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The 1st All Russian Mathematical Olympiad
1961年第一届全俄数学奥林匹克
  1. Given 12 vertices and 16 edges arranged as follows:
    Draw any curve which does not pass through any vertex. Prove that the curve cannot intersect each edge just once. Intersection means that the curve crosses the edge from one side to the other. For example, a circle which had one of the edges as tangent would not intersect that edge.
  2. Given a rectangle ABCD with AC length e and four circles centers A, B, C, D and radii a, b, c, d respectively, satisfying a+c=b+d<e. Prove you can inscribe a circle inside the quadrilateral whose sides are the two outer common tangents to the circles center A and C, and the two outer common tangents to the circles center B and D.
  3. Prove that any 39 successive natural numbers include at least one whose digit sum is divisible by 11.
  4. (a) Arrange 7 stars in the 16 places of a 4 x 4 array, so that no 2 rows and 2 columns contain all the stars.
    (b) Prove this is not possible for <7 stars.
  5. (a) Given a quadruple (a, b, c, d) of positive reals, transform to the new quadruple (ab, bc, cd, da). Repeat arbitarily many times. Prove that you can never return to the original quadruple unless a=b=c=d=1.
    (b) Given n a power of 2, and an n-tuple (a1, a2, ... , an) transform to a new n-tuple (a1a2, a2a3, ... , an-1an, ana1). If all the members of the original n-tuple are 1 or -1, prove that with sufficiently many repetitions you obtain all 1s.
  6. (a) A and B move clockwise with equal angular speed along circles center P and Q respectively. C moves continuously so that AB=BC=CA. Establish C's locus and speed.
    *(b) ABC is an equilateral triangle and P satisfies AP=2, BP=3. Establish the maximum possible value of CP.
  7. Given an m x n array of real numbers. You may change the sign of all numbers in a row or of all numbers in a column. Prove that by repeated changes you can obtain an array with all row and column sums non-negative.
  8. Given n<1 points, some pairs joined by an edge (an edge never joins a point to itself). Given any two distinct points you can reach one from the other in just one way by moving along edges. Prove that there are n-1 edges.
  9. Given any natural numbers m, n and k. Prove that we can always find relatively prime natural numbers r and s such that rm+sn is a multiple of k.
  10. A and B play the following game with N counters. A divides the counters into 2 piles, each with at least 2 counters. Then B divides each pile into 2 piles, each with at least one counter. B then takes 2 piles according to a rule which both of them know, and A takes the remaining 2 piles. Both A and B make their choices in order to end up with as many counters as possible. There are 3 possibilities for the rule:
     R1 B takes the biggest heap (or one of them if there is more than one) and the smallest heap (or one of them if there is more than one).
     R2 B takes the two middling heaps (the two heaps that A would take under R1).
     R3 B has the choice of taking either the biggest and smallest, or the two middling heaps.
    For each rule, how many counters will A get if both players play optimally ?
  11. Given three arbitary infinite sequences of natural numbers, prove that we can find unequal natural numbers m, n such that for each sequence the mth member is not less than the nth member.
  12. 120 unit squares are arbitarily arranged in a 20 x 25 rectangle (both position and orientation is arbitary). Prove that it is always possible to place a circle of unit diameter inside the rectangle without intersecting any of the squares.
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