shaded squares has to equal 8 (number of white squares equals 16). (You can get this either by trying numbers that add up to 24 or by dividing 24 into three parts, with 1/3 of 24 being shaded and 2/3 of 24 not being shaded.) There are already 5 squares shaded grey, so 3 more must be shaded.
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Level 1-2, 2006
5. C) 6
20 students – 8 boys = 12 girls. 12 ¸ 2 = 6.
Since two girls sit at each desk, we need 6 desks.
6. E) 60 m
From A to B + B to C + C to D + D to A:
5 m + 10 m + 20 m + 25 m = 60 m.
7. A) 6
8. E) 9
In the pattern we skip one number, so after 7 comes 9.
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