Olympiad problems

2012 National Olympiad First Round, 21

The angle bisector of vertex $A$ of $\triangle ABC$ cuts $[BC]$ at $D$. The circle passing through $A$ and touching to $BC$ at $D$ meets $[AB]$ and $[AC]$ at $P$ and $Q$, respectively. $AD$ and $PQ$ meet at $T$. If $|AB|=5, |BC|=6, |CA|=7$, then $\frac{|AT|}{|TD|}=?$ $ \textbf{(A)}\ \frac75 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \frac72 \qquad \textbf{(E)}\ 4$

2014 Iran Geometry Olympiad (senior), 4:

Tags: geometry , circumcircle , reflection

A tangent line to circumcircle of acute triangle $ABC$ ($AC>AB$) at $A$ intersects with the extension of $BC$ at $P$. $O$ is the circumcenter of triangle $ABC$.Point $X$ lying on $OP$ such that $\measuredangle AXP=90^\circ$.Points $E$ and $F$ lying on $AB$ and $AC$,respectively,and they are in one side of line $OP$ such that $ \measuredangle EXP=\measuredangle ACX $ and $\measuredangle FXO=\measuredangle ABX $. $K$,$L$ are points of intersection $EF$ with circumcircle of triangle $ABC$.prove that $OP$ is tangent to circumcircle of triangle $KLX$. Author:Mehdi E'tesami Fard , Iran

2017 ASDAN Math Tournament, 8

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Compute $$\int_0^1\frac{2xe^x-1}{2x^2e^x+2}dx.$$

2003 All-Russian Olympiad, 3

Tags: geometry , rectangle , combinatorics

Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$

2003 Hungary-Israel Binational, 2

Tags: circumcircle , trigonometry , geometry

Let $ABC$ be an acute-angled triangle. The tangents to its circumcircle at $A, B, C$ form a triangle $PQR$ with $C \in PQ$ and $B \in PR$. Let $C_{1}$ be the foot of the altitude from $C$ in $\Delta ABC$ . Prove that $CC_{1}$ bisects $\widehat{QC_{1}P}$ .

2013 Ukraine Team Selection Test, 3

Tags: algebra , number theory , prime number , polynomial

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

2021 Israel TST, 3

Tags: geometry , incenter , circumcircle

In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.

1983 AMC 12/AHSME, 19

Tags: trigonometry , angle bisector , trig identities , law of cosines , geometry

Point $D$ is on side $CB$ of triangle $ABC$. If \[ \angle{CAD} = \angle{DAB} = 60^\circ,\quad AC = 3\quad\mbox{ and }\quad AB = 6, \] then the length of $AD$ is $\text{(A)} \ 2 \qquad \text{(B)} \ 2.5 \qquad \text{(C)} \ 3 \qquad \text{(D)} \ 3.5 \qquad \text{(E)} \ 4$

1996 National High School Mathematics League, 5

Tags: function

On $[1,2]$ if two functions $f(x)=x^2+px+q$ and $g(x)=x+\frac{1}{x^2}$ get their minumum value at the same point, then the maximum value of $f(x)$ on $[1,2]$ is $\text{(A)}4+\frac{11}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(B)}4-\frac{5}{2}\sqrt[3]{2}+\sqrt[3]{4}$ $\text{(C)}1-\frac{1}{2}\sqrt[3]{2}+\sqrt[3]{4}\qquad\text{(D)}$ none above

1972 IMO Longlists, 36

Tags: geometry

A finite number of parallel segments in the plane are given with the property that for any three of the segments there is a line intersecting each of them. Prove that there exists a line that intersects all the given segments.

2023 Chile Classification NMO Juniors, 4

Tags: geometry

In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.

2008 iTest Tournament of Champions, 5

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It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b,c)$ such that $a\leq b<c$ and \[\dfrac{a(a+1)}2+\dfrac{b(b+1)}2=\dfrac{c(c+1)}2.\] For instance, $(3,5,6)$ is a PTTN ($6+15=21$). Here we call both $a$ and $b$ $\textit{legs}$ of the PTTN. Find the smallest natural number $n$ such that $n$ is a leg of $\textit{at least}$ six distinct PTTNs.

