Olympiad problems

2011 Morocco National Olympiad, 3

Tags: function , algebra

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

1981 Austrian-Polish Competition, 8

Tags: combinatorial geometry , combinatorics , plane , line , parallel

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

1971 Bulgaria National Olympiad, Problem 3

Tags: geometry , combinatorial geometry , combinatorics

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

2022 Baltic Way, 6

Tags: combinatorics

Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?

2006 Hanoi Open Mathematics Competitions, 3

Tags: number theory , system of equations , diophantine equation

Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

1999 Tournament Of Towns, 2

Tags: composite , number theory , odd

Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite. (V Senderov)

2002 Federal Math Competition of S&M, Problem 1

Tags: inequalities

For any positive numbers $a,b,c$ and natural numbers $n,k$ prove the inequality $$\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.$$

1946 Moscow Mathematical Olympiad, 106

Tags: acute , maximum , geometry , angle

What is the largest number of acute angles that a convex polygon can have?

2023 Baltic Way, 5

Tags: inequalities

Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$.

2006 Switzerland Team Selection Test, 1

Tags: number theory

Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$. Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.

2010 Stanford Mathematics Tournament, 7

Tags:

Find all the integers $x$ in $[20, 50]$ such that $6x + 5 \equiv -19 \mod 10,$ that is, $10$ divides $(6x + 15) + 19.$

1974 IMO Longlists, 24

Tags: least common multiple , greatest common divisor , number theory

Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

2014 Vietnam Team Selection Test, 5

Tags: polynomial , algebra

Find all polynomials $P(x),Q(x)$ which have integer coefficients and satify the following condtion: For the sequence $(x_n )$ defined by \[x_0=2014,x_{2n+1}=P(x_{2n}),x_{2n}=Q(x_{2n-1}) \quad n\geq 1\] for every positive integer $m$ is a divisor of some non-zero element of $(x_n )$

2017 BMT Spring, 15

Tags: geometry , equilateral

In triangle $ABC$, the angle at $C$ is $30^o$, side $BC$ has length $4$, and side $AC$ has length $5$. Let $ P$ be the point such that triangle $ABP$ is equilateral and non-overlapping with triangle $ABC$. Find the distance from $C$ to $ P$.

2022 Cyprus JBMO TST, 4

Tags: number theory , pigeonhole principle

Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that \[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.

1986 IMO Longlists, 21

Tags: geometry

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

2014 Contests, 3

Tags: circumcircle , parallelogram , geometry

Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds. Prove that \[ED=AC-AB \iff R=2r+r_a.\]

2003 Balkan MO, 2

Tags: geometry , circumcircle , ratio , trapezoid , trigonometry , calculus , geometric transformation

Let $ABC$ be a triangle, and let the tangent to the circumcircle of the triangle $ABC$ at $A$ meet the line $BC$ at $D$. The perpendicular to $BC$ at $B$ meets the perpendicular bisector of $AB$ at $E$. The perpendicular to $BC$ at $C$ meets the perpendicular bisector of $AC$ at $F$. Prove that the points $D$, $E$ and $F$ are collinear. [i]Valentin Vornicu[/i]

2009 Portugal MO, 3

Tags: combinatorics

Duarte wants to draw a square whose side's length is $2009$ cm and which is divided in $2009\times2009$ squares whose side's length is $1$ cm and whose sides are parallel to the original square's one, without taking the pencil out of the paper. Starting on one of the vertex of the giant square, what is the length of the shortest line that allows him to make this drawing?

2011 Saudi Arabia Pre-TST, 3.2

Tags: diophantine equation , diophantine , number theory

Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.

2004 China Girls Math Olympiad, 8

Tags: pigeonhole principle , rectangle , combinatorics , geometry

When the unit squares at the four corners are removed from a three by three squares, the resulting shape is called a cross. What is the maximum number of non-overlapping crosses placed within the boundary of a $ 10\times 11$ chessboard? (Each cross covers exactly five unit squares on the board.)

Novosibirsk Oral Geo Oly IX, 2023.4

Tags: geometry , trapezoid , isosceles trapezoid , isosceles

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.

2023 CCA Math Bonanza, L4.4

Tags:

Let $ABC$ be a triangle with side lengths $AB=6, BC=7, CA=8$ and circumcircle $\omega.$ Denote $M$ to be the midpoint of $BC.$ Let $P$ be the intersection of the tangent to $\omega$ at $A$ and $BC.$ The line parallel to $BC$ passing through $A$ intersects $\omega$ at another point $D.$ The tangent to $\omega$ passing through $P$ that is not $PA$ intersects $DM$ at a point $Q.$ Denote $J$ to be the intersection of $(BMQ)$ and $AQ.$ Extend $BJ$ to intersect $AC$ at $E.$ Compute $\tfrac{BJ}{JE}.$ [i]Lightning 4.4[/i]

1996 Estonia Team Selection Test, 3

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$: $(i)$ $f(x)=-f(-x);$ $(ii)$ $f(x+1)=f(x)+1;$ $(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$

2007 Purple Comet Problems, 25

Tags:

Let $x$ be a positive integer less than $200$, and let $y$ be obtained by writing the base 10 digits of $x$ in reverse order. Given that $x$ and $y$ satisfy $11x^2+363y=7xy+6571$, find $x$.