An ant is currently on a vertex of the top face on a 6-sided die. The ant wants to travel to the opposite vertex of the die (the vertex that is farthest from the start), and the ant can travel along edges of the die to other vertices that are on the top face of the die. Every second, the ant picks a valid edge to move along, and the die randomly flips to an adjacent face. If the ant is on any of the bottom vertices after the flip, it is crushed and dies. What is the probability that the ant makes it to its target? (If the ant makes it to the target and the die rolls to crush it, it achieved its dreams before dying, so this counts.) [i]Proposed by Lohith Tummala[/i]

How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?

Determine all positive integers that cannot be written as $\frac{a}{b} + \frac{a+1}{b+1}$ where $a$ and $b$ are positive integers.

Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?

For $a,n$ positive integers we show number of different integer 10-tuples $ (x_1,x_2,...,x_{10})$ on $ (mod n)$ satistfying $x_1x_2...x_{10}=a (mod n)$ with $f(a,n)$. Let $a,b$ given positive integers , a) Prove that there exist a positive integer $c$ such that for all $n\in \mathbb{Z^+}$ $$\frac {f(a,cn)}{f(b,cn)}$$is constant b) Find all $(a,b)$ pairs such that minumum possible value of $c$ is 27 where $c$ satisfying condition in $(a)$

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.

A regular $100$-gon was cut into several parallelograms and two triangles. Prove that these triangles are congruent.

If $w = a + bi$, where $a$ and $b$ are real numbers, then $\Re(w) = a$ and $\Im(w) = b$. Let $z=c+di$, where $c, d \ge 0$. If \begin{align*} \Re(z) + \Im (z) & = 7, \\ \Re(z^2) + \Im(z^2) & = 17, \end{align*} then compute $\left | \Re\left (z^3 \right ) + \Im \left (z^3 \right ) \right |$. [i]Proposed by Lewis Chen[/i]

Prove that there do not exist pairwise distinct complex numbers $a, b, c,$ and $d$ such that $$a^3-bcd=b^3-acd=c^3-abd=d^3-abc.$$

Given a non-isoceles triangle $ABC$ inscribes a circle $(O,R)$ (center $O$, radius $R$). Consider a varying line $l$ such that $l\perp OA$ and $l$ always intersects the rays $AB,AC$ and these intersectional points are called $M,N$. Suppose that the lines $BN$ and $CM$ intersect, and if the intersectional point is called $K$ then the lines $AK$ and $BC$ intersect. $1$, Assume that $P$ is the intersectional point of $AK$ and $BC$. Show that the circumcircle of the triangle $MNP$ is always through a fixed point. $2$, Assume that $H$ is the orthocentre of the triangle $AMN$. Denote $BC=a$, and $d$ is the distance between $A$ and the line $HK$. Prove that $d\leq\sqrt{4R^2-a^2}$ and the equality occurs iff the line $l$ is through the intersectional point of two lines $AO$ and $BC$.

Let $A_1A_2A_3 \cdots A_{2013}$ be a cyclic $2013$-gon. Prove that for every point $P$ not the circumcenter of the $2013$-gon, there exists a point $Q\neq P$ such that $\frac{A_iP}{A_iQ}$ is constant for $i \in \{1, 2, 3, \cdots, 2013\}$. [i]Proposed by Robin Park[/i]

The positive integers $a, b, c$ are such that $$gcd \,\,\, (a, b, c) = 1,$$ $$gcd \,\,\,(a, b + c) > 1,$$ $$gcd \,\,\,(b, c + a) > 1,$$ $$gcd \,\,\,(c, a + b) > 1.$$ Determine the smallest possible value of $a + b + c$. Clarification: gcd stands for greatest common divisor.

Given a prime $p>3$ and an odd integer $n>0$, show that the equation $$xyz=p^n(x+y+z)$$ has at least $3(n+1)$ different solutions up to symmetry. (That is, if $(x',y',z')$ is a solution and $(x'',y'',z'')$ is a permutation of the previous, they are considered to be the same solution.)

A triangle with sides of $ 5$, $ 12$, and $ 13$ has both an inscibed and a circumscribed circle. What is the distance between the centers of those circles? $ \textbf{(A)}\ \frac{3\sqrt{5}}{2}\qquad \textbf{(B)}\ \frac{7}{2}\qquad \textbf{(C)}\ \sqrt{15}\qquad \textbf{(D)}\ \frac{\sqrt{65}}{2}\qquad \textbf{(E)}\ \frac{9}{2}$

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

A rectangle can be divided by parallel lines to its sides into 200 congruent squares, and also in 288 congruent squares. Prove that the rectangle can also be divided into 392 congruent squares.

Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.

The teacher writes numbers $1$ at both ends of the blackboard. The first student adds a $2$ in the middle between them, each next student adds the sum of each two adjacent numbers already on the blackboard between them (hence there are numbers $1, 3, 2, 3, 1$ on the blackboard after the second student, $1, 4, 3, 5, 2, 5, 3, 4, 1$ after the third student etc.) Find the sum of all numbers on the blackboard after the $n$-th student.

Let $x_1,x_2,...,x_n$ ($n \ge 2$) be positive numbers with the sum $1$. Prove that $$\sum_{i=1}^{n} \frac{1}{\sqrt{1-x_i}} \ge n\sqrt{\frac{n}{n-1}} $$

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

Let $ABC$ be an acute triangle with circumcenter $O$, and select $E$ on $\overline{AC}$ and $F$ on $\overline{AB}$ so that $\overline{BE} \perp \overline{AC}$, $\overline{CF} \perp \overline{AB}$. Suppose $\angle EOF - \angle A = 90^{\circ}$ and $\angle AOB - \angle B = 30^{\circ}$. If the maximum possible measure of $\angle C$ is $\tfrac mn \cdot 180^{\circ}$ for some positive integers $m$ and $n$ with $m < n$ and $\gcd(m,n)=1$, compute $m+n$. [i]Proposed by Evan Chen[/i]

Let $P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}$ be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length $1$ or $\sqrt 2$, so that both ends of the segment are in $P$ and do not intersect each other. Furthermore, for each point $(a, b)$ it is true that i) if $a + b$ is a multiple of $3$, then it is an endpoint of exactly $3$ segments. ii) if $a + b$ is an even not multiple of $3$, then it is an endpoint of exactly $2$ segments. iii) if $a + b$ is an odd not multiple of $3$, then it is endpoint of exactly $1$ segment. a) Check that with $n = 6$ it is possible to satisfy all the conditions. b) Show that with $n = 2015$ it is not possible to satisfy all the conditions.

What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$? $\textbf{(A) } 70 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 127 \qquad \textbf{(D) } 159 \qquad \textbf{(E) } 729 $

Show that the maximum number of spheres of radius $1$ that can be placed touching a fixed sphere of radius $1$ so that no pair of spheres has an interior point in common is between $12$ and $14$.

$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$