Olympiad problems

2021 AIME Problems, 13

Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$. Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat{PQ}$ is $120^{\circ}$. Find the distance between the centers of $\omega_1$ and $\omega_2$.

2008 Iran Team Selection Test, 3

Tags: induction , combinatorics

Suppose that $ T$ is a tree with $ k$ edges. Prove that the $ k$-dimensional cube can be partitioned to graphs isomorphic to $ T$.

2014 Harvard-MIT Mathematics Tournament, 10

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[6] Find the number of sets $\mathcal{F}$ of subsets of the set $\{1,\ldots,2014\}$ such that: a) For any subsets $S_1,S_2 \in \mathcal{F}, S_1 \cap S_2 \in \mathcal{F}$. b) If $S \in \mathcal{F}$, $T \subseteq \{1,\ldots,2014\}$, and $S \subseteq T$, then $T \in \mathcal{F}$.

2019 Baltic Way, 14

Tags: geometry

Let $ABC$ be a triangle with $\angle ABC = 90^{\circ}$, and let $H$ be the foot of the altitude from $B$. The points $M$ and $N$ are the midpoints of the segments $AH$ and $CH$, respectively. Let $P$ and $Q$ be the second points of intersection of the circumcircle of the triangle $ABC$ with the lines $BM$ and $BN$, respectively. The segments $AQ$ and $CP$ intersect at the point $R$. Prove that the line $BR$ passes through the midpoint of the segment $MN$.

2018 ELMO Shortlist, 1

Tags: number theory , algebra , polynomial , induction

Determine all nonempty finite sets of positive integers $\{a_1, \dots, a_n\}$ such that $a_1 \cdots a_n$ divides $(x + a_1) \cdots (x + a_n)$ for every positive integer $x$. [i]Proposed by Ankan Bhattacharya[/i]

2003 Iran MO (3rd Round), 15

Tags: linear algebra , matrix , inequalities , vector , analytic geometry , modular arithmetic

Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$. ( for example the diffrence of this two row is only in one index 110 100) [i]Edited by Myth[/i]

2001 Moldova National Olympiad, Problem 6

Tags: inequality , inequalities , algebra , sequence

Set $a_n=\frac{2n}{n^4+3n^2+4},n\in\mathbb N$. Prove that $\frac14\le a_1+a_2+\ldots+a_n\le\frac12$ for all $n$.

Kvant 2023, M2754

Tags: algebra , polynomial

Given are reals $a, b$. Prove that at least one of the equations $x^4-2b^3x+a^4=0$ and $x^4-2a^3x+b^4=0$ has a real root. Proposed by N. Agakhanov

2008 Princeton University Math Competition, A9/B10

Tags: combinatorics

How many spanning trees does the following graph (with $6$ vertices and $9$ edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) [img]https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png[/img]

2022 Czech-Austrian-Polish-Slovak Match, 3

Tags: geometry

Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $l$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$/is tangent to a fixed circle, independent of the choice of $l$.

2002 AMC 10, 18

Tags: algebra , polynomial , quadratic

For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$

2002 Singapore Senior Math Olympiad, 2

Tags: geometry , distance , product , circumcircle , incircle

The vertices of a triangle inscribed in a circle are the points of tangency of a triangle circumscribed about the circle. Prove that the product of the perpendicular distances from any point on the circle to the sides of the inscribed triangle is the same as the product of the perpendicular distances from the same point to the sides of the circumscribed triangle.

2010 AMC 10, 6

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For positive numbers $ x$ and $ y$ the operation $ \spadesuit(x,y)$ is defined as \[ \spadesuit(x,y)\equal{}x\minus{}\frac1y\]What is $ \spadesuit(2,\spadesuit(2,2))$? $ \textbf{(A)}\ \frac23 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \frac43 \qquad \textbf{(D)}\ \frac53 \qquad \textbf{(E)}\ 2$

MOAA Gunga Bowls, 2021.20

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In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$. [i]Proposed by Andrew Wen[/i]

2017 Kazakhstan National Olympiad, 2

Tags: inequalities

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$

2020 Purple Comet Problems, 4

Tags: geometry

The gure below shows a large circle with area $120$ containing a circle with half of the radius of the large circle and six circles with a quarter of the radius of the large circle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/7/9/064a05feb9bd67896c079a5141bf7556d7165b.png[/img]

2006 Bulgaria Team Selection Test, 2

Tags: algebra , inequalities

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

2000 Czech And Slovak Olympiad IIIA, 5

Tags: 3d geometry , tetrahedron , geometry , volume

Monika made a paper model of a tetrahedron whose base is a right-angled triangle. When she cut the model along the legs of the base and the median of a lateral face corresponding to one of the legs, she obtained a square of side a. Compute the volume of the tetrahedron.

2003 Tournament Of Towns, 4

Tags: combinatorics

Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.

2015 Middle European Mathematical Olympiad, 3

Tags: permutation , combinatorics , optimization

There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the maximal sum of steps of all students that they can achieve.

2014 Danube Mathematical Competition, 3

Tags: coprime , arithmetic progression , number theory , arithmetic sequence

Given any integer $n \ge 2$, show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

2019 PUMaC Individual Finals A, B, B1

Tags: number theory

Find all pairs of nonnegative integers $(n, m)$ such that $2^n = 7^m + 9$.

2014 Contests, 3

Tags: quadratic residue , divisibility , hensel , surjective

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

1994 Portugal MO, 4

Tags: combinatorics

To date, in each Mathematics Olympiad Final, no participant has been able to solve all the problems, but every problem has been solved by at least one participant. Prove that in each Final, there was a participant $A$ who solved a problem $P_A$ and another participant $B$ who solved a problem $P_B$ such that $A$ did not solve $P_B$ and $B$ did not solve $P_A$.

1976 Spain Mathematical Olympiad, 6

Tags: matrices , linear algebra , matrix

Given a square matrix $M$ of order $n$ over the field of numbers real, find, as a function of $M$, two matrices, one symmetric and one antisymmetric, such that their sum is precisely $ M$.