Olympiad problems

2010 Tournament Of Towns, 6

In acute triangle $ABC$, an arbitrary point $P$ is chosen on altitude $AH$. Points $E$ and $F$ are the midpoints of sides $CA$ and $AB$ respectively. The perpendiculars from $E$ to $CP$ and from $F$ to $BP$ meet at point $K$. Prove that $KB = KC$.

2021 Moldova EGMO TST, 7

Tags: circumcircle , geometry

A triangle $ABC$ has the orthocenter $H$ different from the vertexes and the circumcenter $O$. Let $M, N$ and $P$ be the circumcenters of triangles $HBC, HCA$ and $HAB$. Prove that the lines $AM, BN, CP$ and $OH$ are concurrent.

1996 Israel National Olympiad, 3

Tags: trigonometry , median , angle bisector , geometry , altitude , angle , concurrency

The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

2025 Romania National Olympiad, 1

Tags: geometry , reflection , circumcircle , centroid , complex number geometry

Let $M$ be a point in the plane, distinct from the vertices of $\triangle ABC$. Consider $N,P,Q$ the reflections of $M$ with respect to lines $AB, BC$ and $CA$, in this order. a) Prove that $N, P ,Q$ are collinear if and only if $M$ lies on the circumcircle of $\triangle ABC$. b) If $M$ does not lie on the circumcircle of $\triangle ABC$ and the centroids of triangles $\triangle ABC$ and $\triangle NPQ$ coincide, prove that $\triangle ABC$ is equilateral.

2017 India IMO Training Camp, 2

Tags: number theory

Let $a,b,c,d$ be pairwise distinct positive integers such that $$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+d}+\frac{d}{d+a}$$ is an integer. Prove that $a+b+c+d$ is [b]not[/b] a prime number.

2014 Switzerland - Final Round, 4

Tags: combinatorics , tiling , tiles

The checkered plane (infinitely large house paper) is given. For which pairs (a,, b) one can color each of the squares with one of $a \cdot b$ colors, so that each rectangle of size $ a \times b$ or $b \times a$, placed appropriately in the checkered plane, always contains a unit square with each color ?

2021 Israel TST, 3

Tags: inequalities

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*} holds for any $8$ real numbers $a,b,c,d,x,y,z,t$? Edit: Fixed a mistake! Thanks @below.

1971 AMC 12/AHSME, 35

Tags: ratio , geometry

Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. The ratio of the area of the first circle to the sum of areas of all other circles in the sequence, is $\textbf{(A) }(4+3\sqrt{2}):4\qquad\textbf{(B) }9\sqrt{2}:2\qquad\textbf{(C) }(16+12\sqrt{2}):1\qquad$ $\textbf{(D) }(2+2\sqrt{2}):1\qquad \textbf{(E) }3+2\sqrt{2}):1$

2010 ISI B.Stat Entrance Exam, 6

Tags: algebra , polynomial

Consider the equation $n^2+(n+1)^4=5(n+2)^3$ (a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation. (b) Does the equation have a solution in positive integers?

1954 AMC 12/AHSME, 19

Tags:

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle: $ \textbf{(A)}\ \text{are always equal to }60^\circ \\ \textbf{(B)}\ \text{are always one obtuse angle and two unequal acute angles} \\ \textbf{(C)}\ \text{are always one obtuse angle and two equal acute angles} \\ \textbf{(D)}\ \text{are always acute angles} \\ \textbf{(E)}\ \text{are always unequal to each other}$

2012 Lusophon Mathematical Olympiad, 1

Tags: combinatorics , combinatorial geometry , collinear , geometry , incenter

Arnaldo and Bernaldo train for a marathon along a circular track, which has in its center a mast with a flag raised. Arnaldo runs faster than Bernaldo, so that every $30$ minutes of running, while Arnaldo gives $15$ laps on the track, Bernaldo can only give $10$ complete laps. Arnaldo and Bernaldo left at the same moment of the line and ran with constant velocities, both in the same direction. Between minute $1$ and minute $61$ of the race, how many times did Arnaldo, Bernaldo and the mast become collinear?

2024-25 IOQM India, 18

Tags:

Let $p,q$ be two-digit number neither of which are divisible by $10$. Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p,q$ very, a computer prints $r$ on the screen if $\gcd(p,q) = 1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).

