Math Olympiad : Training 2
3:49 AM
Muhammad Yusuf
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1. If f(x) = x2 +3 and g(x) = 2x - 1, express f(g(x)) as an explicit function of x, using no parentheses in your answer.
2. Determine the number of pairs (m, n) of positive integers for which
m3 + n = 1000000008 = 109 + 8
3. The streets in a city form a grid, with some running north-south, and others east- west. Alice, Bob, Carol, and Don live at four street intersections which are vertices of a rectangle, with Alice and Don at opposite vertices. The school that they all attend is at an intersection inside the rectangle, and they all walk to it by their shortest possible route. Alice walks 10 blocks, Bob walks 12 blocks, and Carol walks 15 blocks. How many blocks does Don walk to get to school?
4. What is the number of noncongruent rectangles with integer sides and area 315?
5. A rectangular sheet of paper is folded in half, with the crease parallel to the shorter edge. It happens that the ratio of longer side to shorter side remains unchanged. What is this ratio?
6. How many lines in the plane are at distance 2 from the point (0, 0) and also at distance 3 from the point (6, 0)?
7. What is the remainder when x135 + x125 - x115 + x5 + 1 is divided by x3 - x?
8. A ‘turn" is a flip of a pair of fair coins. What is the probability that, in 4 turns, you
obtain a pair of heads at least once?
9. For positive integers a, b, and c, what is the smallest possible value of a + b + c for which abc = 7(a + b + c)?
10. The exterior angles of a triangle are in the ratio 3 : 5 : 6. What is the ratio, similarly expressed and reduced to lowest terms, of the interior angles adjacent to them, in the same order?
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