Olympiad problems

2017 Greece Junior Math Olympiad, 3

Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

2015 Brazil Team Selection Test, 4

Tags: geometry , angle bisector

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

2002 National Olympiad First Round, 5

Tags: inequalities , triangle inequality , geometry

The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude? $ \textbf{a)}\ 4 \qquad\textbf{b)}\ 7 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ 23 $

1997 Chile National Olympiad, 7

Tags: combinatorics

In a career in mathematics, $7$ courses are taught, among which students can choose the ones you want. Determine the number of students in the career, knowing that: $\bullet$ No two students have chosen the same courses. $\bullet$ Any two students have at least one course in common. $\bullet$ If the race had one more student, it would not be possible to do both.

Fractal Edition 2, P3

Tags:

Several football teams participated in a tournament where each team played exactly one game against every other team. Is it possible that there were exactly $2024$ games in total?

2013 Stanford Mathematics Tournament, 7

Tags: 3d geometry , pythagorean theorem , geometry

A fly and an ant are on one corner of a unit cube. They wish to head to the opposite corner of the cube. The fly can fly through the interior of the cube, while the ant has to walk across the faces of the cube. How much shorter is the fly's path if both insects take the shortest path possible?

2008 Putnam, A1

Tags: abstract algebra , algebra , function , limit , calculus , integration

Let $ f: \mathbb{R}^2\to\mathbb{R}$ be a function such that $ f(x,y)\plus{}f(y,z)\plus{}f(z,x)\equal{}0$ for real numbers $ x,y,$ and $ z.$ Prove that there exists a function $ g: \mathbb{R}\to\mathbb{R}$ such that $ f(x,y)\equal{}g(x)\minus{}g(y)$ for all real numbers $ x$ and $ y.$

2009 IberoAmerican Olympiad For University Students, 2

Tags: invariant , linear algebra , analytic geometry , vector , induction , matrix

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$, $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$. Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.

2022 CMIMC Integration Bee, 12

Tags: integration , calculus

\[\int_{\pi/4}^{\pi/2} \tan^{-1}\left(\tan^2(x)\right)\sin(2x)\,\mathrm dx\] [i]Proposed by Vlad Oleksenko[/i]

1999 All-Russian Olympiad, 5

Tags: number theory , modular arithmetic

Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers. Prove that at least three of these numbers are equal.

2003 AIME Problems, 1

Tags: function

The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$

2009 Estonia Team Selection Test, 3

Tags: regular polygon , polyhedron , 3d geometry , convex , geometry

Find all natural numbers $n$ for which there exists a convex polyhedron satisfying the following conditions: (i) Each face is a regular polygon. (ii) Among the faces, there are polygons with at most two different numbers of edges. (iii) There are two faces with common edge that are both $n$-gons.

2014 AMC 12/AHSME, 14

Tags: geometric series , arithmetic sequence , ratio

Let $a<b<c$ be three integers such that $a,b,c$ is an arithmetic progression and $a,c,b$ is a geometric progression. What is the smallest possible value of $c$? $\textbf{(A) }-2\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }4\qquad \textbf{(E) }6\qquad$

2015 Macedonia National Olympiad, Problem 4

Tags: geometry

Let $k_1$ and $k_2$ be two circles and let them cut each other at points $A$ and $B$. A line through $B$ is cutting $k_1$ and $k_2$ in $C$ and $D$ respectively, such that $C$ doesn't lie inside of $k_2$ and $D$ doesn't lie inside of $k_1$. Let $M$ be the intersection point of the tangent lines to $k_1$ and $k_2$ that are passing through $C$ and $D$, respectively. Let $P$ be the intersection of the lines $AM$ and $CD$. The tangent line to $k_1$ passing through $B$ intersects $AD$ in point $L$. The tangent line to $k_2$ passing through $B$ intersects $AC$ in point $K$. Let $KP \cap MD \equiv N$ and $LP \cap MC \equiv Q$. Prove that $MNPQ$ is a parallelogram.

2013 Harvard-MIT Mathematics Tournament, 29

Tags: hmmt , inequalities

Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.

2019 Sharygin Geometry Olympiad, 5

Tags: geometry

Let $A, B, C$ and $D$ be four points in general position, and $\omega$ be a circle passing through $B$ and $C$. A point $P$ moves along $\omega$. Let $Q$ be the common point of circles $\odot (ABP)$ and $\odot (PCD)$ distinct from $P$. Find the locus of points $Q$.

2014 CHKMO, 3

Tags: diophantine equation , number theory

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

2006 Estonia Math Open Senior Contests, 6

Tags: rotation , geometry , pigeonhole principle , geometric transformation , combinatorics

Kati cut two equal regular $ n\minus{}gons$ out of paper. To the vertices of both $ n\minus{}gons$, she wrote the numbers 1 to $ n$ in some order. Then she stabbed a needle through the centres of these $ n\minus{}gons$ so that they could be rotated with respect to each other. Kati noticed that there is a position where the numbers at each pair of aligned vertices are different. Prove that the $ n\minus{}gons$ can be rotated to a position where at least two pairs of aligned vertices contain equal numbers.

2013 ELMO Shortlist, 3

Tags: induction , inequalities , number theory

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2018 ASDAN Math Tournament, 10

Tags: algebra test

Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that $$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$

2005 Flanders Math Olympiad, 4

Tags: number theory

If $n$ is an integer, then find all values of $n$ for which $\sqrt{n}+\sqrt{n+2005}$ is an integer as well.

1999 Mongolian Mathematical Olympiad, Problem 6

Tags: number theory

Show that there exists a positive integer $n$ such that the decimal representations of $3^n$ and $7^n$ both start with the digits $10$.

1984 AMC 12/AHSME, 2

Tags:

If $x,y$ and $y - \frac{1}{x}$ are not 0, then \[\frac{x - \frac{1}{y}}{y - \frac{1}{x}}\] equals $\textbf{(A) }1\qquad\textbf{(B) } \frac{x}{y}\qquad\textbf{(C) }\frac{y}{x}\qquad\textbf{(D) }\frac{x}{y} - \frac{y}{x}\qquad\textbf{(E) } xy - \frac{1}{xy}$

2012 Princeton University Math Competition, B3

Tags: algebra

Evaluate $\sqrt[3]{26 + 15\sqrt3} + \sqrt[3]{26 - 15\sqrt3}$

2023 BMT, 5

Tags: geometry

Triangle $\vartriangle ABC$ has side lengths $AB = 8$, $BC = 15$, and $CA = 17$. Circles $\omega_1$ and $\omega_2$ are externally tangent to each other and within $\vartriangle ABC$. The radius of circle $\omega_2$ is four times the radius of circle $\omega_1$. Circle $\omega_1$ is tangent to $\overline{AB}$ and $\overline{BC}$, and circle $\omega_2$ is tangent to $\overline{BC}$ and $\overline{CA}$. Compute the radius of circle $\omega_1$.