Olympiad problems

2004 CentroAmerican, 2

Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$. $a)$ Prove that $\angle BPC=90^{\circ}$. $b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.

1977 Chisinau City MO, 137

Tags: equal angles , geometry , angle

Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertex, divide this angle into four equal parts.

2020 Turkey EGMO TST, 3

Tags: combinatorics

There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.

1999 Slovenia National Olympiad, Problem 2

Tags: algebra , polynomial

Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$.

2024 LMT Fall, 17

Tags: guts , algebra

Suppose $x$, $y$, $z$ are pairwise distinct real numbers satisfying \[ x^2+3y =y^2 +3z = z^2+3x. \]Find $(x+y)(y+z)(z+x)$.

2001 China Team Selection Test, 2

Tags: number theory

If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number. Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.

2005 Switzerland - Final Round, 4

Tags: number theory , greatest common divisor , gcd

Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.

2009 Paraguay Mathematical Olympiad, 5

Tags:

In a triangle $ABC$, let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$. Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$. Find all possible values of the length $BD$.

2017 International Zhautykov Olympiad, 2

Tags: number theory , divisor , prime

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2021 AIME Problems, 6

Tags:

For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy $$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$

2008 China Second Round Olympiad, 2

Tags: function , algebra

Let $f(x)$ be a periodic function with periods $T$ and $1$($0<T<1$).Prove that: (1)If $T$ is rational,then there exists a prime $p$ such that $\frac{1}{p}$ is also a period of $f$; (2)If $T$ is irrational,then there exists a strictly decreasing infinite sequence {$a_n$},with $1>a_n>0$ for all positive integer $n$,such that all $a_n$ are periods of $f$.

2014 Harvard-MIT Mathematics Tournament, 6

Tags: hmmt , quadratic , geometry , geometric transformation , dilation , inequalities , complex number

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

2018 AMC 10, 18

Tags:

Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? $\textbf{(A)} \text{ 60} \qquad \textbf{(B)} \text{ 72} \qquad \textbf{(C)} \text{ 92} \qquad \textbf{(D)} \text{ 96} \qquad \textbf{(E)} \text{ 120}$

2016 China Second Round Olympiad, Q10

Tags: algebra , function

Let $f(x)$ is an odd function on $R$ , $f(1)=1$ and $f(\frac{x}{x-1})=xf(x)$ $(\forall x<0)$. Find the value of $f(1)f(\frac{1}{100})+f(\frac{1}{2})f(\frac{1}{99})+f(\frac{1}{3})f(\frac{1}{98})+\cdots +f(\frac{1}{50})f(\frac{1}{51}).$

2016 South African National Olympiad, 3

Tags: geometry

The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$ and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.

2014 District Olympiad, 4

Tags: floor function , ceiling function , number theory

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

2005 Bulgaria Team Selection Test, 5

Tags: geometry , incenter , circumcircle , trigonometry , trapezoid , geometric transformation , reflection

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

2002 IMO Shortlist, 6

Tags: algebra , polynomial , laurentiu panaitopol

Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

ICMC 7, 3

Tags: combinatorics

There are 105 users on the social media platform Mathsenger, every pair of which has a direct messaging channel. Prove that each messaging channel may be assigned one of 100 encryption keys, such that no 4 users have the 6 pairwise channels between them all being assigned the same encryption key. [i]Proposed by Fredy Yip[/i]

Novosibirsk Oral Geo Oly IX, 2019.6

Tags: geometry , polyline

A square with side $1$ contains a non-self-intersecting polyline of length at least $200$. Prove that there is a straight line parallel to the side of the square that has at least $101$ points in common with this polyline.

2020 Moldova EGMO TST, 3

Tags: sequence , algebra

Let the sequence $a_n$, $n\geq2$, $a_n=\frac{\sqrt[3]{n^3+n^2-n-1}}{n} $. Find the greatest natural number $k$ ,such that $a_2 \cdot a_3 \cdot . . .\cdot a_k <8$

2025 AIME, 7

Tags: probability , combinatorics , divisor

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2014 India Regional Mathematical Olympiad, 4

Tags: combinatorics

Is it possible to write the numbers $17$,$18$,$19$,...$32$ in a $4*4$ grid of unit squares with one number in each square such that if the grid is divided into four $2*2$ subgrids of unit squares ,then the product of numbers in each of the subgrids divisible by $16$?

1995 All-Russian Olympiad Regional Round, 11.5

Tags: algebra , inequalities , trigonometry

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

2005 China Team Selection Test, 1

Tags: geometry

Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.