Olympiad problems

2022 Pan-African, 4

Find all functions $f$ and $g$ defined from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that for all $x, y > 0$ the two equations hold $$ (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 $$ $$ (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1) $$ [i]Note: $\mathbb{R}_{>0}$ denotes the set of positive real numbers.[/i]

1992 AMC 8, 7

Tags:

The digit-sum of $998$ is $9+9+8=26$. How many 3-digit whole numbers, whose digit-sum is $26$, are even? $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

MathLinks Contest 4th, 7.2

Tags: geometry

Let $\Omega$ be the incircle of a triangle $ABC$. Suppose that there exists a circle passing through $B$ and $C$ and tangent to $\Omega$ in $A'$. Suppose the similar points $B'$, $C'$ exist. Prove that the lines $AA', BB'$ and $CC'$ are concurrent.

Indonesia Regional MO OSP SMA - geometry, 2018.3

Tags: geometry , parallel , equal circles

Let $ \Gamma_1$ and $\Gamma_2$ be two different circles with the radius of same length and centers at points $O_1$ and $O_2$, respectively. Circles $\Gamma_1$ and $\Gamma_2$ are tangent at point $P$. The line $\ell$ passing through $O_1$ is tangent to $\Gamma_2$ at point $A$. The line $\ell$ intersects $\Gamma_1$ at point $X$ with $X$ between $A$ and $O_1$. Let $M$ be the midpoint of $AX$ and $Y$ the intersection of $PM$ and $\Gamma_2$ with $Y\ne P$. Prove that $XY$ is parallel to $O_1O_2$.

2007 Cuba MO, 7

Tags: combinatorics , combinatorial geometry , geometry

Prove that given $n$ points in the plane, not all aligned, there exists a line that passes through exactly two of them. [hide=original wording]Demostrar que dados n puntos en el plano, no todos alineados, existe una recta que pasa por exactamente dos de ellos.[/hide]

2007 Croatia Team Selection Test, 8

Tags: number theory

Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.

2013 Saudi Arabia Pre-TST, 1.3

Tags: combinatorics

Ten students take a test consisting of $4$ different papers in Algebra, Geometry, Number Theory and Combinatorics. First, the proctor distributes randomly the Algebra paper to each student. Then the remaining papers are distributed one at a time in the following order: Geometry, Number Theory, Combinatorics in such a way that no student receives a paper before he finishes the previous one. In how many ways can the proctor distribute the test papers given that a student may for example nish the Number Theory paper before another student receives the Geometry paper, and that he receives the Combinatorics paper after that the same other student receives the Combinatorics papers.

1998 Spain Mathematical Olympiad, 2

Tags: geometry , 3d geometry , modular arithmetic , number theory

Find all four-digit numbers which are equal to the cube of the sum of their digits.

1982 IMO Longlists, 6

Tags: geometry

On the three distinct lines $a, b$, and $c$ three points $A, B$, and $C$ are given, respectively. Construct three collinear points $X, Y,Z$ on lines $a, b, c$, respectively, such that $\frac{BY}{AX} = 2$ and $ \frac{CZ}{AX} = 3$.

2012 Kazakhstan National Olympiad, 1

Tags: algebra , difference of squares , special factorizations , number theory

Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.

Estonia Open Junior - geometry, 2010.1.2

Tags: geometry , equal angles , equal segments , angle

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

2022 Azerbaijan JBMO TST, C5?

Tags: shortlist , combinatorics , coloring , board

Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves? Proposed by [i]Nikola Velov, Macedonia[/i]

1967 All Soviet Union Mathematical Olympiad, 085

Tags: number theory , digit

a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal $999...9$ ($1967$ of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals $1010$, than the given number was divisible by $10$.

1987 China Team Selection Test, 3

Tags: induction , algebra

Let $r_1=2$ and $r_n = \prod^{n-1}_{k=1} r_i + 1$, $n \geq 2.$ Prove that among all sets of positive integers such that $\sum^{n}_{k=1} \frac{1}{a_i} < 1,$ the partial sequences $r_1,r_2, ... , r_n$ are the one that gets nearer to 1.

1969 IMO Shortlist, 42

Tags: algebra , combinatorics , subset , intersection

$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.

1991 All Soviet Union Mathematical Olympiad, 546

Tags: geometry , combinatorics , combinatorial geometry , minimum

The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons? [missing figure]

2025 Macedonian Mathematical Olympiad, Problem 4

Tags: number theory

Let $P(x)=a x^{75}+b$ be a polynomial where $a$ and $b$ are coprime integers in the set $\{1,2,\dots,151\}$, and suppose it satisfies the following condition: there exists at most one prime $p$ such that for every positive integer $k$, $p\mid P(k)$. Prove that for every prime $q \neq p$ there exists a positive integer $k$ for which $q^2 \mid P(k).$

2005 Singapore Senior Math Olympiad, 3

Tags: sum , subset , combinatorics

Let $S$ be a subset of $\{1,2,3,...,24\}$ with $n(S)=10$. Show that $S$ has two $2$-element subsets $\{x,y\}$ and $\{u,v\}$ such that $x+y=u+v$

2006 Estonia Math Open Senior Contests, 10

Tags: inequalities , algebra

Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.

2017 Israel Oral Olympiad, 5

Tags: combinatorics , chase , ratio

A mink is standing in the center of a field shaped like a regular polygon. The field is surrounded by a fence, and the mink can only exit through the vertices of the polygon. A dog is standing on one of the vertices, and can move along the fence. The mink wants to escape the field, while the dog tries to prevent it. Each of them moves with constant velocity. For what ratio of velocities could the mink escape if: a. The field is a regular triangle? b. The field is a square?

1988 IMO Shortlist, 30

Tags: geometry , inradius , trigonometry , area of a triangle

A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that \[ BM^{2} \equal{} X \cot \left( \frac {B}{2}\right) \] where X is the area of triangle $ ABC.$

1967 IMO Shortlist, 4

Tags: geometry , perimeter , square , dissection , triangulation

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

2001 Tournament Of Towns, 7

Tags: combinatorics

Several boxes are arranged in a circle. Each box may be empty or may contain one or several chips. A move consists of taking all the chips from some box and distributing them one by one into subsequent boxes clockwise starting from the next box in the clockwise direction. (a) Suppose that on each move (except for the first one) one must take the chips from the box where the last chip was placed on the previous move. Prove that after several moves the initial distribution of the chips among the boxes will reappear. (b) Now, suppose that in each move one can take the chips from any box. Is it true that for every initial distribution of the chips you can get any possible distribution?

2001 IberoAmerican, 3

Tags: geometry , pigeonhole principle , algebra

Show that it is impossible to cover a unit square with five equal squares with side $s<\frac{1}{2}$.

2000 Slovenia National Olympiad, Problem 1

Tags: digit

In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.