Let $ABC$ be an acute triangle and $O$ be its circumcenter. Let $D$ be the midpoint of $[AB]$. The circumcircle of $\triangle ADO$ meets $[AC]$ at $A$ and $E$. If $|AE|=7$, $|DE|=8$, and $m(\widehat{AOD}) = 45^\circ$, what is the area of $\triangle ABC$? $ \textbf{(A)}\ 56\sqrt 3 \qquad\textbf{(B)}\ 56 \sqrt 2 \qquad\textbf{(C)}\ 50 \sqrt 2 \qquad\textbf{(D)}\ 84 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ x_1,x_2,\ldots,x_n $ be positive real numbers and $ x_{n+1} = x_1 + x_2 + \cdots + x_n $. Prove that \[ \sum_{k=1}^n \sqrt { x_k (x_{n+1} - x_k)} \leq \sqrt { \sum_{k=1}^n x_{n+1}(x_{n+1}-x_k)}. \] [i]Mircea Becheanu[/i]

Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.

Can we divide an equilateral triangle $\vartriangle ABC$ into $2011$ small triangles using $122$ straight lines? (there should be $2011$ triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)

Let $S$ be the set of ordered pairs $(x,y)$ of positive integers for which $x+y\le 20$. Evaluate \[\sum_{(x, y) \in S} (-1)^{x+y}xy.\] [i]Proposed by Andrew Wen[/i]

Given an integer $n \geq 2$ determine the integral part of the number $ \sum_{k=1}^{n-1} \frac {1} {({1+\frac{1} {n}}) \dots ({1+\frac {k} {n})}}$ - $\sum_{k=1}^{n-1} (1-\frac {1} {n}) \dots(1-\frac{k}{n})$

Let $p$ be a positive real number given. Find the minimun vale of $x^3+y^3$, knowing that $x$ and $y$ are positive real numbers such that $xy(x+y)=p$.

For which $k$ do there exist $k$ pairwise distinct primes $p_1,p_2,\ldots ,p_k$ such that \[p_1^2+p_2^2+\ldots +p_k^2=2010? \]

A Magician and a Detective play a game. The Magician lays down cards numbered from $1$ to $52$ face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise. Prove that the Detective can guarantee a win if and only if she is allowed to ask at least $50$ questions. [i]Proposed by Anant Mudgal[/i]

Let $n > 1$ be an integer. There are $n$ boxes in a row, and there are $n + 1$ identical stones. A [i]distribution [/i] is a way to distribute the stones over the boxes, in which every stone is in exactly one of the boxes. We say that two of such distributions are a [i]stone’s throw away[/i] from each other if we can obtain one distribution from the other by moving exactly one stone from one box to another. The [i]cosiness [/i] of a distribution $a$ is defined as the number of distributions that are a stone’s throw away from $a$. Determine the average cosiness of all possible distributions.

Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every $ 90$ seconds, and Robert runs clockwise and completes a lap every $ 80$ seconds. Both start from the start line at the same time. At some random time between $ 10$ minutes and $ 11$ minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? $ \textbf{(A)}\ \frac{1}{16}\qquad \textbf{(B)}\ \frac18\qquad \textbf{(C)}\ \frac{3}{16} \qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac{5}{16}$

How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

In an acute non-isosceles triangle $ABC$, the inscribed circle touches side $BC$ at point $T, Q$ is the midpoint of altitude $AK$, $P$ is the orthocenter of the triangle formed by the bisectors of angles $B$ and $C$ and line $AK$. Prove that the points $P, Q$ and $T$ lie on the same line. (D. Prokopenko)

Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$?

The polynomial $A(x) = x^2 + ax + b$ with integer coefficients has the following property: for each prime $p$ there is an integer $k$ such that $A(k)$ and $A(k + 1)$ are both divisible by $p$. Proof that there is an integer $m$ such that $A(m) = A(m + 1) = 0$.

A student is asked to measure $30.0 {\text{g}}$ of methanol $(d=0.7914 \text{g/mL at 25}^{\circ}\text{C})$ but has only a graduated cylinder with which to measure it. What volume of methanol should the student use to obtain the required ${30.0 \text{g}}$? ${ \textbf{(A)}\ 23.7 \text{mL} \qquad\textbf{(B)}\ 30.0 \text{mL} \qquad\textbf{(C)}\ 32.4 \text{mL} \qquad\textbf{(D)}\ 37.9 \text{mL} }$

