MathDB

1

Part of 1961 All-Soviet Union Olympiad

Problems(3)

No curve intersecting each segment only once

Source: 1961 All-Soviet Union Olympiad

8/4/2015
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]
combinatoricsgraph theory
Clocks with hand endpoint forming an equilateral triangle

Source: 1961 All-Russian Olympiad

8/4/2015
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.)
geometryEquilateral TriangleRussia
Three sequences have a simultaneous relation

Source: 1961 All-Soviet Union Olympiad

8/4/2015
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.
algebraSequences