For every positive integer $m$, a $m\times 2018$ rectangle consists of unit squares (called "cell") is called [i]complete[/i] if the following conditions are met: i. In each cell is written either a "$0$", a "$1$" or nothing; ii. For any binary string $S$ with length $2018$, one may choose a row and complete the empty cells so that the numbers in that row, if read from left to right, produce $S$ (In particular, if a row is already full and it produces $S$ in the same manner then this condition ii. is satisfied). A [i]complete[/i] rectangle is called [i]minimal[/i], if we remove any of its rows and then making it no longer [i]complete[/i]. a. Prove that for any positive integer $k\le 2018$ there exists a [i]minimal[/i] $2^k\times 2018$ rectangle with exactly $k$ columns containing both $0$ and $1$. b. A [i]minimal[/i] $m\times 2018$ rectangle has exactly $k$ columns containing at least some $0$ or $1$ and the rest of columns are empty. Prove that $m\le 2^k$.

Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.

Prove that there exists some positive real number $\lambda$ such that for any $D_{>1}\in\mathbb{R}$, one can always find an acute triangle $\triangle ABC$ in the Cartesian plane such that [list] [*] $A, B, C$ lie on lattice points; [*] $AB, BC, CA>D$; [*] $S_{\triangle ABC}<\frac{\sqrt 3}{4}D^2+\lambda\cdot D^{4/5}$.

For any interval $\mathcal{A}$ in the real number line not containing zero, define its [i]reciprocal[/i] to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[m,n]$ is $10^{10}$ times the length of its reciprocal. [i]Proposed by David Altizio[/i]

Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino? [asy] size(4cm); draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0)); [/asy]

Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

(a) Let $k,n\ge 1$.Find the number of sequences $\phi=S_0,S_1,\ldots,S_k$ of subsets of $[n]=\{1,2,3,\ldots,n\}$ if for all $1\le i\le k$ we have either (i)$S_{i-1}\subset S_i$ and $|S_i-S_{i-1}|$,or (ii)$S_i\subset S_{i-1}$ and $|S_{i-1}-S_i|=1$. (b) Suppose that we add the additional condition that $S_k=\phi$.Show that now the number $f_k(n)$ of sequences is given by$f_k(n)=\frac{1}{2^n}\sum_{i=0}^n\binom ni (n-2i)^k$. Note that $f_k(n)=0$ if $k$ is odd.

Determine all graphs $G$ with the following two properties: $\bullet$ G contains at least one Hamilton path. $\bullet$ For any pair of vertices, $u, v \in G$, if there is a Hamilton path from $u$ to $v$ then the edge $uv$ is in the graph $G$

Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$. Let $N$ be the reflection of $M$ in $BC$. The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$. Determine the value of $\angle AKC$. (A.Blinkov)

If $ a,b,n$ are positive integers, number of solutions of the equaition $ a^2 \plus{} b^4 \equal{} 5^n$ is $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Let $ABC$ be an equilateral triangle. Let $A_1,B_1,C_1$ be interior points of $ABC$ such that $BA_1=A_1C$, $CB_1=B_1A$, $AC_1=C_1B$, and $$\angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$$ Let $BC_1$ and $CB_1$ meet at $A_2,$ let $CA_1$ and $AC_1$ meet at $B_2,$ and let $AB_1$ and $BA_1$ meet at $C_2.$ Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2, BB_1B_2$ and $CC_1C_2$ all pass through two common points. (Note: a scalene triangle is one where no two sides have equal length.) [i]Proposed by Ankan Bhattacharya, USA[/i]

There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$. 1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$? 2) What is the number of students from club $A$?

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could [i]not[/i] be the total value of the four coins, in cents? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 25 \qquad \textbf{(C)}\ 35 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 55$

Real numbers $ a$ and $ b$ satisfy the equations $ 3^{a} \equal{} 81^{b \plus{} 2}$ and $ 125^{b} \equal{} 5^{a \minus{} 3}$. What is $ ab$? $ \textbf{(A)} \minus{} \!60 \qquad \textbf{(B)} \minus{} \!17 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 60$

Let $ABC$ be a triangle with $\angle BCA=20.$ Let points $D\in(BC), F\in(AC)$ be such that $CD=DF=FB=BA.$ Find $\angle ADF.$

Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.

Is it true that for any polynomial $P(x)$ with real coefficients of degree $2023$, there exists a natural number $n$ such that the equation $P(x) = n^{-100}$ has no rational root?

The sequence $(a_n)$ is defined with the recursion $a_{n + 1} = 5a^6_n + 3a^3_{n-1} + a^2_{n-2}$ for $n\ge 2$ and the set of initial values $\{a_0, a_1, a_2\} = \{2013, 2014, 2015\}$. (That is, the initial values are these three numbers in any order.) Show that the sequence contains no sixth power of a natural number.

Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence: $$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$ where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that $$\lim_{n\to\infty}a_n=0$$ [i]Proposed by Laurențiu Modan[/i]

There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? $\text{(A)}\ 12\% \qquad \text{(B)}\ 20\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 60\% \qquad \text{(E)}\ 66\frac{2}{3}\% $

Let $S$ be a set and let $\circ$ be a binary operation on $S$ satisfying two laws $$x\circ x=x \text{ for all } x \text{ in } S, \text{ and}$$ $$(x \circ y) \circ z= (y\circ z) \circ x \text{ for all } x,y,z \text{ in } S.$$ Show that $\circ$ is associative and commutative.