1969 IMO Shortlist, 20

Tags: polygon , lattice points , area , geometry

$(FRA 3)$ A polygon (not necessarily convex) with vertices in the lattice points of a rectangular grid is given. The area of the polygon is $S.$ If $I$ is the number of lattice points that are strictly in the interior of the polygon and B the number of lattice points on the border of the polygon, find the number $T = 2S- B -2I + 2.$

2021 MOAA, 8

Tags: accuracy

Will has a magic coin that can remember previous flips. If the coin has already turned up heads $m$ times and tails $n$ times, the probability that the next flip turns up heads is exactly $\frac{m+1}{m+n+2}$. Suppose that the coin starts at $0$ flips. The probability that after $10$ coin flips, heads and tails have both turned up exactly $5$ times can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2010 Baltic Way, 20

Tags: induction , number theory

Determine all positive integers $n$ for which there exists an infinite subset $A$ of the set $\mathbb{N}$ of positive integers such that for all pairwise distinct $a_1,\ldots , a_n \in A$ the numbers $a_1+\ldots +a_n$ and $a_1a_2\ldots a_n$ are coprime.

2003 AMC 10, 16

Tags: modular arithmetic , algorithm , number theory , greatest common divisor , euclidean algorithm

What is the units digit of $ 13^{2003}$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

2010 Poland - Second Round, 2

Tags: 3d geometry , tetrahedron , circumcircle , incenter , geometry

The orthogonal projections of the vertices $A, B, C$ of the tetrahedron $ABCD$ on the opposite faces are denoted by $A', B', C'$ respectively. Suppose that point $A'$ is the circumcenter of the triangle $BCD$, point $B'$ is the incenter of the triangle $ACD$ and $C'$ is the centroid of the triangle $ABD$. Prove that tetrahedron $ABCD$ is regular.

2013 NIMO Problems, 4

Tags: trigonometry

Find the positive integer $N$ for which there exist reals $\alpha, \beta, \gamma, \theta$ which obey \begin{align*} 0.1 &= \sin \gamma \cos \theta \sin \alpha, \\ 0.2 &= \sin \gamma \sin \theta \cos \alpha, \\ 0.3 &= \cos \gamma \cos \theta \sin \beta, \\ 0.4 &= \cos \gamma \sin \theta \cos \beta, \\ 0.5 &\ge \left\lvert N-100 \cos2\theta \right\rvert. \end{align*}[i]Proposed by Evan Chen[/i]

2004 Indonesia MO, 3

Tags:

In how many ways can we change the sign $ *$ with $ \plus{}$ or $ \minus{}$, such that the following equation is true? \[ 1 *2*3*4*5*6*7*8*9*10\equal{}29\]

1993 Austrian-Polish Competition, 8

Tags: polynomial , functional , functional equation , algebra

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

2020 Purple Comet Problems, 15

Tags: geometry

Daniel had a string that formed the perimeter of a square with area $98$. Daniel cut the string into two pieces. With one piece he formed the perimeter of a rectangle whose width and length are in the ratio $2 : 3$. With the other piece he formed the perimeter of a rectangle whose width and length are in the ratio $3 : 8$. The two rectangles that Daniel formed have the same area, and each of those areas is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

1964 AMC 12/AHSME, 10

Tags: geometry

Given a square side of length $s$. On a diagonal as base a triangle with three unequal sides is constructed so that its area equals that of the square. The length of the altitude drawn to the base is: ${{ \textbf{(A)}\ s\sqrt{2} \qquad\textbf{(B)}\ s/\sqrt{2} \qquad\textbf{(C)}\ 2s \qquad\textbf{(D)}\ 2\sqrt{s} }\qquad\textbf{(E)}\ 2/ \sqrt{s} } $

1989 AMC 8, 17

Tags:

The number $\text{N}$ is between $9$ and $17$. The average of $6$, $10$, and $\text{N}$ could be $\text{(A)}\ 8 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

2022 JHMT HS, 8

Tags: algebra , complex number

Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying \[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]

2024 TASIMO, 6

Tags: abstract algebra , number theory , complete residue system

We call a positive integer $n\ge 4$[i] beautiful[/i] if there exists some permutation $$\{x_1,x_2,\dots ,x_{n-1}\}$$ of $\{1,2,\dots ,n-1\}$ such that $\{x^1_1,\ x^2_2,\ \dots,x^{n-1}_{n-1}\}$ gives all the residues $\{1,2,\dots, n-1\}$ modulo $n$. Prove that if $n$ is beautiful then $n=2p,$ for some prime number $p.$