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]

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 $gcd(k,d(a_1)d(a_2)\ldots d(a_{2019})) =1$

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 \( n \).

$p$ 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.

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

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 $\angle BLA = \angle CKD$.

$ABC$ 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 $\angle DAE$.

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.

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 $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$

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]

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

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

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 $BDE$.

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]

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]

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

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

Let's call a corner the figure that is obtained by removing one cell from a $2 \times 2$ square. Cut the $6 \times 6$ square into corners so that no two of them form a $2 \times 3$ or $3 \times 2$ rectangle together.

Three spheres are constructed so that the edges $[AB], [BC], [AD]$ of the tetrahedron $ABCD$ are their respective diameters. Prove that the spheres cover all the tetrahedron.

Let $a,b,c,$ and $d$ be the lengths of sides $MN,NP,PQ,$ and $QM$, respectively, of quadrilateral $MNPQ$. If $A$ is the area of $MNPQ$, then $\textbf{(A) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is convex}$ $\textbf{(B) }A=\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$ $\textbf{(C) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a rectangle}$ $\textbf{(D) }A\le\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$ $\textbf{(E) }A\ge\left(\frac{a+c}{2}\right)\left(\frac{b+d}{2}\right)\text{ if and only if }MNPQ\text{ is a parallelogram}$

The nine delegates to the Economic Cooperation Conference include 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the sleepers are from the same country is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

For which $ n\in \mathbb{N}$ do there exist rational numbers $ a,b$ which are not integers such that both $ a \plus{} b$ and $ a^n \plus{} b^n$ are integers?

The triangle $ABC$ is $| BC | = a$ and $| AC | = b$. On the ray starting from vertex $C$ and passing the midpoint of side $AB$ , choose any point $D$ other than vertex $C$. Let $K$ and $L$ be the projections of $D$ on the lines $AC$ and $BC$, respectively, $K$ and $L$. Find the ratio $| DK | : | DL |$.

For each side of a given polygon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than $2$.

Let $n > 1$ be a natural number. We write out the fractions $\frac{1}{n}$, $\frac{2}{n}$, $\dots$ , $\dfrac{n-1}{n}$ such that they are all in their simplest form. Let the sum of the numerators be $f(n)$. For what $n>1$ is one of $f(n)$ and $f(2015n)$ odd, but the other is even?