In the diagram below, square $ABCD$ has side length $4$. Two congruent square $EGIK$ and $FHJL$ are drawn such that $AE=FB=BG=HC=CI=JD=DK=LA=1$ and $EF=GH=IJ=KL=2$. Compute the area of the region that lies in both $EGIK$ and $FHJL$.

Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

Let $c>1$ be a real constant. For the sequence $a_1,a_2,...$ we have: $a_1=1$, $a_2=2$, $a_{mn}=a_m a_n$, and $a_{m+n}\leq c(a_m+a_n)$. Prove that $a_n=n$.

Let $ ABC$ be a triangle with $ \angle BAC \equal{} 90^\circ$. A circle is tangent to the sides $ AB$ and $ AC$ at $ X$ and $ Y$ respectively, such that the points on the circle diametrically opposite $ X$ and $ Y$ both lie on the side $ BC$. Given that $ AB \equal{} 6$, find the area of the portion of the circle that lies outside the triangle. [asy]import olympiad; import math; import graph; unitsize(20mm); defaultpen(fontsize(8pt)); pair A = (0,0); pair B = A + right; pair C = A + up; pair O = (1/3, 1/3); pair Xprime = (1/3,2/3); pair Yprime = (2/3,1/3); fill(Arc(O,1/3,0,90)--Xprime--Yprime--cycle,0.7*white); draw(A--B--C--cycle); draw(Circle(O, 1/3)); draw((0,1/3)--(2/3,1/3)); draw((1/3,0)--(1/3,2/3)); label("$A$",A, SW); label("$B$",B, down); label("$C$",C, left); label("$X$",(1/3,0), down); label("$Y$",(0,1/3), left);[/asy]

How many subsets of the set $\{1, 2, \ldots, 11\}$ have median 6? [i]Proposed by Michael Tang

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 5$

In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$. (Maria Rozhkova)

Given that $5^3+5^3 + 5^3 + 5^3 + 5^3 = 5^J$ and $3^2 + 3^2 + 3^2 = 3^N$ , what is the value of $J^ N$ ?

Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

Let $x_0, x_1, . . . , x_{2014}$ be a sequence of real numbers, which for all $i < j$ satisfy $x_i + x_j \le 2j$. Determine the largest possible value of the sum $x_0 + x_1 + · · · + x_{2014}$.

The position [i]vs.[/i] time graph for an object moving in a straight line is shown below. What is the instantaneous velocity at $\text{t = 2 s}$? [asy] size(300); // x-axis draw((0,0)--(12,0)); // x-axis tick marks draw((4,0)--(4,-0.2)); draw((8,0)--(8,-0.2)); draw((12,0)--(12,-0.2)); // x-axis labels label("1",(4,0),2*S); label("2",(8,0),2*S); label("3",(12,0),2*S); //y-axis draw((0,-3)--(0,5)); //y-axis tick marks draw((-0.2,-2)--(0,-2)); draw((-0.2,0)--(0,0)); draw((-0.2,2)--(0,2)); draw((-0.2,4)--(0,4)); // y-axis labels label("-2",(0,-2),2*W); label("0",(0,0),2*W); label("2",(0,2),2*W); label("4",(0,4),2*W); // Axis Labels label("Time (s)",(10,0),N); label(rotate(90)*"Position (m)",(0,1),6*W); //Line draw((0,4)--(12,-2)); [/asy] (a) $-\text{2 m/s}$ (b) $-\frac{1}{2} \ \text{m/s}$ (c) $\text{0 m/s}$ (d) $\text{2 m/s}$ (e) $\text{4 m/s}$

$ A$ takes $ m$ times as long to do a piece of work as $ B$ and $ C$ together; $ B$ takes $ n$ times as long as $ C$ and $ A$ together; and $ C$ takes $ x$ times as long as $ A$ and $ B$ together. Then $ x$, in terms of $ m$ and $ n$, is: $ \textbf{(A)}\ \frac {2mn}{m \plus{} n} \qquad \textbf{(B)}\ \frac {1}{2(m \plus{} n)} \qquad \textbf{(C)}\ \frac {1}{m \plus{} n \minus{} mn} \qquad \textbf{(D)}\ \frac {1 \minus{} mn}{m \plus{} n \plus{} 2mn} \qquad \textbf{(E)}\ \frac {m \plus{} n \plus{} 2}{mn \minus{} 1}$

Given $n\ge 3$. consider a sequence $a_1,a_2,...,a_n$, if $(a_i,a_j,a_k)$ with i+k=2j (i<j<k) and $a_i+a_k\ne 2a_j$, we call such a triple a $NOT-AP$ triple. If a sequence has at least one $NOT-AP$ triple, find the least possible number of the $NOT-AP$ triple it contains.

