Olympiad problems

1976 Bundeswettbewerb Mathematik, 2

Two congruent squares $Q$ and $Q'$ are given in the plane. Show that they can be divided into parts $T_1, T_2, \ldots , T_n$ and $T'_1 , T'_2 , \ldots , T'_n$, respectively, such that $T'_i$ is the image of $T_i$ under a translation for $i=1,2, \ldots, n.$

2015 ASDAN Math Tournament, 5

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Compute the number of zeros at the end of $2015!$.

2001 India IMO Training Camp, 2

Tags: greatest common divisor , number theory

A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.

2009 Chile National Olympiad, 5

Tags: combinatorics

Let $A$ and $B$ be two cubes. Numbers $1,2,...,14$, are assigned in any order, to the faces and vertices of cube $A$. Then each edge of cube $A$ is assigned the average of the numbers assigned to the two faces that contain it. Finally assigned to each face of the cube $B$ the sum of the numbers associated with the vertices, the face and the edges on the corresponding face of cube $A$. If $S$ is the sum of the numbers assigned to the faces of $B$, find the largest and smallest value that $S$ can take.

2001 National Olympiad First Round, 2

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Each of the football teams Istanbulspor, Yesildirek, Vefa, Karagumruk, and Adalet, played exactly one match against the other four teams. Istanbulspor defeated all teams except Yesildirek; Yesildirek defeated Istanbulspor but lost to all the other teams. Vefa defeated all except Istanbulspor. The winner of the game Karagumruk-Adalet is Karagumruk. In how many ways one can order these five teams such that each team except the last, defeated the next team? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of the preceding} $

2020 Brazil Team Selection Test, 8

Tags: inequalities

Let $a_1, a_2,\dots$ be an infinite sequence of positive real numbers such that for each positive integer $n$ we have \[\frac{a_1+a_2+\cdots+a_n}n\geq\sqrt{\frac{a_1^2+a_2^2+\cdots+a_{n+1}^2}{n+1}}.\] Prove that the sequence $a_1,a_2,\dots$ is constant. [i]Proposed by Alex Zhai[/i]

2001 AIME Problems, 3

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Given that \begin{align*} x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523, \text{ and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \text{ when } n \geq 5, \end{align*} find the value of $x_{531}+x_{753}+x_{975}$.

1972 Putnam, A2

Tags: binary operation

Let $S$ be a set with a binary operation $\ast$ such that 1) $a \ast(a\ast b)=b$ for all $a,b\in S$. 2) $(a\ast b)\ast b=a$ for all $a,b\in S$. Show that $\ast$ is commutative and give an example where $\ast$ is not associative.

2017 Indonesia MO, 7

Tags: geometry , parallelogram

Let $ABCD$ be a parallelogram. $E$ and $F$ are on $BC, CD$ respectively such that the triangles $ABE$ and $BCF$ have the same area. Let $BD$ intersect $AE, AF$ at $M, N$ respectively. Prove there exists a triangle whose side lengths are $BM, MN, ND$.

1963 Czech and Slovak Olympiad III A, 1

Tags: geometry , rectangle , 3d geometry , computation

Consider a cuboid$ ABCDA'B'C'D'$ (where $ABCD$ is a rectangle and $AA' \parallel BB' \parallel CC' \parallel DD'$) with $AA' = d$, $\angle ABD' = \alpha, \angle A'D'B = \beta$. Express the lengths x = $AB$, $y = BC$ in terms of $d$ and (acute) angles $\alpha, \beta$. Discuss condition of solvability.