MathDB

2019 Caucasus Mathematical Olympiad

Part of Caucasus Mathematical Olympiad

Subcontests

(7)
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Divisible condition for every subset

Determine if there exist positive integers a1,a2,...,a10a_1,a_2,...,a_{10}, b1,b2,...,b10b_1,b_2,...,b_{10} satisfying the following property: for each non-empty subset SS of {1,2,…,10}\{1,2,\ldots,10\} the sum ∑i∈Sai\sum\limits_{i\in S}a_i divides (12+∑i∈Sbi)\left( 12+\sum\limits_{i\in S}b_i \right).

(another) Divisible condition for every subset

Determine if there exist pairwise distinct positive integers a1,a2,…,a101a_1,a_2,\ldots,a_{101}, b1b_1, b2b_2, \ldots, b101b_{101} satisfying the following property: for each non-empty subset SS of {1,2,…,101}\{1,2,\ldots,101\} the sum ∑i∈Sai\sum\limits_{i\in S}a_i divides (100!+∑i∈Sbi)\left( 100!+\sum\limits_{i\in S}b_i \right).
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Four vertices of special equilateral 8-gon are concyclic

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.

Permutations of 1, 2, ..., 2n with equality condition

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}.
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Determination of the color of the ball

In the kindergarten there is a big box with balls of three colors: red, blue and green, 100 balls in total. Once Pasha took out of the box 30 red, 10 blue, and 20 green balls and played with them. Then he lost five balls and returned the others back into the box. The next day, Sasha took out of the box 8 red, 18 blue, and 48 green balls. Is it possible to determine the color of at least one lost ball?

Paint the largest number in each square

Pasha placed numbers from 1 to 100 in the cells of the square 10×1010\times 10, each number exactly once. After that, Dima considered all sorts of squares, with the sides going along the grid lines, consisting of more than one cell, and painted in green the largest number in each such square (one number could be colored many times). Is it possible that all two-digit numbers are painted green?