When $126$ is added to its reversal, $621,$ the sum is $126 + 621 = 747.$ Find the greatest integer which when added to its reversal yields $1211.$

Let $n$ be a positive integer. Solve the system of equations \begin{align*}x_{1}+2x_{2}+\cdots+nx_{n}&= \frac{n(n+1)}{2}\\ x_{1}+x_{2}^{2}+\cdots+x_{n}^{n}&= n\end{align*} for $n$-tuples $(x_{1},x_{2},\ldots,x_{n})$ of nonnegative real numbers.

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

Twenty cubical blocks are arranged as shown. First, $10$ are arranged in a triangular pattern; then a layer of $6$, arranged in a triangular pattern, is centered on the $10$; then a layer of $3$, arranged in a triangular pattern, is centered on the $6$; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered $1$ through $10$ in some order. Each block in layers $2, 3$ and $4$ is assigned the number which is the sum of the numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block. [asy] size((400)); draw((0,0)--(5,0)--(5,5)--(0,5)--(0,0), linewidth(1)); draw((5,0)--(10,0)--(15,0)--(20,0)--(20,5)--(15,5)--(10,5)--(5,5)--(6,7)--(11,7)--(16,7)--(21,7)--(21,2)--(20,0), linewidth(1)); draw((10,0)--(10,5)--(11,7), linewidth(1)); draw((15,0)--(15,5)--(16,7), linewidth(1)); draw((20,0)--(20,5)--(21,7), linewidth(1)); draw((0,5)--(1,7)--(6,7), linewidth(1)); draw((3.5,7)--(4.5,9)--(9.5,9)--(14.5,9)--(19.5,9)--(18.5,7)--(19.5,9)--(19.5,7), linewidth(1)); draw((8.5,7)--(9.5,9), linewidth(1)); draw((13.5,7)--(14.5,9), linewidth(1)); draw((7,9)--(8,11)--(13,11)--(18,11)--(17,9)--(18,11)--(18,9), linewidth(1)); draw((12,9)--(13,11), linewidth(1)); draw((10.5,11)--(11.5,13)--(16.5,13)--(16.5,11)--(16.5,13)--(15.5,11), linewidth(1)); draw((25,0)--(30,0)--(30,5)--(25,5)--(25,0), dashed); draw((30,0)--(35,0)--(40,0)--(45,0)--(45,5)--(40,5)--(35,5)--(30,5)--(31,7)--(36,7)--(41,7)--(46,7)--(46,2)--(45,0), dashed); draw((35,0)--(35,5)--(36,7), dashed); draw((40,0)--(40,5)--(41,7), dashed); draw((45,0)--(45,5)--(46,7), dashed); draw((25,5)--(26,7)--(31,7), dashed); draw((28.5,7)--(29.5,9)--(34.5,9)--(39.5,9)--(44.5,9)--(43.5,7)--(44.5,9)--(44.5,7), dashed); draw((33.5,7)--(34.5,9), dashed); draw((38.5,7)--(39.5,9), dashed); draw((32,9)--(33,11)--(38,11)--(43,11)--(42,9)--(43,11)--(43,9), dashed); draw((37,9)--(38,11), dashed); draw((35.5,11)--(36.5,13)--(41.5,13)--(41.5,11)--(41.5,13)--(40.5,11), dashed); draw((50,0)--(55,0)--(55,5)--(50,5)--(50,0), dashed); draw((55,0)--(60,0)--(65,0)--(70,0)--(70,5)--(65,5)--(60,5)--(55,5)--(56,7)--(61,7)--(66,7)--(71,7)--(71,2)--(70,0), dashed); draw((60,0)--(60,5)--(61,7), dashed); draw((65,0)--(65,5)--(66,7), dashed); draw((70,0)--(70,5)--(71,7), dashed); draw((50,5)--(51,7)--(56,7), dashed); draw((53.5,7)--(54.5,9)--(59.5,9)--(64.5,9)--(69.5,9)--(68.5,7)--(69.5,9)--(69.5,7), dashed); draw((58.5,7)--(59.5,9), dashed); draw((63.5,7)--(64.5,9), dashed); draw((57,9)--(58,11)--(63,11)--(68,11)--(67,9)--(68,11)--(68,9), dashed); draw((62,9)--(63,11), dashed); draw((60.5,11)--(61.5,13)--(66.5,13)--(66.5,11)--(66.5,13)--(65.5,11), dashed); draw((75,0)--(80,0)--(80,5)--(75,5)--(75,0