Given that $x_1+x_2+x_3=y_1+y_2+y_3=x_1y_1+x_2y_2+x_3y_3=0,$ prove that: \[ \frac{x_1^2}{x_1^2+x_2^2+x_3^2}+\frac{y_1^2}{y_1^2+y_2^2+y_3^2}=\frac{2}{3}\]

Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

Let $ABC$ a triangle with circumcircle $\Gamma$ and $R$ a point inside $ABC$ such that $\angle ABR=\angle RBC$. Let $\Gamma_1$ and $\Gamma_2$ the circumcircles of triangles $ARB$ and $CRB$ respectly. The parallel to $AC$ that pass through $R$, intersect $\Gamma$ in $D$ and $E$, with $D$ on the same side of $BR$ that $A$ and $E$ on the same side of $BR$ that $C$. $AD$ intersect $\Gamma_1$ in $P$ and $CE$ intersect $\Gamma_2$ in $Q$. Prove that $APQC$ is cyclic if and only if $AB=BC$

Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that a) $C, A_1, C_1, A$ are concyclic, b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.

Find the 2000th positive integer that is not the difference between any two integer squares.

[hide=Intro]The answers to each of the ten questions in this section are integers containing only the digits $ 1$ through $ 8$, inclusive. Each answer can be written into the grid on the answer sheet, starting from the cell with the problem number, and continuing across or down until the entire answer has been written. Answers may cross dark lines. If the answers are correctly filled in, it will be uniquely possible to write an integer from $ 1$ to $ 8$ in every cell of the grid, so that each number will appear exactly once in every row, every column, and every marked $2$ by $4$ box. You will get $7$ points for every correctly filled answer, and a $15$ point bonus for filling in every gridcell. It will help to work back and forth between the grid and the problems, although every problem is uniquely solvable on its own. Please write clearly within the boxes. No points will be given for a cell without a number, with multiple numbers, or with illegible handwriting.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/9/b/f4db097a9e3c2602b8608be64f06498bd9d58c.png[/img] [b]1 ACROSS:[/b] Jack puts $ 10$ red marbles, $ 8$ green marbles and 4 blue marbles in a bag. If he takes out $11$ marbles, what is the expected number of green marbles taken out? [b]2 DOWN:[/b] What is the closest integer to $6\sqrt{35}$ ? [b]3 ACROSS: [/b]Alan writes the numbers $ 1$ to $64$ in binary on a piece of paper without leading zeroes. How many more times will he have written the digit $ 1$ than the digit $0$? [b]4 ACROSS:[/b] Integers a and b are chosen such that $-50 < a, b \le 50$. How many ordered pairs $(a, b)$ satisfy the below equation? $$(a + b + 2)(a + 2b + 1) = b$$ [b]5 DOWN: [/b]Zach writes the numbers $ 1$ through $64$ in binary on a piece of paper without leading zeroes. How many times will he have written the two-digit sequence “$10$”? [b]6 ACROSS:[/b] If you are in a car that travels at $60$ miles per hour, $\$1$ is worth $121$ yen, there are $8$ pints in a gallon, your car gets $10$ miles per gallon, a cup of coffee is worth $\$2$, there are 2 cups in a pint, a gallon of gas costs $\$1.50$, 1 mile is about $1.6$ kilometers, and you are going to a coffee shop 32 kilometers away for a gallon of coffee, how much, in yen, will it cost? [b]7 DOWN:[/b] Clive randomly orders the letters of “MIXING THE LETTERS, MAN”. If $\frac{p}{m^nq}$ is the probability that he gets “LMT IS AN EXTREME THING” where p and q are odd integers, and $m$ is a prime number, then what is $m + n$? [b]8 ACROSS:[/b] Joe is playing darts. A dartboard has scores of $10, 7$, and $4$ on it. If Joe can throw $12$ darts, how many possible scores can he end up with? [b]9 ACROSS:[/b] What is the maximum number of bounded regions that $6$ overlapping ellipses can cut the plane into? [b]10 DOWN:[/b] Let $ABC$ be an equilateral triangle, such that $A$ and $B$ both lie on a unit circle with center $O$. What is the maximum distance between $O$ and $C$? Write your answer be in the form $\frac{a\sqrt{b}}{c}$ where $b$ is not divisible by the square of any prime, and $a$ and $c$ share no common factor. What is $abc$ ? PS. You had better use hide for answers.

Let $ a $ and $ b $ be natural numbers with property $ gcd(a,b)=1 $ . Find the least natural number $ k $ such that for every natural number $ r \ge k $ , there exist natural numbers $ m,n >1 $ in such a way that the number $ m^a n^b $ has exactly $ r+1 $ positive divisors.

