$2014$ triangles have non-overlapping interiors contained in a circle of radius $1$. What is the largest possible value of the sum of their areas?

How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? $ \textbf{(A)}\ 496 \qquad\textbf{(B)}\ 500 \qquad\textbf{(C)}\ 512 \qquad\textbf{(D)}\ 516 \qquad\textbf{(E)}\ 560 $

Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.

Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly. a. Prove that if $PO=PO'$ then $O\in(A'B'C')$; b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.

$1+2+3+4+5+6=6+7+8$. What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?

Show that the sum of the squares of the sides of a quadrilateral is at least the sum of the squares of the diagonals. When does equality hold?

Let $\alpha > 0$ be a real number. Compute the limit of the sequence $\{x_n\}_{n\geq 1}$ defined by $$x_n=\begin{cases} \sum \limits_{k=1}^n \sinh \left(\frac{k}{n^2}\right),& \text{when}\ n>\frac{1}{\alpha}\\ 0,& \text{when}\ n\leq \frac{1}{\alpha}\end{cases}$$

$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then $\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$

Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

[color=darkblue]Let $ X$ be a group of people, where any two people are friends or enemies. Each pair of friends from $ X$ doesn't have any common friends, and any two enemies have exactly two common friends. Prove that each person from $ X$ has the same number of friends as others.[/color]

Prove that there do not exist polynomials $ P$ and $ Q$ such that \[ \pi(x)\equal{}\frac{P(x)}{Q(x)}\] for all $ x\in\mathbb{N}$.

Two identical jars are filled with alcohol solutions, the ratio of the volume of alcohol to the volume of water being $p : 1$ in one jar and $q : 1$ in the other jar. If the entire contents of the two jars are mixed together, the ratio of the volume of alcohol to the volume of water in the mixture is $\textbf{(A) }\frac{p+q}{2}\qquad\textbf{(B) }\frac{p^2+q^2}{p+q}\qquad\textbf{(C) }\frac{2pq}{p+q}\qquad\textbf{(D) }\frac{2(p^2+pq+q^2)}{3(p+q)}\qquad\textbf{(E) }\frac{p+q+2pq}{p+q+2}$

Let $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$, $\forall 1 \le k \le 2001$. Is it possible that more than half of the elements are negative?

For each subset $S$ of $\mathbb{N}$, with $r_S (n)$ we denote the number of ordered pairs $(a,b)$, $a,b\in S$, $a\neq b$, for which $a+b=n$. Prove that $\mathbb{N}$ can be partitioned into two subsets $A$ and $B$, so that $r_A(n)=r_B(n)$ for $\forall$ $n\in \mathbb{N}$.

Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after $6$ minutes?

[u]Round 1[/u] [b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive? [b]p2[/b]. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning. [img]https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png[/img] [b]p3.[/b] A $25\times 25$ checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by $4$. For example, the cuts shown on the picture have total length $16$, which is divisible by $4$. [img]https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png[/img] [b]p4.[/b] Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts $N$ hours takes exactly $N$ hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet). [b]p5.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$. [img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png [/img] [u]Round 2[/u] [b]p6.[/b] The sum of $2015$ rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers. (A rational number is one that can be written as $m/n$, where $m$ and $n$ are integers and $n\ne 0$.) [b]p7.[/b] An $N \times N$ table is filled with integers such that numbers in cells that share a side differ by at most $1$. Prove that there is some number that appears in the table at least $N$ times. For example, in the $5 \times 5$ table below the numbers $1$ and $2$ appear at least $5$ times. [img]https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

Let $a,b,b',c,m,q$ be positive integers, where $m>1,q>1,|b-b'|\ge a$. It is given that there exist a positive integer $M$ such that $$S_q(an+b)\equiv S_q(an+b')+c\pmod{m}$$ holds for all integers $n\ge M$. Prove that the above equation is true for all positive integers $n$. (Here $S_q(x)$ is the sum of digits of $x$ taken in base $q$).

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

Let $a_1, a_2, \ldots, a_{11}$ be integers. Prove that there exist numbers $b_1, b_2, \ldots, b_{11}$ such that [list] [*] $b_i$ is equal to $-1,0$ or $1$ for all $i \in \{1, 2,\dots, 11\}$. [*] all numbers can't be zero at a time. [*] the number $N=a_1b_1+a_2b_2+\ldots+a_{11}b_{11}$ is divisible by $2024$. [/list]

On the circle 6 distinct points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $

Let $A = (1, 0)$, $B = (0, 1)$, and $C = (0, 0)$. There are three distinct points, $P, Q, R$, such that $\{A, B, C, P\}$, $\{A, B, C, Q\}$, $\{A, B, C, R\}$ are all parallelograms (vertices unordered). Find the area of $\vartriangle PQR$.

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.