Find the largest number among the following numbers: $ \textbf{(A) }30^{30}\qquad\textbf{(B) }50^{10}\qquad\textbf{(C) }40^{20}\qquad\textbf{(D) }45^{15}\qquad\textbf{(E) }5^{60}$

On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

Find all triples of positive integers $(a,b,p)$ with $a,b$ positive integers and $p$ a prime number such that $2^a+p^b=19^a$

Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that \[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]

Is it true that if a subgroup $G \leq \text{Sym}(\mathbb{N})$ is $n$-transitive for every positive integer $n$, then every group automorphism of $G$ extends to a group automorphism of $\text{Sym}(\mathbb{N})$?

In Tlaxcala, there is a transportation system that works through buses that travel from one city to another in one direction . A set $S$ of cities is said [i]beautiful[/i] if it contains at least three different cities and from each city $A$ in $S$ at least two buses depart, each one goes directly to a different city in $S$ and none of them is $A$ (if there is a direct bus from $A$ to a city $B$ in $S$, there is not necessarily a direct bus from $B$ to $A$). Show that if there exists a beautiful set of cities $S$, then there exists a beautiful $T$ subset of $S$, such that for any two cities in $T$, you can get from one to another by taking buses that only pass through cities in $T$. Note: A bus goes directly from one city to another if it does not pass through any other city.

Let the two tangents from a point $A$ outside a circle $k$ touch $k$ at $M$ and $N$. A line $p$ through $A$ intersects $k$ at $B$ and $C$, and $D$ is the midpoint of $MN$. Prove that $MN$ bisects the angle $BDC$

Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard: [list] [*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line. [/list] Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Let $a$ and $b$ be positive integers. When $a^{2}+b^{2}$ is divided by $a+b,$ the quotient is $q$ and the remainder is $r.$ Find all pairs $(a,b)$ such that $q^{2}+r=1977$.

Let $ a$ and $b$ be positive integers. Prove that there exist positive integers $ x $ and $ y $ such that \[ \binom{x+y}{2} = ax + by . \]

Let $O$ be a center of circle which passes through vertices of quadrilateral $ABCD$, which has perpendicular diagonals. Prove that sum of distances of point $O$ to sides of quadrilateral $ABCD$ is equal to half of perimeter of $ABCD$.

What is the $101$st smallest integer which can represented in the form $3^a+3^b+3^c$, where $a,b,$ and $c$ are integers? [i]Proposed by Dilhan Salgado[/i]

Let $a, b, c$ be real numbers, and define $S_n = a^n + b^n + c^n$ for positive integers $n$. Suppose that $S_1, S_2, S_3$ are integers satisfying $6 | 5S_1 - 3S_2 - 2S_3$. Show that $S_n$ is an integer for all positive integers $n$.

Given the polynomial $p(x) = x^2 + x - 70$, do there exist integers $0<m<n$, so that $p(m)$ is divisible by $n$ and $p(m+1)$ is divisible by $n+1$? [i]Proposed by Nairy Sedrakyan[/i]

In a triangle $ABC$ , $B_{1}$ and $C_{1}$ are the midpoints of $AC$ and $AB$ respectively, and $I$ is the incenter. The lines $B_{1}I$ and $C_{1}I$ meet $AB$ and $AC$ respectively at $C_{2}$ and $B_{2}$ . If the areas of $\Delta ABC$ and $\Delta AB_{2}C_{2}$ are equal, find $\angle{BAC}$ .

Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.

Fill each cell with an integer from $1$-$7$ so each number appears exactly once in each row and column. In each ``cage" of three cells, the three numbers must be valid lengths for the sides of a non-degenerate triangle. Additionally, if a cage has an ``A", the triangle must be acute, and if the cage has an ``R", the triangle must be right. [asy] for(int i = 0; i < 8; ++i){ draw((0,i) -- (7,i)^^(i,0)--(i,7), gray(0.7)); } draw((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle); draw((5.1,6.1) -- (6.1, 6.1) -- (6.1, 5.1) -- (6.9, 5.1) -- (6.9, 6.9) --(5.1, 6.9) -- cycle); label(scale(0.5)*"R", (5.1, 6.9), SE); draw((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle); draw((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle); draw(shift((3,0))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); draw(shift((3,-1))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); label(scale(0.5)*"A", (6.1, 4.9), SE); draw(shift((2,-2))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); draw((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle); label(scale(0.5)*"R", (4.1, 3.9), SE); draw((0.1, 2.1) -- (0.1, 3.9) -- (1.9, 3.9) -- (1.9, 3.1) -- (0.9, 3.1) -- (0.9, 2.1) -- cycle); draw(shift((0, -3))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); label(scale(0.5)*"R", (1.1, 2.9), SE); draw(shift((-2, -6)) * ((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle)); label(scale(0.5)*"A", (0.1, 0.9), SE); draw(shift((0,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); label(scale(0.5)*"A", (4.1, 1.9), SE); draw(shift((2,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); [/asy]

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.

Four faces of tetrahedron $ABCD$ are congruent triangles whose angles form an arithmetic progression. If the lengths of the sides of the triangles are $a < b < c$, determine the radius of the sphere circumscribed about the tetrahedron as a function on $a, b$, and $c$. What is the ratio $c/a$ if $R = a \ ?$

Camila writes down five positive integers. The unique mode of these integers is $2$ greater than their median, and the median is $2$ greater than their arithmetic mean. What is the least possible value for the mode? $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad\textbf{(E) }13$

Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$. b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.

Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get? $\textbf{(A)}\ 200 \qquad \textbf{(B)}\ 202 \qquad \textbf{(C)}\ 220 \qquad \textbf{(D)}\ 380 \qquad \textbf{(E)}\ 398$

Let $M$ be a set of $n \ge 4$ points in the plane, no three of which are collinear. Initially these points are connected with $n$ segments so that each point in $M$ is the endpoint of exactly two segments. Then, at each step, one may choose two segments $AB$ and $CD$ sharing a common interior point and replace them by the segments $AC$ and $BD$ if none of them is present at this moment. Prove that it is impossible to perform $n^3 /4$ or more such moves. [i]Proposed by Vladislav Volkov, Russia[/i]