Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

Assume you have a magical pizza in the shape of an infinite plane. You have a magical pizza cutter that can cut in the shape of an infinite line, but it can only be used $14$ times. To share with as many of your friends as possible, you cut the pizza in a way that maximizes the number of finite pieces (the infinite pieces have infinite mass, so you can’t lift them up). How many finite pieces of pizza do you have?

Let points $A=(0,0,0)$, $B=(1,0,0)$, $C=(0,2,0)$, and $D=(0,0,3)$. Points $E,F,G$, and $H$ are midpoints of line segments $\overline{BD},\overline{AB},\overline{AC}$, and $\overline{DC}$ respectively. What is the area of $EFGH$? $ \textbf{(A)}\ \sqrt2 \qquad\textbf{(B)}\ \frac{2\sqrt5}{3} \qquad\textbf{(C)}\ \frac{3\sqrt5}{4} \qquad\textbf{(D)}\ \sqrt3 \qquad\textbf{(E)}\ \frac{2\sqrt7}{3} $

The first $1997$ natural numbers are written on the blackboard: $1,2,3,\ldots ,1997$. In front of each number, a "$+$" sign or a "$-$" sign will be written in order, from left to right. To decide each sign, a coin is tossed; If it comes up heads, you write "$+$", if it comes up tails, you write "$-$". Once the $1997$ signs are written, the algebraic sum of the expression on the blackboard is carried out and the result is $S$. What is the probability that $S$ is greater than $0$? Clarification: The probability of an event is equal to the number of favorable cases/number of possible cases.

Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$ $$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$ Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)

Maureen is keeping track of the mean of her quiz scores this semester. If Maureen scores an $11$ on the next quiz, her mean will increase by $1$. If she scores an $11$ on each of the next three quizzes, her mean will increase by $2$. What is the mean of her quiz scores currently? $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

Is there a whole number which becomes exactly $57$ times less than itself when one crosses out its first digit?

On the infinite surface of the sea floats a black and bounded oil slick. After every minute the slick and the sea change according to the following law: at each point $P$ of the sea (or of the slick), a disk $D$ of radius $1$ is considered centered on $ P$. If more than half of the area inside the disk $D$ is black, the $P$ point will remain black for the next minute. If more than half of the area inside the disk $D$ is dark blue, the point $P$ will be dark blue for the next minute. In the event that both the clean and the contaminated area within the disk $D$ are the same, its center $P$ will not change color. Can that stain "live" forever or will it disappear at some point?

On a chessboard with $6$ rows and $9$ columns, the Slow Rook is placed in the bottom-left corner and the Blind King is placed on the top-left corner. Then, $8$ Sleeping Pawns are placed such that no two Sleeping Pawns are in the same column, no Sleeping Pawn shares a row with the Slow Rook or the Blind King, and no Sleeping Pawn is in the rightmost column. The Slow Rook can move vertically or horizontally $1$ tile at a time, the Slow Rook cannot move into any tile containing a Sleeping Pawn, and the Slow Rook takes the shortest path to reach the Blind King. How many ways are there to place the Sleeping Pawns such that the Slow Rook moves exactly $15$ tiles to get to the space containing the Blind King? Proposed by Harry Chen (Extile)

Let $x,y,z$ be three positive integers with $\gcd(x,y,z)=1$. If \[x\mid yz(x+y+z),\] \[y\mid xz(x+y+z),\] \[z\mid xy(x+y+z),\] and \[x+y+z\mid xyz,\] show that $xyz(x+y+z)$ is a perfect square. [i]Proposed by usjl[/i]

Let $ABC$ be an acute triangle with circumcenter $O$. Points $E$ and $F$ are chosen on segments $OB$ and $OC$ such that $BE = OF$. If $M$ is the midpoint of the arc $EOA$ and $N$ is the midpoint of the arc $AOF$, prove that $\sphericalangle ENO + \sphericalangle OMF = 2 \sphericalangle BAC$.

Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$. [i]Aprameya Tripathy[/i]

Let \(ABCDE\) be a regular pentagon. Let \(P\) be a variable point on the interior of segment \(AB\) such that \(PA\ne PB\). The circumcircles of \(\triangle PAE\) and \(\triangle PBC\) meet again at \(Q\). Let \(R\) be the circumcenter of \(\triangle DPQ\). Show that as \(P\) varies, \(R\) lies on a fixed line. [i]Proposed by Karthik Vedula[/i]

Ana's monthly salary was $ \$2000$ in May. In June she received a $20 \%$ raise. In July she received a $20 \%$ pay cut. After the two changes in June and July, Ana's monthly salary was $\text{(A)}\ 1920\text{ dollars} \qquad \text{(B)}\ 1980\text{ dollars} \qquad \text{(C)}\ 2000\text{ dollars} \qquad \text{(D)}\ 2020\text{ dollars} \qquad \text{(E)}\ 2040\text{ dollars}$

If $x$ is a real number such that $(x - 3)(x - 1)(x + 1)(x + 3) + 16 = 116^2$, what is the largest possible value of $x$?

