Olympiad problems

2011 Tournament of Towns, 5

In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

2024 Putnam, A2

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For which real polynomials $p$ is there a real polynomial $q$ such that \[ p(p(x))-x=(p(x)-x)^2q(x) \] for all real $x$?

1999 AIME Problems, 12

Tags: geometry , incenter , trigonometry , angle bisector

The inscribed circle of triangle $ABC$ is tangent to $\overline{AB}$ at $P,$ and its radius is 21. Given that $AP=23$ and $PB=27,$ find the perimeter of the triangle.

Novosibirsk Oral Geo Oly VIII, 2019.7

Tags: acute , square , geometry

The square was cut into acute -angled triangles. Prove that there are at least eight of them.

2020 Iran MO (3rd Round), 1

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find all functions from the reals to themselves. such that for every real $x,y$. $$f(y-f(x))=f(x)-2x+f(f(y))$$

2020-21 KVS IOQM India, 24

Tags: geometry , circles , area , tangent circle

Two circles $S_1$ and $S_2$, of radii $6$ units and $3$ units respectively, are tangent to each other, externally. Let $AC$ and $BD$ be their direct common tangents with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest Integer.

2020 LMT Spring, 8

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Let $a,b$ be real numbers satisfying $a^{2} + b^{2} = 3ab = 75$ and $a>b$. Compute $a^{3}-b^{3}$.

2023 pOMA, 5

Tags: combinatorial geometry , combinatorics

Let $n\ge 2$ be a positive integer, and let $P_1P_2\dots P_{2n}$ be a polygon with $2n$ sides such that no two sides are parallel. Denote $P_{2n+1}=P_1$. For some point $P$ and integer $i\in\{1,2,\ldots,2n\}$, we say that $i$ is a $P$-good index if $PP_{i}>PP_{i+1}$, and that $i$ is a $P$-bad index if $PP_{i}<PP_{i+1}$. Prove that there's a point $P$ such that the number of $P$-good indices is the same as the number of $P$-bad indices.

2001 Federal Competition For Advanced Students, Part 2, 3

Tags: angle bisector , geometry

Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.

2012 Iran Team Selection Test, 1

Tags: modular arithmetic , polynomial , number theory , lucas theorem

Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$ , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity. [i]Proposed by Mr.Etesami[/i]

2021 AIME Problems, 3

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Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$$ is divisible by $3$.

1962 All Russian Mathematical Olympiad, 016

Tags: algebra , polynomial

Prove that there are no integers $a,b,c,d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$, and equals $2$ at $x=62$.

2004 Cuba MO, 8

Tags: algebra , functional , functional equation

Determine all functions $f : R_+ \to R_+$ such that: a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$ b) $f(2) = 0$ c) $f(x) \ne 0$ for $0 \le x < 2$.

2012 Federal Competition For Advanced Students, Part 2, 2

Tags: number theory

Solve over $\mathbb{Z}$: \[ x^4y^3(y-x)=x^3y^4-216 \]

1987 Austrian-Polish Competition, 5

Tags: distance , partition , combinatorial geometry , combinatorics

The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.

1961 Polish MO Finals, 5

Tags: geometry , concurrency

Four lines intersecting at six points form four triangles. Prove that the circles circumscribed around out these triangles have a common point.

2023 Germany Team Selection Test, 2

Tags: number theory , sequence

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

2023 USEMO, 6

Tags: algebra

Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$) such that there exist complex numbers $r_1$, $\dots$, $r_n$, not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $m$, find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list] [i]Kaixin Wang[/i]

1969 IMO, 4

Tags: geometry , circumcircle , reflection , tangent

$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.

1992 IMO Longlists, 8

Tags: function , induction , algebra , functional equation

Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation \[f(f(x)) + af(x) = b(a + b)x.\] Prove that there exists a unique solution of this equation.

2020 Iran Team Selection Test, 1

Tags: polynomial , number theory

We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$. [i]Proposed by Masud Shafaie[/i]

2016 239 Open Mathematical Olympiad, 3

Tags: combinatorics

A regular hexagon with a side of $50$ was divided to equilateral triangles with unit side, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice defining a segment of length $2$. As a result of several such operations, exactly one node remains. How many ways is this possible?

2019 Regional Competition For Advanced Students, 3

Tags: combinatorics

Let $n\ge 2$ be a natural number. An $n \times n$ grid is drawn on a blackboard and each field with one of the numbers $-1$ or $+1$ labeled. Then the $n$ row and also the $n$ column sums calculated and the sum $S_n$ of all these $2n$ sums determined. (a) Show that for no odd number $n$ there is a label with $S_n = 0$. (b) Show that if $n$ is an even number, there are at least six different labels with $S_n = 0$.

1994 Baltic Way, 12

Tags: incenter , geometry

The inscribed circle of the triangle $A_1A_2A_3$ touches the sides $A_2A_3,A_3A_1,A_1A_2$ at points $S_1,S_2,S_3$, respectively. Let $O_1,O_2,O_3$ be the centres of the inscribed circles of triangles $A_1S_2S_3, A_2S_3S_1,A_3S_1S_2$, respectively. Prove that the straight lines $O_1S_1,O_2S_2,O_3S_3$ intersect at one point.

2011 Denmark MO - Mohr Contest, 4

Tags: algebra , functional

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .