Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.

Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set \[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}. \]What is the minimum possible value of \[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2 \]$ \textbf{(A)}\ 30\qquad \textbf{(B)}\ 32\qquad \textbf{(C)}\ 34\qquad \textbf{(D)}\ 40\qquad \textbf{(E)}\ 50$

In triangle $ABC, \angle A= 45^o, BH$ is the altitude, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

Prove that if all the edges of the tetrahedron are equal triangles (such a tetrahedron is called equilateral), then its projection on the plane of a face is a triangle.

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$.

$\textbf{(a)}$ On the sides of triangle $ABC$ we consider the points $M\in \overline{BC}$, $N\in \overline{AC}$ and $P\in \overline{AB}$ such that the quadrilateral $MNAP$ with right angles $\angle MNA$ and $\angle MPA$ has an inscribed circle. Prove that $MNAP$ has to be a kite. $\textbf{(b)}$ Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too? [i] (Călin Udrea) [/i]

Let $p$, $q$, and $r$ be primes satisfying \[ pqr = 189999999999999999999999999999999999999999999999999999962. \] Compute $S(p) + S(q) + S(r) - S(pqr)$, where $S(n)$ denote the sum of the decimals digits of $n$. [i]Proposed by Evan Chen[/i]

Eric plans to compete in a triathlon. He can average $ 2$ miles per hour in the $ \tfrac{1}{4}$-mile swim and $ 6$ miles per hour in the $ 3$-mile run. His goal is to finish the triathlon in $ 2$ hours. To accomplish his goal what must his average speed, in miles per hour, be for the $ 15$-mile bicycle ride? $ \textbf{(A)}\ \frac{120}{11} \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ \frac{56}{5} \qquad \textbf{(D)}\ \frac{45}{4} \qquad \textbf{(E)}\ 12$

How many positive integers smaller than $1000$ and not divisible by $5$ and by $7$ are there?

Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

In the country of PUMaC-land, there are $5$ villages and $3$ cities. Vedant is building roads between the $8$ settlements according to the following rules: a) There is at most one road between any two settlements; b) Any city has exactly three roads connected to it; c) Any village has exactly one road connected to it; d) Any two settlements are connected by a path of roads. In how many ways can Vedant build the roads?

Among any four people at a party there is one who has met the three others before the party. Show that among any four people at the party there must be one who has met everyone at the party before the party

$ 1994$ girls are seated at a round table. Initially one girl holds $ n$ tokens. Each turn a girl who is holding more than one token passes one token to each of her neighbours. a.) Show that if $ n < 1994$, the game must terminate. b.) Show that if $ n \equal{} 1994$ it cannot terminate.

The sum of the digits of a natural number is $k{}.$ What is the largest possible sum of digits for[list=a] [*] the square of this number; [*]the fourth power of this number, [/list] given that $k\geqslant 4.$ [i]From the folklore[/i]

Denote by $S(n)$ the sum of digits of $n$. Given a positive integer $N$, we consider the following process: We take the sum of digits $S(N)$, then take its sum of digits $S(S(N))$, then its sum of digits $S(S(S(N)))$... We continue this until we are left with a one-digit number. We call the number of times we had to activate $S(\cdot)$ the [b]depth[/b] of $N$. For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$, and the depth of 45 is 1, since $S(45)=9$. [list=a] [*] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [*] Define $x(n)$ to be the [u]minimal[/u] positive integer with depth $n$. Find the residue of $x(5776)\mod 6$. [*] Find the residue of $x(5776)-x(5708)\mod 2016$. [/list]

$\triangle ABC$ is a triangle and $D,E,F$ are points on $BC$, $CA$, $AB$ respectively. It is given that $BF=BD$, $CD=CE$ and $\angle BAC=48^{\circ}$. Find the angle $\angle EDF$ $ \textbf{(A) }64^{\circ}\qquad\textbf{(B) }66^{\circ}\qquad\textbf{(C) }68^{\circ}\qquad\textbf{(D) }70^{\circ}\qquad\textbf{(E) }72^{\circ} $

Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$. [i]Proposed by Evan Chen[/i]

$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$.

Solve in $R$ the equation: $8x^3-4x^2-4x+1=0$

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Find three distinct positive integers with the least possible sum such that the sum of the reciprocals of any two integers among them is an integral multiple of the reciprocal of the third integer.

How many different ways are there to paint the sides of a tetrahedron with exactly $4$ colors? Each side gets its own color, and two colorings are the same if one can be rotated to get the other.

Let $ a,b,c$ be nonnegative real numbers. Suppose that $ a\plus{}b\plus{}c\ge abc$. Prove that: $ a^2\plus{}b^2\plus{}c^2 \ge abc.$

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

In the blackboard there are drawn $25$ points as shown in the figure. Gastón must choose $4$ points that are vertices of a square. In how many different ways can he make this choice?$$\begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \\\bullet & \bullet & \bullet & \bullet & \bullet \end{matrix}$$