In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$

There are 1$2$ people labeled $1, ..., 12$ working together on $12$ missions, with people $1, ... , i $working on the $i$th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability $1/2$. Given that the first $11$ missions succeed, and the $12$th mission fails, what is the probability that person $12$ is the spy?

Find the remainder of $7^{1010}+8^{2017}$ when divided by $57$.

You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set $\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$? Here $(x, y)$ denotes the greatest common divisor of $x, y$. [i]Proposed by Anton Trygub[/i]

Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.

Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy: [list] [*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$; [*]$f(2024)=1$; [*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$. [/list]

If $\text{A}$ and $\text{B}$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is \[\begin{tabular}[t]{cccc} 9 & 8 & 7 & 6 \\ & A & 3 & 2 \\ & & B & 1 \\ \hline \end{tabular}\] $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{depends on the values of A and B}$

Triangle $ABC$ has median $AF$, and $D$ is the midpoint of the median. Line $CD$ intersects $AB$ in $E$. Prove that $BD = BF$ implies $AE = DE$!

An envelope contains eight bills: $ 2$ ones, $ 2$ fives, $ 2$ tens, and $ 2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ \$ 20$ or more? $ \textbf{(A)}\ \frac {1}{4}\qquad \textbf{(B)}\ \frac {2}{7}\qquad \textbf{(C)}\ \frac {3}{7}\qquad \textbf{(D)}\ \frac {1}{2}\qquad \textbf{(E)}\ \frac {2}{3}$

The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor? $\textbf{(A) }125,875,000\qquad\textbf{(B) }201,400,000\qquad\textbf{(C) }251,750,000\qquad\textbf{(D) }402,800,000\qquad\textbf{(E) }503,500,000\qquad$

2. What’s the closest number to $169$ that’s divisible by $9$?

Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.

Given a circle $C$ and a diameter $AB$ in it, mark a point $P$ on $AB$ different from the ends. In one of the two arcs determined by $AB$ choose the points $M$ and $N$ such that $\angle APM = 60 ^ \circ = \angle BPN$. The segments $MP$ and $NP$ are drawn to obtain three curvilinear triangles; $APM $, $MPN$ and $NPB$ (the sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of $C$.

Let $O$ be the point $(0,0)$. Let $A$, $B$, $C$ be three points in the plane such that $AO=15$, $BO = 15$, and $CO = 7$, and such that the area of triangle $ABC$ is maximal. What is the length of the shortest side of $ABC$?

Call a positive real number special if it has a decimal representation that consists entirely of digits $ 0$ and $ 7$. For example, $ \frac{700}{99} \equal{} 7.\overline{07} \equal{} 7.070707\cdots$ and $ 77.007$ are special numbers. What is the smallest $ n$ such that $ 1$ can be written as a sum of $ n$ special numbers? $ \textbf{(A)}\ 7\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad\\ \textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.}$

Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2-y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.

$PA_1A_2...A_n$ is a pyramid. The base $A_1A_2...A_n$ is a regular n-gon. The apex $P$ is placed so that the lines $PA_i$ all make an angle $60^{\cdot}$ with the plane of the base. For which $n$ is it possible to find $B_i$ on $PA_i$ for $i = 2, 3, ... , n$ such that $A_1B_2 + B_2B_3 + B_3B_4 + ... + B_{n-1}B_n + B_nA_1 < 2A_1P$?

Prove that circles constructed on the sides of a convex quadrilateral as diameters completely cover this quadrilateral.

Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence. [list] [*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence. [*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list] [i]Mitchell Lee and Benjamin Gunby.[/i]

Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.

Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?

we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.

Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.