MathDB

2021 Tuymaada Olympiad

Part of Tuymaada Olympiad

Subcontests

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Knights and Liars again

Some manors of Lipshire county are connected by roads. The inhabitants of manors connected by a road are called neighbours. Is it always possible to settle in each manor a knight (who always tells truth) or a liar (who always lies) so that every inhabitant can say ”The number of liars among my neighbours is at least twice the number of knights”?

Counting monotonous paths in $n\times n$ square

An n×nn\times n square (nn is a positive integer) consists of n2n^2 unit squares.A \emph{monotonous path} in this square is a path of length 2n2n beginning in the left lower corner of the square,ending in its right upper corner and going along the sides of unit squares. For each kk, 0≤k≤2n−10\leq k\leq 2n-1, let SkS_k be the set of all the monotonous paths such that the number of unit squares lying below the path leaves remainder kk upon division by 2n−12n-1.Prove that all SkS_k contain equal number of elements.
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Numbers dividing their pairwise sums

For nn distinct positive integers all their n(n−1)/2n(n-1)/2 pairwise sums are considered. For each of these sums Ivan has written on the board the number of original integers which are less than that sum and divide it. What is the maximum possible sum of the numbers written by Ivan?

Inequality back to business again!

Positive real numbers a1,…,ak,b1,…,bka_1, \dots, a_k, b_1, \dots, b_k are given. Let A=∑i=1kai,B=∑i=1kbiA = \sum_{i = 1}^k a_i, B = \sum_{i = 1}^k b_i. Prove the inequality (∑i=1kaibiaiB+biA−1)2≥∑i=1kai2aiB+biA⋅∑i=1kbi2aiB+biA. \left( \sum_{i = 1}^k \frac{a_i b_i}{a_i B + b_i A} - 1 \right)^2 \ge \sum_{i = 1}^k \frac{a_i^2}{a_i B + b_i A} \cdot \sum_{i = 1}^k \frac{b_i^2}{a_i B + b_i A}.
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