Olympiad problems

2022/2023 Tournament of Towns, P4

Tags: combinatorics , board

Let $n>1$ be an integer. A rook stands in one of the cells of an infinite chessboard that is initially all white. Each move of the rook is exactly $n{}$ cells in a single direction, either vertically or horizontally, and causes the $n{}$ cells passed over by the rook to be painted black. After several such moves, without visiting any cell twice, the rook returns to its starting cell, with the resulting black cells forming a closed path. Prove that the number of white cells inside the black path gives a remainder of $1{}$ when divided by $n{}$.

2006 Team Selection Test For CSMO, 3

Tags: combinatorics

The set $M= \{1;2;3;\ldots ; 29;30\}$ is divided in $k$ subsets such that if $a+b=n^2, (a,b \in M, a\neq b, n$ is an integer number $)$, then $a$ and $b$ belong different subsets. Determine the minimum value of $k$.

2004 Alexandru Myller, 1

Tags: function , combinatorics , injectivity

Find the number of self-maps of a set of $ 5 $ elements having the property that the preimage of any element of this set has $ 2 $ elements at most. [i]Adrian Zanoschi[/i]

2023 Math Prize for Girls Problems, 10

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Find all integers $x$ between 0 and the prime number 4099 such that $x^3 - 3$ is divisible by 4099.

2018 Latvia Baltic Way TST, P5

Tags: combinatorics

Alice and Bob play a game on a numbered row of $n \ge 5$ squares. At the beginning a pebble is put on the first square and then the players make consecutive moves; Alice starts. During a move a player is allowed to choose one of the following: [list] [*] move the pebble one square forward; [*] move the pebble four squares forward; [*] move the pebble two squares backwards. [/list] All of the possible moves are only allowed if the pebble stays within the borders of the square row. The player who moves the pebble to the last square (a.k.a $n\text{-th}$) wins. Determine for which values of $n$ each of the players has a winning strategy.

2017 Kyiv Mathematical Festival, 4

Tags: game , combinatorics

Two players in turn put two or three coins into their own hats (before the game starts, the hats are empty). Each time, after both players made five moves, they exchange hats.The player wins, if after his move his hat contains one hundred or more coins. Which player has a winning strategy?

MOAA Team Rounds, 2023.5

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Angeline starts with a 6-digit number and she moves the last digit to the front. For example, if she originally had $100823$ she ends up with $310082$. Given that her new number is $4$ times her original number, find the smallest possible value of her original number. [i]Proposed by Angeline Zhao[/i]

1972 AMC 12/AHSME, 14

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A triangle has angles of $30^\circ$ and $45^\circ$. If the side opposite the $45^\circ$ angle has length $8$, then the side opposite the $30^\circ$ angle has length $\textbf{(A) }4\qquad\textbf{(B) }4\sqrt{2}\qquad\textbf{(C) }4\sqrt{3}\qquad\textbf{(D) }4\sqrt{6}\qquad \textbf{(E) }6$

2014 Contests, 2

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$: $xP(x-c) = (x - 2014)P(x)$

1997 Baltic Way, 18

Tags: combinatorics

a) Prove the existence of two infinite sets $A$ and $B$, not necessarily disjoint, of non-negative integers such that each non-negative integer $n$ is uniquely representable in the form $n=a+b$ with $a\in A,b\in B$. b) Prove that for each such pair $(A,B)$, either $A$ or $B$ contains only multiples of some integer $k>1$.

1956 Putnam, A7

Tags: binomial coefficient

Prove that the number of odd binomial coefficients in any finite binomial expansion is a power of $2.$

2009 Saint Petersburg Mathematical Olympiad, 7

Tags: algebra

Discriminants of square trinomials $f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x)$ equals $1$. Prove that $f(x)+h(x)+g(x) \equiv 0$

2005 South East Mathematical Olympiad, 4

Tags: function , modular arithmetic , number theory

Find all positive integer solutions $(a, b, c)$ to the function $a^{2} + b^{2} + c^{2} = 2005$, where $a \leq b \leq c$.

2005 Purple Comet Problems, 8

Tags: number theory , relatively prime

The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

1959 AMC 12/AHSME, 2

Tags: altitude , area

Through a point $P$ inside the triangle $ABC$ a line is drawn parallel to the base $AB$, dividing the triangle into two equal areas. If the altitude to $AB$ has a length of $1$, then the distance from $P$ to $AB$ is: $ \textbf{(A)}\ \frac12 \qquad\textbf{(B)}\ \frac14\qquad\textbf{(C)}\ 2-\sqrt2\qquad\textbf{(D)}\ \frac{2-\sqrt2}{2}\qquad\textbf{(E)}\ \frac{2+\sqrt2}{8} $

2023 Belarus - Iran Friendly Competition, 4

Tags: geometry , incircle

Let $\Gamma$ be the incircle of a non-isosceles triangle $ABC$, $I$ be it’s incenter. Let $A_1, B_1, C_1$ be the tangency points of $\Gamma$ with the sides $BC, AC, AB$ respectively. Let $A_2 = \Gamma \cap AA_1$, $M = C_1B_1 \cap AI$, $P$ and $Q$ be the other (different from $A_1$ and $A_2$) intersection points of $\Gamma$ and $A_1M$, $A_2M$ respectively. Prove that $A$, $P$ and $Q$ are colinear.

2021-2022 OMMC, 15

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Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that $$n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}.$$ [i]Proposed by Justin Lee[/i]

2018 Finnish National High School Mathematics Comp, 2

Tags: angle bisector , sidelengths , geometry

The sides of triangle $ABC$ are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.

KoMaL A Problems 2023/2024, A. 858

Tags: quadratic residue , number theory , diophantine equation

Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$

2008 May Olympiad, 3

Tags: number theory , prime

In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.

2020 Philippine MO, 2

Tags: algebra

Determine all positive integers $k$ for which there exist positive integers $r$ and $s$ that satisfy the equation $$(k^2-6k+11)^{r-1}=(2k-7)^{s}.$$

2020 Tournament Of Towns, 5

Tags: right angle , trapezoid , equal segments , trapezium , geometry

Let $ABCD$ be an inscribed trapezoid. The base $AB$ is $3$ times longer than $CD$. Tangents to the circumscribed circle at the points $A$ and $C$ intersect at the point $K$. Prove that the angle $KDA$ is a right angle. Alexandr Yuran

2012 Dutch IMO TST, 1

Tags: perfect power , number theory , prime , gcd , greatest common divisor

For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2019 Taiwan APMO Preliminary Test, P4

Tags: algebra , floor function , sequence , recursive

We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$ (1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

1975 Dutch Mathematical Olympiad, 3

Tags: inequalities , sum , algebra

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$