Eight consecutive positive integers are divided into 2 sets, such that the sum of the squares of the elements in the first set is equal to the sum of the squares of the elements in the second set. Prove that the sum of the lements in the first set is equal to the sum of the elements in the second one.

Find all non negative integers $a{}$ and $b{}$ satisfying $5^a+6=31^b$ [i]Proposed by Erik Füredi, Budapest[/i]

[b]p1.[/b] Let $A(S)$ denote the average value of a set $S$. Let $T$ be the set of all subsets of the set $\{1, 2, 3, 4, ... , 2012\}$, and let $R$ be $\{A(K)|K \in T \}$. Compute $A(R)$. [b]p2.[/b] Consider the minute and hour hands of the Campanile, our clock tower. During one single day ($12:00$ AM - $12:00$ AM), how many times will the minute and hour hands form a right-angle at the center of the clock face? [b]p3.[/b] In a regular deck of $52$ face-down cards, Billy flips $18$ face-up and shuffles them back into the deck. Before giving the deck to Penny, Billy tells her how many cards he has flipped over, and blindfolds her so she can’t see the deck and determine where the face-up cards are. Once Penny is given the deck, it is her job to split the deck into two piles so that both piles contain the same number of face-up cards. Assuming that she knows how to do this, how many cards should be in each pile when he is done? [b]p4.[/b] The roots of the equation $x^3 + ax^2 + bx + c = 0$ are three consecutive integers. Find the maximum value of $\frac{a^2}{b+1}$. [b]p5.[/b] Oski has a bag initially filled with one blue ball and one gold ball. He plays the following game: first, he removes a ball from the bag. If the ball is blue, he will put another blue ball in the bag with probability $\frac{1}{437}$ and a gold ball in the bag the rest of the time. If the ball is gold, he will put another gold ball in the bag with probability $\frac{1}{437}$ and a blue ball in the bag the rest of the time. In both cases, he will put the ball he drew back into the bag. Calculate the expected number of blue balls after $525600$ iterations of this game. [b]p6.[/b] Circles $A$ and $B$ intersect at points $C$ and $D$. Line $AC$ and circle $B$ meet at $E$, line $BD$ and circle $A$ meet at $F$, and lines $EF$ and $AB$ meet at $G$. If $AB = 10$, $EF = 4$, $FG = 8$, find $BG$. PS. You had better use hide for answers.

Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that \[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\] [i]Proposed by Titu Andreescu, Saudi Arabia[/i]

Let $\mathbb Z_{\ge 0}$ be the set of non-negative integers, and let $f:\mathbb Z_{\ge 0}\times \mathbb Z_{\ge 0} \to \mathbb Z_{\ge 0}$ be a bijection such that whenever $f(x_1,y_1) > f(x_2, y_2)$, we have $f(x_1+1, y_1) > f(x_2 + 1, y_2)$ and $f(x_1, y_1+1) > f(x_2, y_2+1)$. Let $N$ be the number of pairs of integers $(x,y)$ with $0\le x,y<100$, such that $f(x,y)$ is odd. Find the smallest and largest possible values of $N$.

Let $P$ be the point in the interior of $\vartriangle ABC$. Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$. The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$, and $ABC'$ are congruent to $\vartriangle ABC$.

In a pond there are $n \geq 3$ stones arranged in a circle. A princess wants to label the stones with the numbers $1, 2, \dots, n$ in some order and then place some toads on the stones. Once all the toads are located, they start jumping clockwise, according to the following rule: when a toad reaches the stone labeled with the number $k$, it waits for $k$ minutes and then jumps to the adjacent stone. What is the greatest number of toads for which the princess can label the stones and place the toads in such a way that at no time are two toads occupying a stone at the same time? [b]Note:[/b] A stone is considered occupied by two toads at the same time only if there are two toads that are on the stone for at least one minute.

The six-pointed star in the figure is regular: all interior angles of the small triangles are equal. Each of the thirteen marked points is assigned a color, green or red. Prove that there are always three points of the same color, which are the vertices of an equilateral triangle.