MathDB

1961 All-Soviet Union Olympiad

Part of All-Russian Olympiad

Subcontests

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Multiply by the next one - quartets

Consider a quartet of positive numbers (a,b,c,d)(a,b,c,d). In one step, we transform it to (ab,bc,cd,da)(ab,bc,cd,da). Prove that you can never obtain the initial set if neither of a,b,c,da,b,c,d is 11.

Nickolas and Peter divides nuts

Nickolas and Peter divide 2n+12n+1 nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: Stage 1: Peter divides the nuts into 2 heaps, each containing at least 2 nuts. Stage 2: Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: Method 1: Nickolas takes the smallest and largest of the heaps. Method 2: Nickolas takes the two middle size heaps. Method 3: Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice.
Determine the most and the least profitable method for Nickolas.

Multiply by the next one - 2^k-tuples

Consider a 2k2^k-tuple of numbers (a1,a2,…,a2k)(a_1,a_2,\dots,a_{2^k}) all equal to 11 or −1-1. In one step, we transform it to (a1a2,a2a3,…,a2ka1)(a_1a_2,a_2a_3,\dots,a_{2^k}a_1). Prove that eventually, we will obtain a 2k2^k-tuple consisting only of 11's.
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Outer tangents form a circumscribed quadrangle

Consider a rectangle A1A2A3A4A_1A_2A_3A_4 and a circle Ci\mathcal{C}_i centered at AiA_i with radius rir_i for i=1,2,3,4i=1,2,3,4. Suppose that r1+r3=r2+r4<dr_1+r_3=r_2+r_4<d, where dd is the diagonal of the rectangle. The two pairs of common outer tangents of C1\mathcal{C}_1 and C3\mathcal{C}_3, and of C2\mathcal{C}_2 and C4\mathcal{C}_4 form a quadrangle. Prove that this quadrangle has an inscribed circle.

Classic - sum of rows and columns nonnegative

Consider a table with one real number in each cell. In one step, one may switch the sign of the numbers in one row or one column simultaneously. Prove that one can obtain a table with non-negative sums in each row and each column.

Another circle fits

Consider 120120 unit squares arbitrarily situated in a 20×2520\times 25 rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.
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No curve intersecting each segment only once

Consider the figure below, composed of 16 segments. Prove that there is no curve intersecting each segment exactly once. (The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.) [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]

Clocks with hand endpoint forming an equilateral triangle

Points AA and BB move on circles centered at OAO_A and OBO_B such that OAAO_AA and OBBO_BB rotate at the same speed. Prove that vertex CC of the equilateral triangle ABCABC moves along a certain circle at the same angular velocity. (The vertices of ABCABC are oriented clockwise.)

Three sequences have a simultaneous relation

Prove that for any three infinite sequences of natural numbers (an)n≥1(a_n)_{n\ge 1}, (bn)n≥1(b_n)_{n\ge 1}, (cn)n≥1(c_n)_{n\ge 1}, there exist numbers pp and qq such that ap≥aqa_p\ge a_q, bp≥bqb_p\ge b_q and cp≥cqc_p\ge c_q.