Let ${A}$ be a finite set and ${\rightarrow}$ be a binary relation on it such that for any ${a,b,c \in A}$, if ${a\neq b}, {a \rightarrow c}$ and ${b \rightarrow c}$ then either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Let ${B,\,B \subset A}$ be minimal with the property: for any ${a \in A \setminus B}$ there exists ${b \in B}$, such that either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Supposing that ${A}$ has at most ${k}$ elements that are pairwise not in relation ${\rightarrow}$, prove that ${B}$ has at most ${k}$ elements.

Let $k$ and $n$ be integers such that $k\ge 2$ and $k \le n \le 2k-1$. Place rectangular tiles, each of size $1 \times k$, or $k \times 1$ on a $n \times n$ chessboard so that each tile covers exactly $k$ cells and no two tiles overlap. Do this until no further tile can be placed in this way. For each such $k$ and $n$, determine the minimum number of tiles that such an arrangement may contain.

Solve in the set $R$ the equation $$\frac{3x+3}{\sqrt{x}}-\frac{x+1}{\sqrt{x^2-x+1}}=4$$

A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?

King Tin writes the first $n$ perfect squares on the royal chalkboard, but he omits the first (so for n = $3$, he writes $4$ and $9$). His son, Prince Tin, comes along and repeats the following process until only one number remains: [i]He erases the two greatest numbers still on the board, calls them a and b, and writes the value of $\frac{ab-1}{a+b-2}$ on the board. [/i]Let $S(n)$ be the last number that Prince Tin writes on the board. Let $\lim_{n\to \infty} S(n) = r$, meaning that $r$ is the unique number such that for every $\epsilon > 0$ there exists a positive integer $N$ so that $|S(n) - r| < \epsilon$ for all $n > N$. If $r$ can be written in simplest form as $\frac{m}{n}$, find $m + n$.

Find all non-negative integers $n$ for which the equation \[ {\left( x^2 + y^2 \right)}^n = {(xy)}^{2018} \] admits positive integral solutions.

Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.

Which pattern of identical squares could NOT be folded along the lines shown to form a cube? [asy] unitsize(12); draw((0,0)--(0,-1)--(1,-1)--(1,-2)--(2,-2)--(2,-3)--(4,-3)--(4,-2)--(3,-2)--(3,-1)--(2,-1)--(2,0)--cycle); draw((1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)); draw((7,0)--(8,0)--(8,-1)--(11,-1)--(11,-2)--(8,-2)--(8,-3)--(7,-3)--cycle); draw((7,-1)--(8,-1)--(8,-2)--(7,-2)); draw((9,-1)--(9,-2)); draw((10,-1)--(10,-2)); draw((14,-1)--(15,-1)--(15,0)--(16,0)--(16,-1)--(18,-1)--(18,-2)--(17,-2)--(17,-3)--(16,-3)--(16,-2)--(14,-2)--cycle); draw((15,-2)--(15,-1)--(16,-1)--(16,-2)--(17,-2)--(17,-1)); draw((21,-1)--(22,-1)--(22,0)--(23,0)--(23,-2)--(25,-2)--(25,-3)--(22,-3)--(22,-2)--(21,-2)--cycle); draw((23,-1)--(22,-1)--(22,-2)--(23,-2)--(23,-3)); draw((24,-2)--(24,-3)); draw((28,-1)--(31,-1)--(31,0)--(32,0)--(32,-2)--(31,-2)--(31,-3)--(30,-3)--(30,-2)--(28,-2)--cycle); draw((32,-1)--(31,-1)--(31,-2)--(30,-2)--(30,-1)); draw((29,-1)--(29,-2)); label("(A)",(0,-0.5),W); label("(B)",(7,-0.5),W); label("(C)",(14,-0.5),W); label("(D)",(21,-0.5),W); label("(E)",(28,-0.5),W); [/asy]

Consider the harmonic series $\sum_{n\ge1}\frac1n=1+\frac12+\frac13+\ldots$. Prove that every positive rational number can be obtained as an unordered partial sum of this series. (An unordered partial sum may skip some of the terms $\frac1k$.)

