60-odd years of moscow mathematical
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Moscow olympiad problems
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, . . . inside a square with side 5 6 so that they do not overlap, but it is impossible to do this in a square of a smaller side. 34.(8-10). Given a wooden ball, a compass, and a piece of paper, draw on the paper a circle of radius equal to that of the ball. (It is allowed to draw circles on the ball.) 35.(9-10). A square is divided in two ways into 100 parts of equal area. Prove that it is possible to select 100 points such that after each partition there is exactly one point in every part. SELECTED PROBLEMS OF MOSCOW MATHEMATICAL CIRCLES 149 36.(8). Prove that the number ³ 2 4 + 1 4 ´ ³ 4 4 + 1 4 ´ ³ 6 4 + 1 4 ´ ³ 8 4 + 1 4 ´ ³ 10 4 + 1 4 ´ ³ 12 4 + 1 4 ´ ³ 1 4 + 1 4 ´ ³ 3 4 + 1 4 ´ ³ 5 4 + 1 4 ´ ³ 7 4 + 1 4 ´ ³ 9 4 + 1 4 ´ ³ 11 4 + 1 4 ´ is an integer, and find the number by simplification without actual calculations. 37.(8-10). A table is entirely covered with 100 square tablecloths, A round hole burnt through the cover damageing each of the tablecloths. Each tablecloth would have covered the table but for the hole. Prove that some three of the tablecloths completely cover the table. 38.(8-10). Given 40 dice with the sum of the numbers on the opposite faces of each die equal to 7. Place the dice on one another to form a parallelepiped. Is it possible to rotate each die about its vertical axis so that the sums of the numbers on four lateral sides of the parallelepiped were equal? 39.(8-10). Concider two concentric circles, two parallel chords l and m tangent to the inner circle, point A on the outer circle between l and m; tangents to the inner circle through A, and their intersection points C and D with the chords. Prove that the product AC · AD does not depend on the position of A. 40.(8-10). Prove that the difference between the numbers 1 − 1 2 + 1 3 + 1 4 + . . . 1 (n − 1) + 1 n − 1 and 1 − 1 2 + 1 3 + 1 4 + . . . 1 (n − 1) + 1 n − 1 + 1 n is less than 1 (n − 1)!n! . 41.(8-9). Points A, B and C move uniformly along three circles in the same direction with the same angular velocity. How does the center of mass of triangle ABC move? 42.(8-9). Numbers 1, 2, 3, . . . , 1974 are written on a blackboard. In one move we can erase any two numbers of the set and write in their place the absolute value of their difference. After 1973 moves one number is left. What number can it be? 43.(8-9). A and B are the tangent points of straight lines a and b and a circle. We selected point C on line a, and point D on line b. Segment AB meets segment CD at point M . Prove that CA/CM = DB/DM . 44.(8-10). Given p + 1 distinct positive integers for a prime p, prove that among them there is a pair of numbers x and y such that the quotient after the division of the greater of these two numbers by GCD(x, y) is not less than p + 1. 45.(9-10). Prove that Download 1.08 Mb. Do'stlaringiz bilan baham: |
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