Olympiad problems

1998 Tournament Of Towns, 2

John and Mary each have a white $8 \times 8$ square divided into $1 \times 1$ cells. They have painted an equal number of cells on their respective squares in blue. Prove that one can cut up each of the two squares into $2 \times 1 $ dominoes so that it is possible to reassemble John's dominoes into a new square and Mary's dominoes into another square with the same pattern of blue cells. (A Shapovalov)

PEN H Problems, 80

Tags: diophantine equation

Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

VII Soros Olympiad 2000 - 01, 8.5

Tags: combinatorics

Vanya was asked to write on the board an expression equal to $10$, using only the numbers $1$, the signs $+$ and $-$ and brackets (you cannot make up the numbers $11$, $111$, etc., as well as $(-1)$). He knows that the bully Anton will then correct all the $+$ signs to $-$ and vice versa. Help Vanya compose the required expression, which will remain equal to $10$ even after Anton's actions.

2012 Kazakhstan National Olympiad, 3

Tags: incenter , geometric transformation , reflection , geometry

Line $PQ$ is tangent to the incircle of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point

2023 Baltic Way, 1

Tags: algebra

Find all strictly increasing sequences of positive integers $a_1, a_2, \ldots$ with $a_1=1$, satisfying $$3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n}$$ for all positive integers $n$.

2020 Putnam, A4

Tags:

Consider a horizontal strip of $N+2$ squares in which the first and the last square are black and the remaining $N$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color tis neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. Find \[ \lim_{N\to\infty}\frac{w(N)}{N}.\]

1982 Poland - Second Round, 4

Tags: geometry , 3d geometry , sphere

Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space that transforms the set $ A $ into the set $ A $ and the point $ P $ into the point $ Q $. Prove that there is a sphere passing through all points of the set $ A $.

2010 Stanford Mathematics Tournament, 4

Tags: function

If $x^2+\frac{1}{x^2}=7,$ find all possible values of $x^5+\frac{1}{x^5}.$

2003 Regional Competition For Advanced Students, 2

Tags: number theory

Find all prime numbers $ p$ with $ 5^p\plus{}4p^4$ is the square of an integer.

2022 Saudi Arabia IMO TST, 1

Tags: geometry

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.