Olympiad problems

2005 Abels Math Contest (Norwegian MO), 3a

In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12 \angle A $

2014 Vietnam Team Selection Test, 1

Tags: induction , algebra

Find all $ f:\mathbb{Z}\rightarrow\mathbb{Z} $ such that \[ f(2m+f(m)+f(m)f(n))=nf(m)+m \] $ \forall m,n\in\mathbb{Z} $

Kyiv City MO 1984-93 - geometry, 1991.8.3

Tags: parallelogram , geometry , equilateral

On the sides of the parallelogram $ABCD$ outside it are constructed equilateral triangles $ABM$, $BCN$, $CDP$, $ADQ$. Prove that $MNPQ$ is a parallelogram.

1970 IMO Longlists, 19

Tags: inequalities

Let $1<n\in\mathbb{N}$ and $1\le a\in\mathbb{R}$ and there are $n$ number of $x_i, i\in\mathbb{N}, 1\le i\le n$ such that $x_1=1$ and $\frac{x_{i}}{x_{i-1}}=a+\alpha _ i$ for $2\le i\le n$, where $\alpha _i\le \frac{1}{i(i+1)}$. Prove that $\sqrt[n-1]{x_n}< a+\frac{1}{n-1}$.

2016 Dutch IMO TST, 3

Tags: number theory , least common multiple

Find all positive integers $k$ for which the equation: $$ \text{lcm}(m,n)-\text{gcd}(m,n)=k(m-n)$$ has no solution in integers positive $(m,n)$ with $m\neq n$.

2011 ELMO Shortlist, 2

Tags: geometric transformation , trigonometry , reflection , angle bisector , geometry

Let $\omega,\omega_1,\omega_2$ be three mutually tangent circles such that $\omega_1,\omega_2$ are externally tangent at $P$, $\omega_1,\omega$ are internally tangent at $A$, and $\omega,\omega_2$ are internally tangent at $B$. Let $O,O_1,O_2$ be the centers of $\omega,\omega_1,\omega_2$, respectively. Given that $X$ is the foot of the perpendicular from $P$ to $AB$, prove that $\angle{O_1XP}=\angle{O_2XP}$. [i]David Yang.[/i]

2018 Switzerland - Final Round, 9

Tags: geometry , parallelogram , inequalities

Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?

2010 AMC 8, 14

Tags:

What is the sum of the prime factors of $2010$? $ \textbf{(A)}\ 67 \qquad\textbf{(B)}\ 75\qquad\textbf{(C)}\ 77\qquad\textbf{(D)}\ 201\qquad\textbf{(E)}\ 210 $

IMSC 2024, 5

Tags: algebra , polynomial , imsc

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying $$ f(xy)=P(f(x), f(y)) $$ for all $x, y\in\mathbb{R}_{>0}$.\\ [i]Proposed by Navid Safaei, Iran[/i]

2019 Turkey MO (2nd round), 2

Tags: arithmetic sequence , construction , number theory

Let $d(n)$ denote the number of divisors of a positive integer $n$. If $k$ is a given odd number, prove that there exist an increasing arithmetic progression in positive integers $(a_1,a_2,\ldots a_{2019}) $ such that $gcd(k,d(a_1)d(a_2)\ldots d(a_{2019})) =1$

2014 Chile TST IMO, 4

Tags: polynomial , modulo , algebra

Let $ f(n) $ be a polynomial with integer coefficients. Prove that if $ f(-1) $, $ f(0) $, and $ f(1) $ are not divisible by 3, then $ f(n) \neq 0 $ for all integers $ n $.

2016 Turkey Team Selection Test, 9

Tags: number theory , polynomial

$p$ is a prime. Let $K_p$ be the set of all polynomials with coefficients from the set $\{0,1,\dots ,p-1\}$ and degree less than $p$. Assume that for all pairs of polynomials $P,Q\in K_p$ such that $P(Q(n))\equiv n\pmod p$ for all integers $n$, the degrees of $P$ and $Q$ are equal. Determine all primes $p$ with this condition.

2017 ASDAN Math Tournament, 1

Tags:

Compute $$\int_0^6\frac{x-3}{x^2-6x-7}dx.$$

2008 Singapore Senior Math Olympiad, 1

Tags: trapezium , trapezoid , equal angles , geometry

Let $ABCD$ be a trapezium with $AD // BC$. Suppose $K$ and $L$ are, respectively, points on the sides $AB$ and $CD$ such that $\angle BAL = \angle CDK$. Prove that $\angle BLA = \angle CKD$.

2016 Azerbaijan IMO TST First Round, 2

Tags: geometry

$ABC$ be atriangle with sides $AB=20$ , $AC=21$ and $BC=29$. Let $D$ and $E$ be points on the side $BC$ such that $BD=8$ and $EC=9$. Find the angle $\angle DAE$.

2018 European Mathematical Cup, 1

Tags: algebra

Let $a, b, c$ be non-zero real numbers such that $a^2+b+c=\frac{1}{a}, b^2+c+a=\frac{1}{b}, c^2+a+b=\frac{1}{c}.$ Prove that at least two of $a, b, c$ are equal.

2005 Today's Calculation Of Integral, 30

Tags: calculus , integration , algebra , partial fractions , calculus computations

A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$ Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

2004 Nicolae Păun, 1

Tags: function , algebra , decomposition , injectivity

Prove that any function that maps the integers to themselves is a sum of any finite number of injective functions that map the integers to themselves. [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

2021 Peru PAGMO TST, P2

Tags: geometry , rectangle

The bisector of the diagonal $BD$ of a rectangle $ABCD$ (with $AB < BC$) intersects the lines $BC$ and $BA$ at points $E$ and $F$, respectively. The line passing through point $F$ and parallel to segment $AC$ intersects line $CD$ at point $G$. Prove that lines $EG$ and $AC$ are perpendicular

2008 Purple Comet Problems, 3

Tags: percent

There were 891 people voting at precinct 91. There were 20 percent more female voters than male voters. How many female voters were there?

2017 Germany Team Selection Test, 3

Tags: geometry , incenter , reflection , inversion , cyclic quadrilateral

Let $ABC$ be a triangle with $AB = AC \neq BC$ and let $I$ be its incentre. The line $BI$ meets $AC$ at $D$, and the line through $D$ perpendicular to $AC$ meets $AI$ at $E$. Prove that the reflection of $I$ in $AC$ lies on the circumcircle of triangle $BDE$.

2004 Nicolae Coculescu, 1

Tags: equation , algebra

Find all pairs of integers $ (a,b) $ such that the equation $$ |x-1|+|x-a|+|x-b|=1 $$ has exactly one real solution. [i]Florian Dumitrel[/i]

2019 China Second Round Olympiad, 4

Tags: combinatorics , graph theory

Let $V$ be a set of $2019$ points in space where any of the four points are not on the same plane, and $E$ be the set of edges connected between them. Find the smallest positive integer $n$ satisfying the following condition: if $E$ has at least $n$ elements, then there exists $908$ two-element subsets of $E$ such that [list][*]The two edges in each subset share a common vertice, [*]Any of the two subsets do not intersect.[/list]

2018 China Girls Math Olympiad, 5

Tags: algebra , maximum value , complex

Let $\omega \in \mathbb{C}$, and $\left | \omega \right | = 1$. Find the maximum length of $z = \left( \omega + 2 \right) ^3 \left( \omega - 3 \right)^2$.

1987 IMO Longlists, 24

Tags: polynomial , vieta , algebra

Prove that if the equation $x^4 + ax^3 + bx + c = 0$ has all its roots real, then $ab \leq 0.$