A connected checkered figure is drawn on a checkered paper. It is known that the figure can be cut both into $2\times 2$ squares and into (possibly rotated) [url=https://upload.wikimedia.org/wikipedia/commons/thumb/3/30/Tetromino-skew2.svg/1200px-Tetromino-skew2.svg.png]skew-tetrominoes[/url]. Prove that there is a hole in the figure. [i]Proposed by Y. Markelov and A. Sairanov[/i]

What is the value of $ x$ if $ |x \minus{} 1| \equal{} |x \minus{} 2|$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {3}{2}\qquad \textbf{(E)}\ 2$

Let $n>1$ be an even positive integer. An $2n \times 2n$ grid of unit squares is given, and it is partitioned into $n^2$ contiguous $2 \times 2$ blocks of unit squares. A subset $S$ of the unit squares satisfies the following properties: (i) For any pair of squares $A,B$ in $S$, there is a sequence of squares in $S$ that starts with $A$, ends with $B$, and has any two consecutive elements sharing a side; and (ii) In each of the $2 \times 2$ blocks of squares, at least one of the four squares is in $S$. An example for $n=2$ is shown below, with the squares of $S$ shaded and the four $2 \times 2$ blocks of squares outlined in bold. [asy] size(2.5cm); fill((0,0)--(4,0)--(4,1)--(0,1)--cycle,mediumgrey); fill((0,0)--(0,4)--(1,4)--(1,0)--cycle,mediumgrey); fill((0,3)--(4,3)--(4,4)--(0,4)--cycle,mediumgrey); fill((3,0)--(3,4)--(4,4)--(4,0)--cycle,mediumgrey); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw((1,0)--(1,4)); draw((2,0)--(2,4),linewidth(1)); draw((3,0)--(3,4)); draw((0,1)--(4,1)); draw((0,2)--(4,2),linewidth(1)); draw((0,3)--(4,3)); [/asy] In terms of $n$, what is the minimum possible number of elements in $S$?

A number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}$. What is the probability that $m^n$ has a units digit of $1$? $\textbf{(A) } \frac{1}{5} \qquad \textbf{(B) } \frac{1}{4} \qquad \textbf{(C) } \frac{3}{10} \qquad \textbf{(D) } \frac{7}{20} \qquad \textbf{(E) } \frac{2}{5} $

$G$ is a group that order of each element of it Commutator group is finite. Prove that subset of all elemets of $G$ which have finite order is a subgroup og $G$.

An irregular octahedron has eight faces that are equilateral triangles of side length $2$. However, instead of each vertex having four "neighbors" (vertices that share an edge with it) like in a regular octahedron, for this octahedron, two of the vertices have exactly three neighbors, two of the vertices have exactly four neighbors, and two of the vertices have exactly five neighbors. Compute the volume of this octahedron. [i]Proposed by Connor Gordon[/i]

At most how many points with integer coordinates are there over a circle with center of $(\sqrt{20}, \sqrt{10})$ in the $xy$-plane? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None} $

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.

In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$. [i]Proposed by Harrison Wang[/i]

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.

Cat and Claire are having a discussion about their favorite positive two-digit numbers. [b]Cat:[/b] My number has a $1$ in its tens digit. Is it possible that your number is a multiple of my number? [b]Claire:[/b] No, however, my number is not prime. Additionally, if I told you the two digits of my number, you still wouldn't know my number. [b]Cat:[/b] Aha, my number and your number aren't relatively prime! [b]Claire:[/b] Then our numbers must share the same ones digit! What is the product of Cat and Claire's numbers?

The diameter $\overline{AB}$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and the line $ED$ is perpendicular to the line $AD$. Segment $\overline{AE}$ intersects the circle at point $C$ between $A$ and $E$. What is the area of $\triangle ABC$? $\textbf{(A) \ } \frac{120}{37}\qquad \textbf{(B) \ } \frac{140}{39}\qquad \textbf{(C) \ } \frac{145}{39}\qquad \textbf{(D) \ } \frac{140}{37}\qquad \textbf{(E) \ } \frac{120}{31}$

Is it true that for each $ n$, the regular $ 2n$-gon is a projection of some polyhedron having not greater than $ n \plus{} 2$ faces?

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]