Last summer $100$ students attended basketball camp. Of those attending, $52$ were boys and $48$ were girls. Also, $40$ students were from Jonas Middle School and $60$ were from Clay Middle School. Twenty of the girls were from Jonas Middle School. How many of the boys were from Clay Middle School? $\text{(A)}\ 20 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 40 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 52$

Let $a, b, c, d$ be real numbers satisfying the system of equation $\[(a+b)(c+d)=2 \\ (a+c)(b+d)=3 \\ (a+d)(b+c)=4\]$ Find the minimum value of $a^2+b^2+c^2+d^2$.

A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?

In the center of the square of side $1$ shown in the figure is an ant. At one point the ant starts walking until it touches the left side $(a)$, then continues walking until it reaches the bottom side $(b)$, and finally returns to the starting point. Show that, regardless of the path followed by the ant, the distance it travels is greater than the square root of $2$. [asy] unitsize(2 cm); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); label("$a$", (0,0.5), W); label("$b$", (0.5,0), S); dot((0.5,0.5)); [/asy]

For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$. Find the minimum possible value for $|p(15)|$.

Consider the acute-angled triangle $ ABC$, altitude $ AD$ and point $ E$ - intersection of $ BC$ with diameter from $ A$ of circumcircle. Let $ M,N$ be symmetric points of $ D$ with respect to the lines $ AC$ and $ AB$ respectively. Prove that $ \angle{EMC} \equal{} \angle{BNE}$.

Find $\frac{area(CDF)}{area(CEF)}$ in the figure. [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5.75cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -2, xmax = 21, ymin = -2, ymax = 16; /* image dimensions */ /* draw figures */ draw((0,0)--(20,0)); draw((13.48,14.62)--(7,0)); draw((0,0)--(15.93,9.12)); draw((13.48,14.62)--(20,0)); draw((13.48,14.62)--(0,0)); label("6",(15.16,12.72),SE*labelscalefactor); label("10",(18.56,5.1),SE*labelscalefactor); label("7",(3.26,-0.6),SE*labelscalefactor); label("13",(13.18,-0.71),SE*labelscalefactor); label("20",(5.07,8.33),SE*labelscalefactor); /* dots and labels */ dot((0,0),dotstyle); label("$B$", (-1.23,-1.48), NE * labelscalefactor); dot((20,0),dotstyle); label("$C$", (19.71,-1.59), NE * labelscalefactor); dot((7,0),dotstyle); label("$D$", (6.77,-1.64), NE * labelscalefactor); dot((13.48,14.62),dotstyle); label("$A$", (12.36,14.91), NE * labelscalefactor); dot((15.93,9.12),dotstyle); label("$E$", (16.42,9.21), NE * labelscalefactor); dot((9.38,5.37),dotstyle); label("$F$", (9.68,4.5), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

We have $0\le x_i<b$ for $i=0,1,\ldots,n$ and $x_n>0,x_{n-1}>0$. If $a>b$, and $x_nx_{n-1}\ldots x_0$ represents the number $A$ base $a$ and $B$ base $b$, whilst $x_{n-1}x_{n-2}\ldots x_0$ represents the number $A'$ base $a$ and $B'$ base $b$, prove that $A'B<AB'$.

A circle passes through the three vertices of an isosceles triangle that has two sides of length $ 3$ and a base of length $ 2$. What is the area of this circle? $ \textbf{(A)}\ 2\pi\qquad \textbf{(B)}\ \frac {5}{2}\pi\qquad \textbf{(C)}\ \frac {81}{32}\pi\qquad \textbf{(D)}\ 3\pi\qquad \textbf{(E)}\ \frac {7}{2}\pi$

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[(a-1)(b-1)(c-1)\hspace{0.2in}\text{is a divisor of}\hspace{0.2in}abc-1.\]

Given 5 points on a plane. Let $\lambda$ be the ratio of maximum value between the points to minimum value between the points. Prove that $\lambda\geq2\sin\frac{3}{10}\pi$.

In eight years Henry will be three times the age that Sally was last year. Twenty five years ago their ages added to $83$. How old is Henry now?

Square pyramid $ABCDE$ has base $ABCD,$ which measures $3$ cm on a side, and altitude $\overline{AE}$ perpendicular to the base$,$ which measures $6$ cm. Point $P$ lies on $\overline{BE},$ one third of the way from $B$ to $E;$ point $Q$ lies on $\overline{DE},$ one third of the way from $D$ to $E;$ and point $R$ lies on $\overline{CE},$ two thirds of the way from $C$ to $E.$ What is the area, in square centimeters, of $\triangle PQR?$ $\textbf{(A) } \frac{3\sqrt2}{2} \qquad\textbf{(B) } \frac{3\sqrt3}{2} \qquad\textbf{(C) } 2\sqrt2 \qquad\textbf{(D) } 2\sqrt3 \qquad\textbf{(E) } 3\sqrt2$

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ $\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

Sally has five red cards numbered $ 1$ through $ 5$ and four blue cards numbered $ 3$ through $ 6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

Prove that there are no integers $x,y,z$ such that \[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]

Let $ABC$ be an equilateral triangle with side length $1$. Points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$ respectively such that $\triangle BDE$ is right isosceles, while points $F$ and $G$ lie on $\overline{BC}$ and $\overline{AB}$ respectively such that $\triangle CFG$ is right isosceles. Find the area of the intersection of $\triangle BDE$ and $\triangle CFG$. [i]Proposed by Ishin Shah[/i]

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$

We define a [i]pseudo-inverse[/i] $B\in \mathcal M_n(\mathbb C)$ of a matrix $A\in\mathcal M_n(\mathbb C)$ a matrix which fulfills the relations \[ A = ABA \quad \text{ and } \quad B=BAB. \] a) Prove that any square matrix has at least a pseudo-inverse. b) For which matrix $A$ is the pseudo-inverse unique? [i]Marius Cavachi[/i]

Compute $$\left(20+\frac{1}{23}\right)\cdot\left(23+\frac{1}{20}\right)-\left(20-\frac{1}{23}\right)\cdot\left(23-\frac{1}{20}\right)$$ [i]Proposed by Andy Xu[/i]

Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. M. Kungodjin

In how many ways can you choose several numbers from the numbers $1,2,3,..., 11$ so that among the selected numbers there are not three consecutive numbers?

Find, and write out explicitly, a permutation $\{p(1),p(2),\ldots,p(20)\}$ of $\{1,2,\ldots,20\}$ such that $k+p(k)$ is a power of $2$ for $k=1,2,\ldots,20$, and prove that only one such permutation exists.

Let $I$ and $I_a$ be the incenter and excenter (opposite vertex $A$) of a triangle $ABC$, respectively. Let $A'$ be the point on its circumcircle opposite to $A$, and $A_1$ be the foot of the altitude from $A$. Prove that $\angle IA_1I_a=\angle IA'I_a$. [i](Proposed by Pavel Kozhevnikov)[/i]

In a country, some pairs of cities are connected by one-way roads. It turns out that every city has at least two out-going and two in-coming roads assigned to it, and from every city one can travel to any other city by a sequence of roads. Prove that it is possible to delete a cyclic route so that it is still possible to travel from any city to any other city.