Olympiad problems

2025 Turkey Team Selection Test, 1

In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.

2011 JHMT, 4

Tags: geometry

Compute the largest value of $r$ such that three non-overlapping circles of radius $r$ can be inscribed in a unit square.

Oliforum Contest III 2012, 2

Tags: polynomial , sum of digits , number theory

Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

2014 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , graph theory

Some cities in country are connected with oneway road. It is known that every closed cyclic route, that don`t break traffic laws, consists of even roads. Prove that king of city can place military bases in some cities such that there are not roads between these cities, but for every city without base we can go from city with base by no more than $1$ road. [hide=PS]I think it should be one more condition, like there is cycle that connect all cities [/hide]

1978 Swedish Mathematical Competition, 2

Tags: sum , number theory , digit , algebra

Let $s_m$ be the number $66\cdots 6$ with $m$ digits $6$. Find \[ s_1 + s_2 + \cdots + s_n \]

1991 Arnold's Trivium, 91

Tags: algebra , polynomial , linear algebra , matrix

Find the Jordan normal form of the operator $e^{d/dt}$ in the space of quasi-polynomials $\{e^{\lambda t}p(t)\}$ where the degree of the polynomial $p$ is less than $5$, and of the operator $\text{ad}_A$, $B\mapsto [A, B]$, in the space of $n\times n$ matrices $B$, where $A$ is a diagonal matrix.

2013 Czech And Slovak Olympiad IIIA, 6

Tags: inequalities , algebra

Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$.

1992 Turkey Team Selection Test, 1

Tags: prime number , number theory

Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.

2014 Contests, 3

Tags: number theory , divisibility

Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

1970 Swedish Mathematical Competition, 5

Tags: geometry , square

A $3\times 1$ paper rectangle is folded twice to give a square side $1$. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making $6$ holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?

2015 May Olympiad, 1

Tags: number theory , digit

The teacher secretly thought of a three-digit $S$ number. Students $A, B, C$ and $D$ tried to guess, saying, respectively, $541$, $837$, $291$ and $846$. The teacher told them, “Each of you got it right exactly one digit of $S$ and in the correct position ”. What is the number $S$?

2021 CMIMC, 2.3 1.1

Tags: combinatorics

Adam has a box with $15$ pool balls in it, numbered from $1$ to $15$, and picks out $5$ of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to $1$. How many selections of $5$ balls could he have drawn from the box? [i]Proposed by Adam Bertelli[/i]

2020 AIME Problems, 15

Tags: geometry

Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT=CT=16$, $BC=22$, and $TX^2+TY^2+XY^2=1143$. Find $XY^2$.

2019 Finnish National High School Mathematics Comp, 1

Tags: algebra , equation

Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$

2020-IMOC, N6

Tags: number theory , greatest common divisor

$\textbf{N6.}$ Let $a,b$ be positive integers. If $a,b$ satisfy that \begin{align*} \frac{a+1}{b} + \frac{b+1}{a} \end{align*} is also a positive integer, show that \begin{align*} \frac{a+b}{gcd(a,b)^2} \end{align*} is a Fibonacci number. [i]Proposed by usjl[/i]

2014 Czech and Slovak Olympiad III A, 6

Tags: inequalities

For arbitrary non-negative numbers $a$ and $b$ prove inequality $\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$, and find, where equality occurs. (Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)

2012 France Team Selection Test, 2

Tags: circumcircle , trigonometry , parallelogram , geometric transformation , reflection , ratio , geometry

Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.

2023 Turkey Junior National Olympiad, 1

Tags: combinatorics

Initially, there are $n$ red boxes numbered with the numbers $1,2,\dots ,n$ and $n$ white boxes numbered with the numbers $1,2,\dots ,n$ on the table. At every move, we choose $2$ different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is $6$ more than the number of balls from the white one or the number of balls from the white one is $16$ more than the number of balls from the red one. With that given information find all possible values of $n$

2020 Thailand TST, 1

Tags: binomial coefficient , combinatorics , trivial

The infinite sequence $a_0,a _1, a_2, \dots$ of (not necessarily distinct) integers has the following properties: $0\le a_i \le i$ for all integers $i\ge 0$, and \[\binom{k}{a_0} + \binom{k}{a_1} + \dots + \binom{k}{a_k} = 2^k\] for all integers $k\ge 0$. Prove that all integers $N\ge 0$ occur in the sequence (that is, for all $N\ge 0$, there exists $i\ge 0$ with $a_i=N$).

2000 Swedish Mathematical Competition, 4

Tags: geometry , lattice , area , geometric inequalities , 3d geometry , combinatorial geometry

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

2009 Canadian Mathematical Olympiad Qualification Repechage, 10

Tags: function , combinatorics

Ten boxes are arranged in a circle. Each box initially contains a positive number of golf balls. A move consists of taking all of the golf balls from one of the boxes and placing them into the boxes that follow it in a counterclockwise direction, putting one ball into each box. Prove that if the next move always starts with the box where the last ball of the previous move was placed, then after some number of moves, we get back to the initial distribution of golf balls in the boxes.

1951 AMC 12/AHSME, 1

Tags: percent

The percent that $ M$ is greater than $ N$ is: $ \textbf{(A)}\ \frac {100(M \minus{} N)}{M} \qquad\textbf{(B)}\ \frac {100(M \minus{} N)}{N} \qquad\textbf{(C)}\ \frac {M \minus{} N}{N} \qquad\textbf{(D)}\ \frac {M \minus{} N}{M}$ $ \textbf{(E)}\ \frac {100(M \plus{} N)}{N}$

Denmark (Mohr) - geometry, 2009.4

Tags: geometry , square , equal segments

Let $E$ be an arbitrary point different from $A$ and $B$ on the side $AB$ of a square $ABCD$, and let $F$ and $G$ be points on the segment $CE$ so that $BF$ and $DG$ are perpendicular to $CE$. Prove that $DF = AG$.

1969 IMO Longlists, 6

Tags: binomial coefficient , trigonometry , combinatorics

$(BEL 6)$ Evaluate $\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right)^{10}$ in two different ways and prove that $\dbinom{10}{1}-\dbinom{10}{3}+\frac{1}{2}\dbinom{10}{5}=2^4$

2014 AMC 8, 1

Tags:

Harry and Terry are each told to calculate $8-(2+5)$. Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$. If Harry's answer is $H$ and Terry's answer is $T$, what is $H-T$? $\textbf{(A) }-10\qquad\textbf{(B) }-6\qquad\textbf{(C) }0\qquad\textbf{(D) }6\qquad \textbf{(E) }10$