MathDB

3

Part of 2019 Caucasus Mathematical Olympiad

Problems(2)

Four vertices of special equilateral 8-gon are concyclic

Source: IV Caucasus Mathematic Olympiad

4/7/2019
Points A′A' and B′B' lie inside the parallelogram ABCDABCD and points C′C' and D′D' lie outside of it, so that all sides of 8-gon AA′BB′CC′DD′AA'BB'CC'DD' are equal. Prove that A′A', B′B', C′C', D′D' are concyclic.
geometry
Permutations of 1, 2, ..., 2n with equality condition

Source: IV Caucasus Mathematic Olympiad

4/7/2019
Find all positive integers n⩾2n\geqslant 2 such that there exists a permutation a1a_1, a2a_2, a3a_3, \ldots, a2na_{2n} of the numbers 1,2,3,…,2n1, 2, 3, \ldots, 2n satisfying a1⋅a2+a3⋅a4+…+a2n−3⋅a2n−2=a2n−1⋅a2n.a_1\cdot a_2 + a_3\cdot a_4 + \ldots + a_{2n-3} \cdot a_{2n-2} = a_{2n-1} \cdot a_{2n}.
algebrapermutations