60-odd years of moscow mathematical
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Moscow olympiad problems
n
is less than 10 100 . Is the set of such positive integers n finite or infinite? 19.(7-9). Prove that no digit is repeated 5 000 001 times in a row in the first 10 million digits of the decimal representation of √ 2 (the integer part included). 20.(7-9). There are 1 000 airports in the land Shvambrania, and the distances between every two airports are distinct. Suppose an airplane departs from each airport and flies to the nearest airport. What greatest number of airplanes can land in an arbitrary airport if Shvambrania is a) a plane? b) a sphere? 21.(9-10). Several spherical holes are made in a cheese cube. Prove that it is possible to cut the cheese into convex polyhedrons so that there is exactly one hole inside each of the polyhedrons. 22.(10). Let sin α = 3 5 . Prove that sin 25α = n 5 25 , where n is an integer not divisible by 5. 23.(7-10). Three bulbs — one blue, one green, and one red — are somehow connected by wires to n switches. Each switch can be in one of three positions. For any position of all the switches exactly one bulb is turned on, but if all n switches are simultaneously flipped (each by its own of the 2 possible ways), another bulb is turned on. Prove that the color of the bulb which is turned on is determined by one fixed switch and does not depend in any way on the other switches. 24.(8-10). A grasshopper hops on an infinite chessboard with squares of side 1 moving with each hop a distance of α to the right and β upwards. Prove that if numbers α, β and α β are irrational, then the grasshopper will necessarily reach a black square. 25.(9-10). Prove that tan 3π 11 + 4 sin 2π 11 = √ 11. 26.(8-10). Solve in positive integers: 520(xyzt + xy + xz + zt + 1) = 577(yzt + y + z). 27.(7-8). Prove that if at all times at least one of ten uniformly functioning alarm clocks shows correct time, then at least one of them always shows correct time. 28.(9-10). The space is divided into identical and identically oriented parallelepipeds. a) Prove that for each parallelepiped at least 14 of the parallelepipeds have a common point with it. b) What is the least number of parallelepipeds that have a common point with a given parallelepiped if the parallelepipeds are still identical but not equally oriented? 29.(8-10). A triangular lamina of area 1 is cut into 4 parts (three triangles and 1 quadrilateral) by two straight cuts. Three parts have the same area. Find the area of every part. 30.(8-9). Prove that if the arithmetic mean of the first 10 10 10 digits in the decimal expression of 2 − √ 2 is between 4 1 3 and 4 2 3 , then the same is true for √ 2 − 1. 31.(8-10). Prove that at any given moment there is a point on the surface of the Sun (considered as a sphere) from which one can see not more than 3 planets (out of 9 known ones). 32.(7-9). There are two containers: the first one has 1l of water in it, the second one is empty. We pore half of the water from the first container into the second one; then we pore one third of the water from the second container back into the first container; then we pore one fourth of the water from the first container into the second container, and so on. How much water is there in the first container after 12345 refills? 33.(9-10)*. Prove that it is possible to arrange infinitely many squares with sides 1 2 , 1 3 , 1 4 , 1 5 , . . . , 1 Download 1.08 Mb. Do'stlaringiz bilan baham: |
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