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

[u]Round 1[/u] [b]p1.[/b] Five girls and three boys are sitting in a room. Suppose that four of the children live in California. Determine the maximum possible number of girls that could live somewhere outside California. [b]p2.[/b] A $4$-meter long stick is rotated $60^o$ about a point on the stick $1$ meter away from one of its ends. Compute the positive difference between the distances traveled by the two endpoints of the stick, in meters. [b]p3.[/b] Let $f(x) = 2x(x - 1)^2 + x^3(x - 2)^2 + 10(x - 1)^3(x - 2)$. Compute $f(0) + f(1) + f(2)$. [u]Round 2[/u] [b]p4.[/b] Twenty boxes with weights $10, 20, 30, ... , 200$ pounds are given. One hand is needed to lift a box for every $10$ pounds it weighs. For example, a $40$ pound box needs four hands to be lifted. Determine the number of people needed to lift all the boxes simultaneously, given that no person can help lift more than one box at a time. [b]p5.[/b] Let $ABC$ be a right triangle with a right angle at $A$, and let $D$ be the foot of the perpendicular from vertex$ A$ to side $BC$. If $AB = 5$ and $BC = 7$, compute the length of segment $AD$. [b]p6.[/b] There are two circular ant holes in the coordinate plane. One has center $(0, 0)$ and radius $3$, and the other has center $(20, 21)$ and radius $5$. Albert wants to cover both of them completely with a circular bowl. Determine the minimum possible radius of the circular bowl. [u]Round 3[/u] [b]p7.[/b] A line of slope $-4$ forms a right triangle with the positive x and y axes. If the area of the triangle is 2013, find the square of the length of the hypotenuse of the triangle. [b]p8.[/b] Let $ABC$ be a right triangle with a right angle at $B$, $AB = 9$, and $BC = 7$. Suppose that point $P$ lies on segment $AB$ with $AP = 3$ and that point $Q$ lies on ray $BC$ with $BQ = 11$. Let segments $AC$ and $P Q$ intersect at point $X$. Compute the positive difference between the areas of triangles $AP X$ and $CQX$. [b]p9.[/b] Fresh Mann and Sophy Moore are racing each other in a river. Fresh Mann swims downstream, while Sophy Moore swims $\frac12$ mile upstream and then travels downstream in a boat. They start at the same time, and they reach the finish line 1 mile downstream of the starting point simultaneously. If Fresh Mann and Sophy Moore both swim at $1$ mile per hour in still water and the boat travels at 10 miles per hour in still water, find the speed of the current. [u]Round 4[/u] [b]p10.[/b] The Fibonacci numbers are defined by $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. The first few terms of the Fibonacci sequence are $0$, $1$, $1$, $2$, $3$, $5$, $8$, $13$. Every positive integer can be expressed as the sum of nonconsecutive, distinct, positive Fibonacci numbers, for example, $7 = 5 + 2$. Express $121$ as the sum of nonconsecutive, distinct, positive Fibonacci numbers. (It is not permitted to use both a $2$ and a $1$ in the expression.) [b]p11.[/b] There is a rectangular box of surface area $44$ whose space diagonals have length $10$. Find the sum of the lengths of all the edges of the box. [b]p12.[/b] Let $ABC$ be an acute triangle, and let $D$ and $E$ be the feet of the altitudes to $BC$ and $CA$, respectively. Suppose that segments $AD$ and $BE$ intersect at point $H$ with $AH = 20$ and $HD = 13$. Compute $BD \cdot CD$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c4h2809420p24782524]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$? [i]Proposed by Oleksiy Masalitin[/i]

The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

In a triangle $ABC$ , we have a point $O$ on $BC$ . Now show that there exists a line $l$ such that $l||AO$ and $l$ divides the triangle $ABC$ into two halves of equal area .

Let $ ABCD$ be a convex quadrilateral and let $ N$ on side $ AD$ and $ M$ on side $ BC$ be points such that $ \dfrac{AN}{ND}\equal{}\dfrac{BM}{MC}$. The lines $ AM$ and $ BN$ intersect at $ P$, while the lines $ CN$ and $ DM$ intersect at $ Q$. Prove that if $ S_{ABP}\plus{}S_{CDQ}\equal{}S_{MNPQ}$, then either $ AD\parallel BC$ or $ N$ is the midpoint of $ DA$.

The figure below has a 1 × 1 square, a 2 × 2 square, a 3 × 3 square, a 4 × 4 square, and a 5 × 5 square. Each of the larger squares shares a corner with the 1 × 1 square. Find the area of the region covered by these five squares. [center][img]https://snag.gy/1AfJWt.jpg[/img][/center]

Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\ \frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$

Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$. [i]Z. Daroczy, Gy. Maksa[/i]