How many moles of ions are present in $250$ mL of a $4.4$ M solution of sodium sulfate? ${ \textbf{(A)}\hspace{.05in}1.1 \qquad\textbf{(B)}\hspace{.05in}2.2 \qquad\textbf{(C)}\hspace{.05in}3.3 \qquad\textbf{(D)}\hspace{.05in}13}\qquad $

Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

In regular triangular pyramid $P-ABC$, $PO$ is its height, $M$ is the midpoint of $PO$. Draw the plane that passes $AM$ and parallel to $BC$. Now the triangular pyramid is divided into two parts. Find the ratio of their volume.

Let's call a power of two [i]compact[/i] if it can be represented as the sum of no more than $10^9$ not necessarily distinct factorials of positive integer numbers. Prove that the set of compact powers of two is finite.

a) Given $5$ points on a plane, no three of which lie on one line. Prove that four of these points can be taken as vertices of a convex quadrilateral. b) Inside a square, consider a convex quadrilateral and inside the quadrilateral, take a point $A$. It so happens that no three of the $9$ points — the vertices of the square, of the quadrilateral and $A$ — lie on one line. Prove that $5$ of these points are vertices of a convex pentagon.

Daniel chooses a positive integer $n$ and tells Ana. With this information, Ana chooses a positive integer $k$ and tells Daniel. Daniel draws $n$ circles on a piece of paper and chooses $k$ different points on the condition that each of them belongs to one of the circles he drew. Then he deletes the circles, and only the $k$ points marked are visible. From these points, Ana must reconstruct at least one of the circumferences that Daniel drew. Determine which is the lowest value of $k$ that allows Ana to achieve her goal regardless of how Daniel chose the $n$ circumferences and the $k$ points.

Find the minimum positive integer $n\ge 3$, such that there exist $n$ points $A_1,A_2,\cdots, A_n$ satisfying no three points are collinear and for any $1\le i\le n$, there exist $1\le j \le n (j\neq i)$, segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i+1}$, where $A_{n+1}=A_1$

There is an isosceles triangle which follows the following: $ \bar{AB}=\bar{AC}=5, \bar{BC}=6 $ D,E are points on $ \bar{AC} $ which follows $ \bar{AD}=1, \bar{EC}=2 $ If the extent of $ \triangle $ BDE = S, Find 15S.

Circle $\omega$, centered at $X$, is internally tangent to circle $\Omega$, centered at $Y$, at $T$. Let $P$ and $S$ be variable points on $\Omega$ and $\omega$, respectively, such that line $PS$ is tangent to $\omega$ (at $S$). Determine the locus of $O$ -- the circumcenter of triangle $PST$.

Find all $a$ such that for any positive integer $n$, the number $an(n+2)(n+3)(n+4)$ is an integer. (Author: O. Podlipski) [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=427802](similar to Problem 5 of grade 9)[/url] Same problem for grades 10 and 11

Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area$(S_1) = 441$ and area$(S_2) = 440$. [asy] size(250); real a=15, b=5; real x=a*b/(a+b), y=a/((a^2+b^2)/(a*b)+1); pair A=(0,b), B=(a,0), C=origin, X=(y,0), Y=(0, y*b/a), Z=foot(Y, A, B), W=foot(X, A, B); draw(A--B--C--cycle); draw(W--X--Y--Z); draw(shift(-(a+b), 0)*(A--B--C--cycle^^(x,0)--(x,x)--(0,x))); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$A$", (A.x-a-b,A.y), dir(point--A)); label("$B$", (B.x-a-b,B.y), dir(point--B)); label("$C$", (C.x-a-b,C.y), dir(point--C)); label("$S_1$", (x/2-a-b, x/2)); label("$S_2$", intersectionpoint(W--Y, X--Z)); dot(A^^B^^C^^(-a-b,0)^^(-b,0)^^(-a-b,b));[/asy]

There is a unique function $f: \mathbb{N} \to \mathbb{R}$ such that $f(1) > 0$ and such that \[\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right) = 1\] for all $n \ge 1$. What is $f(2018^{2019})$?

Find the greatest value of positive integer $ x$ , such that the number $ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$ is a perfect square .

In acute triangle $ABC$, $AB > AC$. $D$ and $E$ are the midpoints of $AB$, $AC$ respectively. The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$. The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$. Prove that $AP = AQ$.