2015 Purple Comet Problems, 5

Tags: geometry , rectangle

The diagram below shows a rectangle with one side divided into seven equal segments and the opposite side divided in half. The rectangle has area 350. Find the area of the shaded region. For Diagram go to purplecomet.org/welcome/practice, the $2015$ middle school contest, and #5.

2018 APMO, 5

Tags: polynomial , number theory , algebra

Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.

2011 All-Russian Olympiad Regional Round, 11.2

Tags: number theory

2011 non-zero integers are given. It is known that the sum of any one of them with the product of the remaining 2010 numbers is negative. Prove that if all numbers are split arbitrarily into two groups, the sum of the two products will also be negative. (Authors: N. Agahanov & I. Bogdanov)

1991 Canada National Olympiad, 2

Tags: number theory

Let $n$ be a fixed positive integer. Find the sum of all positive integers with the property that in base $2$ each has exactly $2n$ digits, consisting of $n$ 1's and $n$ 0's. (The first digit cannot be $0$.)

2013 Purple Comet Problems, 24

Tags: modular arithmetic

Find the remainder when $333^{333}$ is divided by $33$.

2025 Malaysian IMO Team Selection Test, 4

Tags: geometry

Let $ABC$ be a triangle, with incenter $I$ and $A$-excenter $J$. The lines $BI$, $CI$, $BJ$ and $CJ$ intersect the circumcircle of $ABC$ at $P$, $Q$, $R$ and $S$ respectively. Let $IM$, $JN$ be diameters in the circumcircles of triangles $IPQ$ and $JRS$ respectively. Prove that $\angle BAM+\angle CAN=180^{\circ}$. [i]Proposed by Ivan Chan Kai Chin[/i]

2006 Bulgaria National Olympiad, 1

Tags: modular arithmetic , number theory

Let $p$ be a prime such that $p^2$ divides $2^{p-1}-1$. Prove that for all positive integers $n$ the number $\left(p-1\right)\left(p!+2^n\right)$ has at least $3$ different prime divisors. [i]Aleksandar Ivanov[/i]

2010 IMO Shortlist, 5

Tags: graph theory , tournament graphs , vertex degree , combinatorics

$n \geq 4$ players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players $bad$ if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let $w_i$ and $l_i$ be respectively the number of wins and losses of the $i$-th player. Prove that \[\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.\] [i]Proposed by Sung Yun Kim, South Korea[/i]

2017 Iran Team Selection Test, 5

Tags: combinatorics

$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s. [i]Proposed by Aryan Tajmir[/i]

1970 AMC 12/AHSME, 7

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Inside square $ABCD$ with side $s$, quarter-circle arcs with radii $s$ and centers at $A$ and $B$ are drawn. These arcs intersect at point $X$ inside the square. How far is $X$ from side $CD$? $\textbf{(A) }\frac{1}{2}s(\sqrt{3}+4)\qquad\textbf{(B) }\frac{1}{2}s\sqrt{3}\qquad\textbf{(C) }\frac{1}{2}s(1+\sqrt{3})\qquad$ $\textbf{(D) }\frac{1}{2}s(\sqrt{3}-1)\qquad \textbf{(E) }\frac{1}{2}s(2-\sqrt{3})$

2005 Georgia Team Selection Test, 3

Tags: calculus , derivative , cauchy inequality , inequalities

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

2012 Belarus Team Selection Test, 1

Tags: number theory , perfect cube

For $n$ positive integers $a_1,...,a_n$ consider all their pairwise products $a_ia_j$, $1 \le i < j \le n$. Let $N$ be the number of those products which are the cubes of positive integers. Find the maximal possible value of $N$ if it is known that none of $a_j$ is a cube of an integer. (S. Mazanik)

2001 Hungary-Israel Binational, 6

Tags: combinatorics

Let be given $32$ positive integers with the sum $120$, none of which is greater than $60.$ Prove that these integers can be divided into two disjoint subsets with the same sum of elements.