Olympiad problems

2005 District Olympiad, 2

Let $ABC$ be a triangle inscribed in a circle of center $O$ and radius $R$. Let $I$ be the incenter of $ABC$, and let $r$ be the inradius of the same triangle, $O\neq I$, and let $G$ be its centroid. Prove that $IG\perp BC$ if and only if $b=c$ or $b+c=3a$.

1994 Baltic Way, 4

Tags: algebra

Is there an integer $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is a rational number?

1977 IMO Longlists, 53

Tags: number theory

Find all pairs of integers $a$ and $b$ for which \[7a+14b=5a^2+5ab+5b^2\]

2019 Jozsef Wildt International Math Competition, W. 25

Tags: function , inequalities , gamma function , summation

Let $x_i$, $y_i$, $z_i$, $w_i \in \mathbb{R}^+, i = 1, 2,\cdots n$, such that$$\sum \limits_{i=1}^nx_i=nx,\ \sum \limits_{i=1}^ny_i=ny,\ \sum \limits_{i=1}^nw_i=nw $$ $$\Gamma \left(z_i\right)\geq \Gamma \left(w_i\right),\ \sum \limits_{i=1}^n\Gamma \left(z_i\right)=n\Gamma^* (z)$$Then$$\sum \limits_{i=1}^n \frac{\left(\Gamma \left(x_i\right)+\Gamma \left(y_i\right)\right)^2}{\Gamma \left(z_i\right)-\Gamma \left(w_i\right)}\geq n\frac{\left(\Gamma \left(x\right)+\Gamma \left(y\right)\right)^2}{\Gamma^* \left(z\right)-\Gamma \left(w\right)}$$

2016 Brazil Team Selection Test, 4

Tags: algebra

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

2023 AIME, 11

Tags:

Find the number of subsets of ${1,2,3,...,10}$ that contain exactly one pair of consecutive integers. Examples of such subsets are ${1,2,5}$ and ${1,3,6,7,10}$.

2017 China Team Selection Test, 5

Tags: number theory , greatest common divisor

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

2023 Bulgaria National Olympiad, 2

Tags: angle bisector , circumcircle , geometry

Let $ABC$ be an acute triangle and $A_{1}, B_{1}, C_{1}$ be the touchpoints of the excircles with the segments $BC, CA, AB$ respectively. Let $O_{A}, O_{B}, O_{C}$ be the circumcenters of $\triangle AB_{1}C_{1}, \triangle BC_{1}A_{1}, \triangle CA_{1}B_{1}$ respectively. Prove that the lines through $O_{A}, O_{B}, O_{C}$ respectively parallel to the internal angle bisectors of $\angle A,\angle B, \angle C$ are concurrent.

2019 India IMO Training Camp, P3

Tags: function , number theory

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

2002 Iran MO (3rd Round), 5

Tags: radical axis , power of a point , circumcircle , perpendicular bisector , geometry

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.

2011 Bundeswettbewerb Mathematik, 3

Tags: combinatorics

When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out: (1) If two teams play against each other, there are no more team playing against both of them. (2) If two teams do not play against each other, then there is always exactly two other teams playing against them both. Prove that all teams play the same number of games.

2008 Princeton University Math Competition, A3

Tags: algebra

A sequence $\{a_i\}$ is defined by $a_1 = c$ for some $c > 0$ and $a_{n+1} = a_n + \frac{n}{a_n}$. Prove that $\frac{a_n}{n}$ converges and find its limit.

MathLinks Contest 6th, 6.3

Tags: geometry

Let $C_1, C_2$ and $C_3$ be three circles, of radii $2, 4$ and $6$ respectively. It is known that each of them are tangent exteriorly with the other two circles. Let $\Omega_1$ and $\Omega_2$ be two more circles, each of them tangent to all of the $3$ circles above, of radius $\omega_1$ and $\omega_2$ respectively. Prove that $\omega_1 + \omega_2 = 2\omega_1\omega_2$.

2009 Today's Calculation Of Integral, 476

Tags: parabola , function , calculus computations , conic , geometry , calculus , integration

Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\equal{}1\minus{}|x|$. Find the equation of the parabola such that the area of the region surrounded by the parabola and the $ x$ axis is maximal.

2002 Iran MO (2nd round), 1

Tags: permutation , divisibility , combinatorics

Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

2001 ITAMO, 2

Tags: combinatorics

In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$, and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament?

2012 Romanian Master of Mathematics, 2

Tags: trigonometry , circumcircle , complex number , geometry

Given a non-isosceles triangle $ABC$, let $D,E$, and $F$ denote the midpoints of the sides $BC,CA$, and $AB$ respectively. The circle $BCF$ and the line $BE$ meet again at $P$, and the circle $ABE$ and the line $AD$ meet again at $Q$. Finally, the lines $DP$ and $FQ$ meet at $R$. Prove that the centroid $G$ of the triangle $ABC$ lies on the circle $PQR$. [i](United Kingdom) David Monk[/i]

2024 Benelux, 4

Tags: number theory

For each positive integer $n$, let $rad(n)$ denote the product of the distinct prime factors of $n$. Show that there exists integers $a,b > 1$ such that $gcd(a,b)=1$ and $$rad(ab(a+b)) < \frac{a+b}{2024^{2024}}$$. For example, $rad(20)=rad(2^2\cdot 5)=2\cdot 5=10$.

CNCM Online Round 3, 1

Tags: v4913 orz

Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions? [i]Proposed by Albert Wang (awang2004)[/i]

1997 India Regional Mathematical Olympiad, 3

Tags: inequalities

Solve for real $x$: \[ \frac{1}{[x]} + \frac{1}{[2x]} = x - [x] + \frac{1}{3}. \]

2011 Morocco National Olympiad, 2

Tags: algebra , quadratic

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

2013 Stanford Mathematics Tournament, 19

Tags: geometry , trigonometry

A triangle with side lengths $2$ and $3$ has an area of $3$. Compute the third side length of the triangle.

2015 Korea Junior Math Olympiad, 5

Tags: geometry , circumcircle , incenter

Let $I$ be the incenter of an acute triangle $\triangle ABC$, and let the incircle be $\Gamma$. Let the circumcircle of $\triangle IBC$ hit $\Gamma$ at $D, E$, where $D$ is closer to $B$ and $E$ is closer to $C$. Let $\Gamma \cap BE = K (\not= E)$, $CD \cap BI = T$, and $CD \cap \Gamma = L (\not= D)$. Let the line passing $T$ and perpendicular to $BI$ meet $\Gamma$ at $P$, where $P$ is inside $\triangle IBC$. Prove that the tangent to $\Gamma$ at $P$, $KL$, $BI$ are concurrent.

1999 Bulgaria National Olympiad, 3

Tags: number theory

Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$.

2010 AMC 10, 2

Tags: percent

Makayla attended two meetings during her 9-hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 25 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 35$