Olympiad problems

2012 Balkan MO, 1

Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$. Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.

2017 Purple Comet Problems, 18

Tags: analytic geometry

In the $3$-dimensional coordinate space nd the distance from the point $(36, 36, 36)$ to the plane that passes through the points $(336, 36, 36)$, $(36, 636, 36)$, and $(36, 36, 336)$.

2007 China Team Selection Test, 3

Tags: least common multiple , relatively prime , number theory

Let $ n$ be a positive integer, let $ A$ be a subset of $ \{1, 2, \cdots, n\}$, satisfying for any two numbers $ x, y\in A$, the least common multiple of $ x$, $ y$ not more than $ n$. Show that $ |A|\leq 1.9\sqrt {n} \plus{} 5$.

2011 Philippine MO, 4

Tags: function , algebra

Find all (if there is one) functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$, \[f(f(x))+xf(x)=1.\]

Indonesia MO Shortlist - geometry, g1

Tags: collinear , geometry , angle bisector

Given triangle $ABC$, $AL$ bisects angle $\angle BAC$ with $L$ on side $BC$. Lines $LR$ and $LS$ are parallel to $BA$ and $CA$ respectively, $R$ on side $AC$ and$ S$ on side $AB$, respectively. Through point $B$ draw a perpendicular on $AL$, intersecting $LR$ at $M$. If point $D$ is the midpoint of $BC$, prove that that the three points $A, M, D$ lie on a straight line.

2021 Balkan MO Shortlist, G1

Tags: shortlist , geometry , concurrency

Let $ABC$ be a triangle with $AB < AC < BC$. On the side $BC$ we consider points $D$ and $E$ such that $BA = BD$ and $CE = CA$. Let $K$ be the circumcenter of triangle $ADE$ and let $F$, $G$ be the points of intersection of the lines $AD$, $KC$ and $AE$, $KB$ respectively. Let $\omega_1$ be the circumcircle of triangle $KDE$, $\omega_2$ the circle with center $F$ and radius $FE$, and $\omega_3$ the circle with center $G$ and radius $GD$. Prove that $\omega_1$, $\omega_2$, and $\omega_3$ pass through the same point and that this point of intersection lies on the line $AK$.

2014 Contests, 1

Tags: algebra , equation

Show that there are no positive real numbers $x, y, z$ such $(12x^2+yz)(12y^2+xz)(12z^2+xy)= 2014x^2y^2z^2$ .

2021 JBMO TST - Turkey, 2

Tags: number theory

For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$ is an integer?

2004 National Olympiad First Round, 32

Tags:

If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ 30 $

2007 Indonesia TST, 3

Tags: function , algebra , polynomial , induction , number theory

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

2014 IPhOO, 8

Tags:

A plane, flying at a height of $3000$ meters above the level ground below, receives a signal from the airport where the pilot intends to land. Using a vertical dipole antenna, the airport's air traffic control system is capable of transmitting 110-watt, 24 MHz signals. When the plane's horizontal position is 5 kilometers from the airport, what is the intensity of the signal at the plane's receiving antenna, in $\text{W}/\text{m}^2$? (The height of the transmitting antenna is negligible.) [i]Problem proposed by Kimberly Geddes[/i]

1990 Austrian-Polish Competition, 9

Tags: sum , algebra , partition , combinatorics

$a_1, a_2, ... , a_n$ is a sequence of integers such that every non-empty subsequence has non-zero sum. Show that we can partition the positive integers into a finite number of sets such that if $x_i$ all belong to the same set, then $a_1x_1 + a_2x_2 + ... + a_nx_n$ is non-zero.

1998 All-Russian Olympiad, 1

Tags: analytic geometry , vieta , algebra

Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.

2005 JBMO Shortlist, 1

Tags: power of a point , radical axis , geometry

Let $ABC$ be an acute-angled triangle inscribed in a circle $k$. It is given that the tangent from $A$ to the circle meets the line $BC$ at point $P$. Let $M$ be the midpoint of the line segment $AP$ and $R$ be the second intersection point of the circle $k$ with the line $BM$. The line $PR$ meets again the circle $k$ at point $S$ different from $R$. Prove that the lines $AP$ and $CS$ are parallel.

2005 National High School Mathematics League, 13

Tags:

Define sequence $(a_n)$: $a_0=1,a_{n+1}=\frac{7a_n+\sqrt{45a_n^2-36}}{2},n\in\mathbb{N}$. [b](a)[/b] Prove that for all $n\in\mathbb{N}$, $a_n$ is a positive integer. [b](b)[/b] Prove that for all $n\in\mathbb{N}$, $a_na_{n+1}-1$ is a perfect square.

2021 Iberoamerican, 4

Tags: algebra

Let $a,b,c,x,y,z$ be real numbers such that \[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \] Show that $a^2+b^2+c^2=x^2+y^2+z^2$.

MOAA Team Rounds, 2019.6

Tags: floor function , ceiling function , algebra , team

Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

MOAA Team Rounds, 2021.18

Tags: team

Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

2008 Germany Team Selection Test, 2

Tags: binomial coefficient , number theory , divisibility

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

2016 Japan Mathematical Olympiad Preliminary, 5

Tags: geometry , congruent triangles

Let $ABCD$ be a quadrilateral with $AC=20$, $AD=16$. We take point $P$ on segment $CD$ so that triangle $ABP$ and $ACD$ are congruent. If the area of triangle $APD$ is $28$, find the area of triangle $BCP$. Note that $XY$ expresses the length of segment $XY$.

2022 Harvard-MIT Mathematics Tournament, 6

Tags: combinatorics

The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divide , divisible , power

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

2024 China National Olympiad, 1

Tags: number theory

Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. [i]Proposed by Yinghua Ai[/i]

2002 Federal Math Competition of S&M, Problem 4

Tags: combinatorics , rectangle , geometry

Is it possible to cut a rectangle $2001\times2003$ into pieces of the form [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS82L2RjZTZjNzc0M2YxMzM1ZDIzZTY2Zjc2NGJlMWJlMWUwMmU2ZWRlLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNS0xMyBhdCAzLjQ2LjQ2IFBNLnBuZw==[/img] each consisting of three unit squares?

2014 India IMO Training Camp, 3

Tags: function , number theory

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.