A ring is of the shape of a hoola-hoop of negligible thickness. A ring of radius $R$ carries a current $I$. Prove that the magnetic field at a given point in the plane of the ring at a distance $a$ from the center, due to the magnetic field of the ring, is \[ B = \dfrac {\mu_0}{2\pi} \cdot IR \cdot \displaystyle\int_{0}^{\pi} \dfrac {R - a \cos \theta}{\sqrt{\left( a^2 + R^2 - 2aR \cos \theta \right)^3}} \, \mathrm{d}\theta. \] [i]Problem proposed by Ahaan Rungta[/i]

Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is divided by $1000$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in a set $X$.) [i]Proposed by David Altizio[/i]

Let $S_n $ be sum of squares of the coefficient of the polynomial $(1+x)^n$. Prove that $S_{2n} +1$ is not divisible by $3.$

Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays. (A Kanel)

The battery life on a computer decreases at a rate proportional to the display brightness. Austin starts off his day with both his battery life and brightness at $100\%$. Whenever his battery life (expressed as a percentage) reaches a multiple of $25$, he also decreases the brightness of his display to that multiple of $25$. If left at $100\%$ brightness, the computer runs out of battery in $1$ hour. Compute the amount of time, in minutes, it takes for Austin’s computer to reach $0\%$ battery using his modified scheme.

Prove that for any positive integer $M$ there exists an integer divisible by $M$ such that the sum of its digits (in its decimal representation) is odd. (D Fomin, St Petersburg)

In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar? $\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}8 \qquad \textbf{(C)}\hspace{.05in}9 \qquad \textbf{(D)}\hspace{.05in}10 \qquad \textbf{(E)}\hspace{.05in}12$

Let $ABC$ be an acute triangle with $AB<BC$. Let $I$ be the incenter of $ABC$, and let $\omega$ be the circumcircle of $ABC$. The incircle of $ABC$ is tangent to the side $BC$ at $K$. The line $AK$ meets $\omega$ again at $T$. Let $M$ be the midpoint of the side $BC$, and let $N$ be the midpoint of the arc $BAC$ of $\omega$. The segment $NT$ intersects the circumcircle of $BIC$ at $P$. Prove that $PM\parallel AK$.

Let $n$ be a positive integer, $n\geq 2$. For each $t\in \mathbb{R}$, $t\neq k\pi$, $k\in\mathbb{Z}$, we consider the numbers \[ x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}. \] Prove that if $x_n(t) = y_n(t) =0$ if and only if $\tan {\frac {nt}2} = n \tan {\frac t2}$. [i]Constantin Buse[/i]

The larger of two consecutive odd integers is three times the smaller. What is their sum? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

How many functions $f : f\{1, 2, 3, 4, 5\}\longrightarrow\{1, 2, 3, 4, 5\}$ satisfy $f(f(x)) = f(x)$ for all $x\in\{ 1,2, 3, 4, 5\}$?

Let $a_1 = 20$, $a_2 = 16$, and for $k \ge 3$, let $a_k = \sqrt[3]{k-a_{k-1}^3-a_{k-2}^3}$. Compute $a_1^3+a_2^3+\cdots + a_{10}^3$.

Let $Z$ be the set of all integers. Find all the function $f: Z->Z$ such that $f(4x+3y)=f(3x+y)+f(x+2y)$ For all integers $x,y$

For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\ge 2$, let $f(n)$ be the minimal $k$ for which there exists a set $S$ of $n$ positive integers such that $s\left(\sum_{x\in X} x\right)=k$ for any nonempty subset $X\subset S$. Prove that there are constants $0<C_1<C_2$ with \[C_1 \log_{10} n \le f(n) \le C_2 \log_{10} n.\]

Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.

Prove that there exists a set $X \subseteq \mathbb{N}$ which contains exactly 2022 elements such that for every distinct $a, b, c \in X$ the following equality: \[ \gcd(a^n+b^n, c) = 1 \] is satisfied for every positive integer $n$.

A square of side $n$ ($n$ natural) is divided into $n^2$ squares of side $1$. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length. (a) Prove that there are $8$ rectangles of width $n - 1$. (b) Determine the number of rectangles with width $n -k$ ($0\le k \le n -1,k$ integer). (c) Determine a formula for $1^3 + 2^3 +...+ n^3$.

$\vartriangle ABC$ is divided by a straight line $BD$ into two triangles. Prove that the sum of the radii of circles inscribed in triangles $ABD$ and $DBC$ is greater than the radius of the circle inscribed in $\vartriangle ABC$.