Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and \[(10a+b)^{c/d}=e^{f/g}.\] Given that $h>i$, evaluate $10h+i$. [i]Proposed by James Lin[/i]

Two three-dimensional objects are said to have the same coloring if you can orient one object (by moving or turning it) so that it is indistinguishable from the other. For example, suppose we have two unit cubes sitting on a table, and the faces of one cube are all black except for the top face which is red, and the the faces of the other cube are all black except for the bottom face, which is colored red. Then these two cubes have the same coloring. In how many different ways can you color the edges of a regular tetrahedron, coloring two edges red, two edges black, and two edges green? (A regular tetrahedron has four faces that are each equilateral triangles. The figure below depicts one coloring of a tetrahedron, using thick, thin, and dashed lines to indicate three colors.)

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by [i]Morteza Saghafian[/i] and [i]Amir Jafari[/i]

The expressions $ a\plus{}bc$ and $ (a\plus{}b)(a\plus{}c)$ are: $ \textbf{(A)}\ \text{always equal} \qquad \textbf{(B)}\ \text{never equal} \qquad \textbf{(C)}\ \text{equal whenever }a\plus{}b\plus{}c\equal{}1 \\ \textbf{(D)}\ \text{equal when }a\plus{}b\plus{}c\equal{}0 \qquad \textbf{(E)}\ \text{equal only when }a\equal{}b\equal{}c\equal{}0$

Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in \{1,2, 3, \ldots, 2027\},$ $$f(n)=\begin{cases} 1 & \text{if } $n$ \text{ is a perfect square}, \\ 0 & \text{otherwise.} \end{cases}$$ Suppose that $\tfrac{a}{b}$ is the coefficient of $x^{2025}$ in $f,$ where $a$ and $b$ are integers such that $\gcd(a,b)=1.$ Compute the unique integer $r$ between $0$ and $2026$ (inclusive) such that $a-rb$ is divisible by $2027.$ (Note that $2027$ is prime.)

In the country of PUMACsboro, there are $n$ distinct cities labelled $1$ through $n$. There is a rail line going from city $i$ to city $j$ if and only if $i<j$; you can only take this rail line from city $i$ to city $j$. What is the smallest possible value of $n$ such that if each rail line's track is painted orange or black, you can always take the train between $2019$ cities on tracks that are all the same color? (This means there are some cities $c_1,c_2,\dots,c_{2019}$ such that there is a rail line going from city $c_i$ to $c_{i+1}$ for all $1\leq i\leq 2018$ and their rail lines' tracks are either all orange or all black.)

Paul is in the desert and has a pile of gypsum crystals. No matter how he divides the pile into two nonempty piles, at least one of the resulting piles has a number of crystals that, when written in base $10,$ has a sum of digits at least $7.$ Given that Paul’s initial pile has at least two crystals, compute the smallest possible number of crystals in the initial pile.

For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$.

In a country, mathematicians chose an $\alpha> 2$ and issued coins in denominations of 1 ruble, as well as $\alpha ^k$ rubles for each positive integer k. $\alpha$ was chosen so that the value of each coins, except the smallest, was irrational. Is it possible that any natural number of rubles can be formed with at most 6 of each denomination of coins?

Prove that from a set of ten distinct two-digit numbers, it is always possible to find two disjoint subsets whose members have the same sum.

Given natural number $k$ with a property "if $n$ is divisible by $k$, than the number, obtained from $n$ by reversing the order of its digits is also divisible by $k$". Prove that the $k$ is a divisor of $99$.

Let triangle $\triangle{ABC}$ have $AB=90$ and $AC=66$. Suppose that the line $IG$ is perpendicular to side $BC$, where $I$ and $G$ are the incenter and centroid, respectively. Find the length of $BC$.

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

Let $\triangle$$ABC$ a triangle with circumcircle $\Gamma$. Suppose there exist points $R$ and $S$ on sides $AB$ and $AC$, respectively, such that $BR=RS=SC$. A tangent line through $A$ to $\Gamma$ meet the line $RS$ at $P$. Let $I$ the incenter of triangle $\triangle$$ARS$. Prove that $PA=PI$

Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$

Find how many are the numbers of $2013$ digits $d_1d_2…d_{2013}$ with odd digits $d_1,d_2,…,d_{2013}$ such that the sum of $1809$ terms $$d_1 \cdot d_2+d_2\cdot d_3+…+d_{1809}\cdot d_{1810}$$ has remainder $1$ when divided by $4$ and the sum of $203$ terms $$d_{1810}\cdot d_{1811}+d_{1811}\cdot d_{1812}+…+d_{2012}\cdot d_{2013}$$ has remainder $1$ when dividing by $4$.

$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$ $(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$

Let $ M $ be a point in the interior of a triangle $ ABC, $ let $ D $ be the intersection of $ AM $ with $ BC, $ let $ E $ be the intersection of $ M $ with AC, let $ F $ be the intersection of $ CM $ with $ AB. $ Knowing that the expression $$ \frac{MA}{MD}\cdot \frac{MB}{ME}\cdot \frac{MC}{MF} $$ is minimized, describe the point $ M. $

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

An event occurs many years ago. It occurs periodically in $x$ consecutive years, then there is a break of $y$ consecutive years. We know that the event occured in $1964$, $1986$, $1996$, $2008$ and it didn't occur in $1976$, $1993$, $2006$, $2013$. What is the first year in that the event will occur again?

Two circles, $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ meet $\omega_1$ again at $C$, and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ meets the circle $\omega$ through $A, O_1,O_2$ again at $F$. Prove that the length of the segment $EF$ is equal to the diameter of $\omega$.

Let $a>0$. Show that $$ \lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}$$ lies between $e^a$ and $e^{a+1}.$

Find the greatest value of $C$ for which, for any $x, y, z,u$, and such that for $0\le x\le y \le z\le u$, holds the inequality $$(x + y +z + u)^2 \ge Cyz .$$

How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

$m\times n$ grid is tiled by mosaics $2\times2$ and $1\times3$ (horizontal and vertical). Prove that the number of ways to choose a $1\times2$ rectangle (horizontal and vertical) such that one of its cells is tiled by $2\times2$ mosaic and the other cell is tiled by $1\times3$ mosaic [horizontal and vertical] is an even number.