This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 8

2019 All-Russian Olympiad, 5
Tags: ARMO
In kindergarten, nurse took $n>1$ identical cardboard rectangles and distributed them to $n$ children; every child got one rectangle. Every child cut his (her) rectangle into several identical squares (squares of different children could be different). Finally, the total number of squares was prime. Prove that initial rectangles was squares.

2019 All-Russian Olympiad, 6
Tags: ARMO
There is point $D$ on edge $AC$ isosceles triangle $ABC$ with base $BC$. There is point $K$ on the smallest arc $CD$ of circumcircle of triangle $BCD$. Ray $CK$ intersects line parallel to line $BC$ through $A$ at point $T$. Let $M$ be midpoint of segment $DT$. Prove that $\angle AKT=\angle CAM$.

2019 All-Russian Olympiad, 7
Tags: ARMO
Among 16 coins there are 8 heavy coins with weight of 11 g, and 8 light coins with weight of 10 g, but it's unknown what weight of any coin is. One of the coins is anniversary. How to know, is anniversary coin heavy or light, via three weighings on scales with two cups and without any weight?

2019 All-Russian Olympiad, 1
Tags: ARMO , combinatorics , combinatorial geometry
There are 5 points on plane. Prove that you can chose some of them and shift them such that distances between shifted points won't change and as a result there will be symetric by some line set of 5 points.

2019 All-Russian Olympiad, 1
Tags: ARMO
There is located real number $f(A)$ in any point A on the plane. It's known that if $M$ will be centroid of triangle $ABC$ then $f(M)=f(A)+f(B)+f(C)$. Prove that $f(A)=0$ for all points A.

2019 All-Russian Olympiad, 4
Tags: ARMO , combinatorics , graph theory
10000 children came to a camp; every of them is friend of exactly eleven other children in the camp (friendship is mutual). Every child wears T-shirt of one of seven rainbow's colours; every two friends' colours are different. Leaders demanded that some children (at least one) wear T-shirts of other colours (from those seven colours). Survey pointed that 100 children didn't want to change their colours [translator's comment: it means that any of these 100 children (and only them) can't change his (her) colour such that still every two friends' colours will be different]. Prove that some of other children can change colours of their T-shirts such that as before every two friends' colours will be different.

2019 All-Russian Olympiad, 2
Tags: ARMO
Find minimal natural $n$ for which there exist integers $a_1, a_2,\ldots, a_n$ such that quadratic trinom $$x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)$$ has at least one integral root.

2019 All-Russian Olympiad, 7
Tags: ARMO
There are non-constant polynom $P(x)$ with integral coefficients and natural number $n$. Suppose that $a_0=n$, $a_k=P(a_{k-1})$ for any natural $k$. Finally, for every natural $b$ there is number in sequence $a_0, a_1, a_2, \ldots$ that is $b$-th power of some natural number that is more than 1. Prove that $P(x)$ is linear polynom.