Olympiad problems

2024 Putnam, A2

Tags:

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

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 $f(a) = a$ .

2009 HMNT, 5

Tags: combinatorics

The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height $a$ to a mountain of height $b$ takes $(b - a)^2$ time. Suppose that you start on the mountain of height $1$ and that you can move up, down, left, or right to get from one mountain to the next. What is the minimum amount of time you need to get to the mountain of height $49$? [img]https://cdn.artofproblemsolving.com/attachments/0/6/10b07a2b2ae4ba750cfffc3dc678880333c2de.png[/img]