Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$, there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$

The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$

A positive integer is called $\emph{lucky}$ if it is divisible by $7$, and the sum of its digits is also divisible by $7$. Fix a positive integer $n$. Show that there exists some lucky integer $l$ such that $\left|n - l\right| \leq 70$.

Find all natural pairs $ (x,y)$, such that $ x$ and $ y$ are relative prime and satisfy equality: $ 2x^2 \plus{} 5xy \plus{} 3y^2 \equal{} 41x \plus{} 62y \plus{} 21$.

Compute the number of ways to fill each cell in a $8 \times 8$ square grid with one of the letters $H, M,$ or $T$ such that every $2 \times 2$ square in the grid contains the letters $H, M, M, T$ in some order.

All cells of the $m\times n$ table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values $m = 2, 3, 4,$ find all $n$ such that the mentioned coloring is possible.

Given a list $A$ of $n$ real numbers, the following algorithm, known as $\textit{insertion sort}$, sorts the elements of $A$ from least to greatest. \begin{tabular}{l} 1: \textbf{FUNCTION} $IS(A)$ \\ 2: $\quad$ \textbf{FOR} $i=0,\ldots, n-1$: \\ 3: $\quad\quad$ $j \leftarrow i$\\ 4: $\quad\quad$ \textbf{WHILE} $j>0$ \& $A[j-1]>A[j]:$\\ 5: $\quad\quad\quad$ \textbf{SWAP} $A[j], A[j-1]$\\ 6: $\quad\quad\quad$ $j \leftarrow j-1$\\ 7: \textbf{RETURN} $A$ \end{tabular} As $A$ ranges over all permutations of $\{1, 2, \ldots, n\}$, let $f(n)$ denote the expected number of comparisons (i.e., checking which of two elements is greater) that need to be made when sorting $A$ with insertion sort. Evaluate $f(13) - f(12)$.

$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$. The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$. Show that the circumcenter of $CEF$ lies on $K$.

Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the intersection point of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the intersection point of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$. (A. Zaslavsky)

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

In the diagram shown, how many pathways are there from point $A$ to point $B$ if you are only allowed to travel due East, Southeast, or Southwest? [img]https://cdn.artofproblemsolving.com/attachments/9/1/0a1219fb430c402fef4b7555ddff7c88fec47e.jpg[/img]

A spherical planet has the equator of length $1$. On this planet, $N$ circular roads of length $1$ each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for : (a) $N=3$ ([i]4 points)[/i] (b) $N=4$ ([i]6 points)[/i] [i]Alexandr Berdnikov [/i]

The complex numbers $z_1$ and $z_2$ are given such that $z_1=-1+i$ and $z_2=2+4i$. Find the complex number $z_3$ such that $z_1,z_2,z_3$ are the points of an equilateral triangle. How many solutions do we have ?

Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]

Find all functions $f:\mathbb{Z} \to \mathbb{Z}$ such that for all $m\in\mathbb{Z}$: [list][*] $f(m+8) \le f(m)+8$, [*] $f(m+11) \ge f(m)+11$.[/list]

An alien from the planet OMO Centauri writes the first ten prime numbers in arbitrary order as U, W, XW, ZZ, V, Y, ZV, ZW, ZY, and X. Each letter represents a nonzero digit. Each letter represents the same digit everywhere it appears, and different letters represent different digits. Also, the alien is using a base other than base ten. The alien writes another number as UZWX. Compute this number (expressed in base ten, with the usual, human digits). [i]Proposed by Luke Robitaille & Eric Shen[/i]

Define the [i]inverse [/i] of triangle $ABC$ with respect to a point $O$ in the following way: construct the circumcircle of $ABC$ and construct lines $AO$, $BO$, and $CO$. Let $A'$ be the other intersection of $AO$ and the circumcircle (if $AO$ is tangent, then let $A' = A$). Similarly define $B'$ and $C'$. Then $A'B'C'$ is the inverse of $ABC$ with respect to $O$. Compute the area of the inverse of the triangle given in the plane by $A(-6, -21)$, $B(-23, 10)$, $C(16, 23)$ with respect to $O(1, 3)$.

$F_{n}$ denotes the Fibonacci Sequence where $F_{1}=0, F_{2}=1, F_{n}=F_{n-1}+F_{n-2},\ \forall \ n \geq 3$ Find$$\sum\limits_{n=3}^{\infty}\frac{18+999F_n}{F_{n-1}\times F_{n+1}}$$ [list=1] [*] 2016 [*] 2017 [*] 2018 [*] None of these [/list]

Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of the square.

Let $s(k)$ denotes sum of digits of positive integer $k$. Prove that there are infinitely many positive integers $n$, which are not divisible by $10$ and satisfies: $$s(n^2) < s(n) - 5$$

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

A chord $PQ$ of the circumcircle of a triangle $ABC$ meets the sides $BC, AC$ at points $A', B'$ respectively. The tangents to the circumcircle at $A$ and $B$ meet at a point $X$, and the tangents at points $P$ and $Q$ meet at point $Y$. The line $XY$ meets $AB$ at a point $C'$. Prove that the lines $AA', BB'$ and $CC'$ concur.

In $\Delta ABC$, the sum of the sides is $2s$ and the radius of the incircle is $r$. Three semicircles with diameters $AB$, $BC$ and $CA$ are drawn on the outside of $ABC$. A circle with radius $t$ touches all three semicircles. Prove that \[ \frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r. \]

Determine whether or not two polynomials $P, Q$ with degree no less than 2018 and with integer coefficients exist such that $$P(Q(x))=3Q(P(x))+1$$ for all real numbers $x$.

Let $A, B, C, D, E$ be five distinct points on a circle $\Gamma$ in the clockwise order and let the extensions of $CD$ and $AE$ meet at a point $Y$ outside $\Gamma$. Suppose $X$ is a point on the extension of $AC$ such that $XB$ is tangent to $\Gamma$ at $B$. Prove that $XY = XB$ if and only if $XY$ is parallel $DE$.