Olympiad problems

2020 Costa Rica - Final Round, 6

$10$ persons sit around a circular table and on the table there are $22$ vases. Two persons can see each other if and only if there are no vases aligned with them. Prove that there are at least two people who can see each other.

2009 Estonia Team Selection Test, 3

Tags: polyhedron , regular polygon , convex , geometry , 3d geometry

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

2018 China Team Selection Test, 3

Tags: geometry

Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.

2017 Saudi Arabia JBMO TST, 1

Tags: inequalities , algebra

Let $a,b,c>0$ and $abc=1$ . Prove that $$ \sqrt{2(1+a^2)(1+b^2)(1+c^2)}\ge 1+a+b+c.$$

IV Soros Olympiad 1997 - 98 (Russia), 11.7

Tags: inequalities , logarithm , algebra

Solve the inequality $$\log_{\frac12} x\ge 16^x$$

2024 Malaysian IMO Team Selection Test, 3

Tags: number theory

Find all primes $p$ such that for any integer $k$, there exist two integers $x$ and $y$ such that $$x^3+2023xy+y^3 \equiv k \pmod p$$ [i]Proposed by Tristan Chaang Tze Shen[/i]

2012 Turkmenistan National Math Olympiad, 3

Tags: logarithm , inequalities

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

1984 Iran MO (2nd round), 7

Tags: geometry

Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)

2010 China Team Selection Test, 3

Tags: pigeonhole principle , ceiling function , number theory

Let $k>1$ be an integer, set $n=2^{k+1}$. Prove that for any positive integers $a_1<a_2<\cdots<a_n$, the number $\prod_{1\leq i<j\leq n}(a_i+a_j)$ has at least $k+1$ different prime divisors.

2019 ELMO Shortlist, A1

Tags: algebra , inequality , inequalities

Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$ [i]Proposed by Milan Haiman[/i]

2025 Belarusian National Olympiad, 11.3

Tags: geometry

An arbitrary triangle $ABC$ is given. Using ruler and compass construct three pairwise tangent circles $w_A$,$w_B$, $w_C$ with equal radii such that $A \in w_A, B \in w_B, C \in w_C$. [i]Matsvei Zorka[/i]

2015 IMO Shortlist, A3

Tags: inequalities , algebra , multivariate polynomial , maximization , n-variable inequality

Let $n$ be a fixed positive integer. Find the maximum possible value of \[ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, \] where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , combinatorial geometry , tiles , tiling

Two right isosceles triangles of legs equal to $1$ are glued together to form either an isosceles triangle - called [i]t-shape[/i] - of leg $\sqrt2$, or a parallelogram - called [i]p-shape[/i] - of sides $1$ and $\sqrt2$. Find all integers $m$ and $n, m, n \ge 2$, such that a rectangle $m \times n$ can be tilled with t-shapes and p-shapes.

2014 Contests, 2

Tags: geometry , 3d geometry

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

2019 JHMT, 7

Tags: geometry

Regular hexagon $ABCDEF$ has side length $\alpha$. Line $\ell$ intersects $A$ and bisects $\overline{CD}$ (and the point of intersection is $M$), line $m$ intersects $C$ and $E$, and line $n$ intersects $B$ and $E$. Lines $n$ and $\ell$ intersect at a point $G$, and lines $m$ and $\ell$ intersect at a point $H$. $[\vartriangle CHM] : [\vartriangle GHE] : [\vartriangle ABG] = a : b : c$ where $[\vartriangle ABC]$ is the area of $\vartriangle ABC$. Find $a + b + c$.

2011 Armenian Republican Olympiads, Problem 3

Tags: number theory

Find all integers $a, m, n, k,$ such that $(a^m+1)(a^n-1)=15^k.$

2001 Cuba MO, 1

Tags: combinatorics

In each square of a $3 \times 3$ board a real number is written. The element of the $i$ -th row and the $j$ -th column is equal to abso;uteof the difference of the sum of the elements of column $j$ and the sum of the elements of row $i$. Prove that every element of the board is equal to the sum or difference of two other elements on the board.

1997 AMC 12/AHSME, 22

Tags:

Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $ \$56$. The absolute difference between the amounts Ashley and Betty had to spend was $ \$19$. The absolute difference between the amounts Betty and Carlos had was $ \$7$, between Carlos and Dick was $ \$5$, between Dick and Elgin was $ \$4$, and between Elgin and Ashley was $ \$11$. How much did Elgin have? $ \textbf{(A)}\ \$6\qquad \textbf{(B)}\ \$7\qquad \textbf{(C)}\ \$8\qquad \textbf{(D)}\ \$9\qquad \textbf{(E)}\ \$10$

1999 Yugoslav Team Selection Test, Problem 1

Tags: polynomial , algebra

For a natural number $n$, let $P(x)$ be the polynomial of $2n$−th degree such that: $P(0) = 1$ and $P(k) = 2^{k-1}$ for $k = 1, 2, . . . , 2n$. Prove that $2P(2n + 1) - P(2n + 2) = 1$. P.S. I tried to prove it by firstly expressing this polynomial using Lagrange interpolation but get bored of computations - it seems like it can be done this way, but I'd like to see more 'clever' solution. :)

2010 Kurschak Competition, 3

Tags: number theory

For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?

2014 EGMO, 3

Tags: quadratic , number theory , combinatorial number theory , divisor

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

1982 Vietnam National Olympiad, 3

Tags: 3d geometry , geometry

Let $ABCDA'B'C'D'$ be a cube (where $ABCD$ and $A'B'C'D'$ are faces and $AA',BB',CC',DD'$ are edges). Consider the four lines $AA', BC, D'C'$ and the line joining the midpoints of $BB'$ and $DD'$. Show that there is no line which cuts all the four lines.

2017 ELMO Shortlist, 2

Tags: number theory

An integer $n>2$ is called [i]tasty[/i] if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers? [i]Proposed by Vincent Huang[/i]

2017 Harvard-MIT Mathematics Tournament, 16

Tags: algebra , complex number

Let $a$ and $b$ be complex numbers satisfying the two equations \begin{align*} a^3 - 3ab^2 & = 36 \\ b^3 - 3ba^2 & = 28i. \end{align*} Let $M$ be the maximum possible magnitude of $a$. Find all $a$ such that $|a| = M$.

Kyiv City MO Juniors Round2 2010+ geometry, 2021.8.2

Tags: geometry , incenter , right triangle

In a triangle $ABC$, $\angle B=90^o$ and $\angle A=60^o$, $I$ is the point of intersection of its angle bisectors. A line passing through the point $I$ parallel to the line $AC$, intersects the sides $AB$ and $BC$ at the points $P$ and $T$ respectively. Prove that $3PI+IT=AC$ . (Anton Trygub)