Olympiad problems

2016 Saudi Arabia GMO TST, 3

Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

2005 France Team Selection Test, 3

Tags: combinatorics

In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken more that by the half of the participants. What is the least value of $n$?

2020 Princeton University Math Competition, 12

Tags: algebra

Given a sequence $a_0, a_1, a_2, ... , a_n$, let its [i]arithmetic approximant[/i] be the arithmetic sequence $b_0, b_1, ... , b_n$ that minimizes the quantity $\sum_{i=0}^{n}(b_i -a_i)^2$, and denote this quantity the sequence’s anti-arithmeticity. Denote the number of integer sequences whose arithmetic approximant is the sequence $4$, $8$, $12$, $16$ and whose anti-arithmeticity is at most $20$.

Kyiv City MO 1984-93 - geometry, 1984.7.3

Tags: geometry , angle

On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.

2006 Kyiv Mathematical Festival, 4

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2006 Sharygin Geometry Olympiad, 26

Tags: geometry , cone , 3d geometry , concyclic , tangent circle

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

2023 May Olympiad, 2

Tags: number theory

We say that a four-digit number $\overline{abcd}$ is [i]slippery [/i] if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?

1970 IMO Longlists, 4

Tags: algebra , system of equations

Solve the system of equations for variables $x,y$, where $\{a,b\}\in\mathbb{R}$ are constants and $a\neq 0$. \[x^2 + xy = a^2 + ab\] \[y^2 + xy = a^2 - ab\]

2008 Baltic Way, 14

Tags: geometry , 3d geometry , combinatorics

Is it possible to build a $ 4\times 4\times4$ cube from blocks of the following shape consisting of $ 4$ unit cubes?

2022 Purple Comet Problems, 22

Tags: geometry

Circle $\omega_1$ has radius $7$ and center $C_1$. Circle $\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\omega_1$ and $\omega_2$ pass through $A_1$ on $\omega_1$ and $A_2$ on $\omega_2$, and let a common external tangent of $\omega_1$ and $\omega_2$ pass through $B_1$ on $\omega_1$ and $B_2$ on $\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$.