John is flipping his favorite bottle, which currently contains $10$ ounces of water. However, his bottle is broken from excessive flipping, so after he performs a flip, one ounce of water leaks out of his bottle. When his bottle contains k ounces of water, he has a $\frac{1}{k+1}$ probability of landing it on its bottom. What is the expected number of number of flips it takes for John’s bottle to land on its bottom ?

Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$. Determine the smallest number of pieces Paul needs to make in order to accomplish this.

One side of a triangle has length $75$. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle

The magician puts out hardly a deck of $52$ cards and announces that $51$ of them will be thrown out of the table, and there will remain three of clubs. The viewer at each step says which count from the edge the card should be thrown out, and the magician chooses to count from the left or right edge, and ejects the corresponding card. At what initial positions of the three of clubs can the success of the focus be guaranteed?

Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$. a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$. b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$. [i]Proposed by Vincent Huang[/i]

Given is a number with $1998$ digits which is divisible by $9$. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$.

Prove that for $0<x<\frac{\pi}{2}$, $$\sin x + \tan x > 2x$$

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that $$ f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3 $$ for all integers $a, b, c$.

100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

Find all functions $f:\mathbb{R}^+ \to \mathbb{R}$ such that for all $x,y,z>0$ $$ 3(x^3+y^3+z^3)\geq f(x+y+z)\cdot f(xy+yz+xz) \geq (x+y+z)(xy+yz+xz). $$

Four frogs are positioned at four points on a straight line such that the distance between any two neighbouring points is 1 unit length. Suppose the every frog can jump to its corresponding point of reflection, by taking any one of the other 3 frogs as the reference point. Prove that, there is no such case that the distance between any two neighbouring points, where the frogs stay, are all equal to 2008 unit length.

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.

In the triangle $ABC$ it is known that $AC = 21$ cm, $BC = 28$ cm and $\angle C = 90^o$. On the hypotenuse $AB$, we construct a square $ABMN$ with center $O$ such that the segment $CO$ intersects the hypotenuse $AB$ at the point $K$. Find the lengths of the segments $AK$ and $KB$.

A rectangular piece of paper is divided into square cells by lines parallel to the sides of the rectangle. $n$ (horizontal) rows of $m$ cells have emerged and $m$ (vertical) columns of $n$ cells have also been formed. There is a number in each cell. Find the largest number in each of the $n$ rows. The smallest maxima of those $n$ rows is called $A$. We also look for the smallest number in each of the $m$ columns. The largest minima of those $m$ columns is called $B$. Prove that $A$ is greater than or equal to $B$. Can you give a simple example where $A = B$?

The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?

Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true? $\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$

Find the number of $5$-digit $n$, s.t. every digit of $n$ is either $0$, $1$, $3$, or $4$, and $n$ is divisible by $15$.

Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.

Consider a triangle ${ABC}$ and let ${M}$ be the midpoint of the side ${BC.}$ Suppose ${\angle MAC=\angle ABC}$ and ${\angle BAM=105^{\circ}.}$ Find the measure of ${\angle ABC}$.

Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$. [i]Proposed by David Altizio[/i]

Let $S=\{a+np : n=0,1,2,3,... \}$ where $a$ is a positive integer and $p$ is a prime. Suppose there exist positive integers $x$ and $y$ st $x^{41}$ and $y^{49}$ are in $S$. Determine if there exists a positive integer $z$ st $z^{2009}$ is in $S$.

Prove that for any positive integer $m$, the equation \[ \frac{n}{m}\equal{}\lfloor\sqrt[3]{n^2}\rfloor\plus{}\lfloor\sqrt{n}\rfloor\plus{}1\] has (at least) a positive integer solution $n_{m}$. [i]Cezar Lupu & Dan Schwarz[/i]

In the Cancun´s league participate $30$ teams. For this tournament we want to divide the $30$ teams in $2$ groups such that: $\textbf{1.}$ Every team plays exactly $82$ games $\textbf{2.}$ The number of gamen between teams of differents groups is equal to the half of games played. Can we do this?

Let $A, B\subset \mathbb{Z}$ be two sets of integers. We say that $A,B$ are [u]mutually repulsive[/u] if there exist positive integers $m,n$ and two sequences of integers $\alpha_1, \alpha_2, \dots, \alpha_n$ and $\beta_1, \beta_2, \dots, \beta_m$, for which there is a [b]unique[/b] integer $x$ such that the number of its appearances in the sequence of sets $A+\alpha_1, A+\alpha_2, \dots, A+\alpha_n$ is [u]different[/u] than the number of its appearances in the sequence of sets $B+\beta_1, \dots, B+\beta_m$. For a given quadruple of positive integers $(n_1,d_1, n_2, d_2)$, determine whether the sets \[A=\{d_1, 2d_1, \dots, n_1d_1\}\] \[B=\{d_2, 2d_2, \dots, n_2d_2\}\] are mutually repulsive. For a set $X\subset \mathbb{Z}$ and $c\in \mathbb{Z}$, we define $X+c=\{x+c\mid x\in X\}$.

The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.