Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell [i]correct [/i] if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be [i]correct[/i]?

Let \(\mathbb{Z}^2\) denote the set of points in the Euclidean plane with integer coordinates. Find all functions \(f : \mathbb{Z}^2 \to [0,1]\) such that for any point \(P\), the value assigned to \(P\) is the average of all the values assigned to points in \(\mathbb{Z}^2\) whose Euclidean distance from \(P\) is exactly 2020.

Find all non negative integer pairs $(a,b)$ such that $3^a5^b-2024$ is a square of a positive integer.

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

Let $P$ be a point and $\omega$ be a circle with center $O$ and radius $r$ . We define the power of the point $P$ with respect to the circle $\omega$ to be $OP^2 - r^2$ , and we denote this by pow $(P, \omega)$. We define the radical axis of two circles $\omega_1$ and $\omega_2$ to be the locus of all points P such that pow $(P,\omega_1) =$ pow $(P,\omega_2)$. It turns out that the pairwise radical axes of three circles are either concurrent or pairwise parallel. The concurrence point is referred to as the radical center of the three circles. In $\vartriangle ABC$, let $I$ be the incenter, $\Gamma$ be the circumcircle, and $O$ be the circumcenter. Let $A_1,B_1,C_1$ be the point of tangency of the incircle of $\vartriangle ABC$ with side $BC,CA, AB$, respectively. Let $X_1,X_2 \in \Gamma$ such that $X_1,B_1,C_1,X_2$ are collinear in this order. Let $M_A$ be the midpoint of $BC$, and define $\omega_A$ as the circumcircle of $\vartriangle X_1X_2M_A$. Define $\omega_B$ ,$\omega_C$ analogously. The goal of this problem is to show that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ lies on line $OI$. (a) Let$ A'_1$ denote the intersection of $B_1C_1$ and $BC$. Show that $\frac{A_1B}{A_1C}=\frac{A'_1B}{A'_1C}$. (b) Prove that $A_1$ lies on $\omega_A$. (c) Prove that $A_1$ lies on the radical axis of $\omega_B$ and $\omega_C$ . (d) Prove that the radical axis of $\omega_B$ and $\omega_C$ is perpendicular to $B_1C_1$. (e) Prove that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ is the orthocenter of $\vartriangle A_1B_1C_1$. (f ) Conclude that the radical center of $\omega_A$, $\omega_B$, $\omega_C$ , $O$, and $I$ are collinear. PS. You had better use hide for answers.

Let $A_1B_3A_2B_1A_3B_2$ be a cyclic hexagon such that $A_1B_1,A_2B_2,A_3B_3$ intersect at one point. Let $C_1=A_1B_1\cap A_2A_3,C_2=A_2B_2\cap A_1A_3,C_3=A_3B_3\cap A_1A_2$. Let $D_1$ be the point on the circumcircle of the hexagon such that $C_1B_1D_1$ touches $A_2A_3$. Define $D_2,D_3$ analogously. Show that $A_1D_1,A_2D_2,A_3D_3$ meet at one point.

In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$.

A SET cards have four characteristics: number, color, shape, and shading, each of which has $3$ values. A SET deck has $81$ cards, one for each combination of these values. A SET is three cards such that, for each characteristic, the values of the three cards for that characteristics are either all the same or all different. In how many ways can you replace each SET card in the deck with another SET card (possibly the same), with no card used twice, such that any three cards that were a SET before are still a SET? (Alternately, a SET card is an ordered $4$-tuple of $0$s, $1$s, and $2$s, and three cards form a SET if their sum is ($0, 0, 0, 0$) mod $3$, for instance, ($0, 1, 2, 2$), ($1, 0, 2, 1$), and ($2, 2, 2, 0$) form a SET. How many permutations of the SET cards maintain SET-ness?)

Let $ABCD$ be a convex quadrilateral. Show that there exists a square $A'B'C'D'$ (Vertices maybe ordered clockwise or counter-clockwise) such that $A \not = A', B \not = B', C \not = C', D \not = D'$ and $AA',BB',CC',DD'$ are all concurrent.

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

Alex, Pei-Hsin, and Edward got together before the contest to send a mailing to all the invited schools. Pei-Hsin usually just stuffs the envelopes, but if Alex leaves the room she has to lick them as well and has a $25\%$ chance of dying from an allergic reaction before he gets back. Licking the glue makes Edward a bit psychotic, so if Alex leaves the room there is a $20\%$ chance that Edward will kill Pei-Hsin before she can start licking envelopes. Alex leaves the room and comes back to find Pei-Hsin dead. What is the probability that Edward was responsible?

Anja has to write $2014$ integers on the board such that arithmetic mean of any of the three numbers is among those $2014$ numbers. Show that this is possible only when she writes nothing but $2014$ equal integers.

Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.

Real numbers $x_1, x_2, x_3, \cdots , x_n$ satisfy $x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2 = 1$. Show that \[ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+ x_1^2 + x_2^2 + x_3^2 + \cdots + x_n^2} < \sqrt{\frac n2} . \]

How many digits are there in the decimal expression of $2^{100}$ ?

What is the greatest integer $k$ which makes the statement "When we take any $6$ subsets with $5$ elements of the set $\{1,2,\dots, 9\}$, there exist $k$ of them having at least one common element." true? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

In an acute-angled triangle $ABC$, point $I$ is the incenter, $H$ is the orthocenter, $O$ is the center of the circumscribed circle, $T$ and $K$ are the touchpoints of the $A$-excircle and incircle with side $BC$ respectively. It turned out that the segment $TI$ is passing through the point $O$. Prove that $HK$ is the angle bisector of $\angle BHC$. (Matvii Kurskyi)

Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.

Determine all integers $n \geq 2$ such that there exists a permutation $x_0, x_1, \ldots, x_{n - 1}$ of the numbers $0, 1, \ldots, n - 1$ with the property that the $n$ numbers $$x_0, \hspace{0.3cm} x_0 + x_1, \hspace{0.3cm} \ldots, \hspace{0.3cm} x_0 + x_1 + \ldots + x_{n - 1}$$ are pairwise distinct modulo $n$.

Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2+b^2+c^2+a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m\rfloor$.

If $x,y$ are integers and $17$ divides both $x^2 -2xy + y^2 -5x + 7y$ and $x^2 - 3xy + 2y^2 + x - y$ , then prove that $17$ divides $xy - 12x + 15y$.

Prove that $(2m)!(2n)!$ is a multiple of $m!n!(m+n)!$ for any non-negative integers $m$ and $n$.

A circle $ C$ has two parallel tangents $ L'$ and$ L"$. A circle $ C'$ touches $ L'$ at $ A$ and $ C$ at $ X$. A circle $ C"$ touches $ L"$ at $ B$, $ C$ at $ Y$ and $ C'$ at $ Z$. The lines $ AY$ and $ BX$ meet at $ Q$. Show that $ Q$ is the circumcenter of $ XYZ$

Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$.