), dashed); draw((80,0)--(85,0)--(90,0)--(95,0)--(95,5)--(90,5)--(85,5)--(80,5)--(81,7)--(86,7)--(91,7)--(96,7)--(96,2)--(95,0), dashed); draw((85,0)--(85,5)--(86,7), dashed); draw((90,0)--(90,5)--(91,7), dashed); draw((95,0)--(95,5)--(96,7), dashed); draw((75,5)--(76,7)--(81,7), dashed); draw((78.5,7)--(79.5,9)--(84.5,9)--(89.5,9)--(94.5,9)--(93.5,7)--(94.5,9)--(94.5,7), dashed); draw((83.5,7)--(84.5,9), dashed); draw((88.5,7)--(89.5,9), dashed); draw((82,9)--(83,11)--(88,11)--(93,11)--(92,9)--(93,11)--(93,9), dashed); draw((87,9)--(88,11), dashed); draw((85.5,11)--(86.5,13)--(91.5,13)--(91.5,11)--(91.5,13)--(90.5,11), dashed); draw((28,6)--(33,6)--(38,6)--(43,6)--(43,11)--(38,11)--(33,11)--(28,11)--(28,6), linewidth(1)); draw((28,11)--(29,13)--(34,13)--(39,13)--(44,13)--(43,11)--(44,13)--(44,8)--(43,6), linewidth(1)); draw((33,6)--(33,11)--(34,13)--(39,13)--(38,11)--(38,6), linewidth(1)); draw((31,13)--(32,15)--(37,15)--(36,13)--(37,15)--(42,15)--(41,13)--(42,15)--(42,13), linewidth(1)); draw((34.5,15)--(35.5,17)--(40.5,17)--(39.5,15)--(40.5,17)--(40.5,15), linewidth(1)); draw((53,6)--(58,6)--(63,6)--(68,6)--(68,11)--(63,11)--(58,11)--(53,11)--(53,6), dashed); draw((53,11)--(54,13)--(59,13)--(64,13)--(69,13)--(68,11)--(69,13)--(69,8)--(68,6), dashed); draw((58,6)--(58,11)--(59,13)--(64,13)--(63,11)--(63,6), dashed); draw((56,13)--(57,15)--(62,15)--(61,13)--(62,15)--(67,15)--(66,13)--(67,15)--(67,13), dashed); draw((59.5,15)--(60.5,17)--(65.5,17)--(64.5,15)--(65.5,17)--(65.5,15), dashed); draw((78,6)--(83,6)--(88,6)--(93,6)--(93,11)--(88,11)--(83,11)--(78,11)--(78,6), dashed); draw((78,11)--(79,13)--(84,13)--(89,13)--(94,13)--(93,11)--(94,13)--(94,8)--(93,6), dashed); draw((83,6)--(83,11)--(84,13)--(89,13)--(88,11)--(88,6), dashed); draw((81,13)--(82,15)--(87,15)--(86,13)--(87,15)--(92,15)--(91,13)--(92,15)--(92,13), dashed); draw((84.5,15)--(85.5,17)--(90.5,17)--(89.5,15)--(90.5,17)--(90.5,15), dashed); draw((56,12)--(61,12)--(66,12)--(66,17)--(61,17)--(56,17)--(56,12), linewidth(1)); draw((61,12)--(61,17)--(62,19)--(57,19)--(56,17)--(57,19)--(67,19)--(66,17)--(67,19)--(67,14)--(66,12), linewidth(1)); draw((59.5,19)--(60.5,21)--(65.5,21)--(64.5,19)--(65.5,21)--(65.5,19), linewidth(1)); draw((81,12)--(86,12)--(91,12)--(91,17)--(86,17)--(81,17)--(81,12), dashed); draw((86,12)--(86,17)--(87,19)--(82,19)--(81,17)--(82,19)--(92,19)--(91,17)--(92,19)--(92,14)--(91,12), dashed); draw((84.5,19)--(85.5,21)--(90.5,21)--(89.5,19)--(90.5,21)--(90.5,19), dashed); draw((84,18)--(89,18)--(89,23)--(84,23)--(84,18)--(84,23)--(85,25)--(90,25)--(89,23)--(90,25)--(90,20)--(89,18), linewidth(1));[/asy] $ \textbf{(A)}\ 55 \qquad\textbf{(B)}\ 83 \qquad\textbf{(C)}\ 114 \qquad\textbf{(D)}\ 137 \qquad\textbf{(E)}\ 144 $

There are $3$ clubs $A,B,C$ with non-empty members. For any triplet of members $(a, b, c)$ with $a \in A, b \in B, c \in C$, two of them are friend and two of them are not friend (here the friend relationship is bidirectional). Prove that one of these statements must be true 1. There exist one student from $A$ that knows all students from $B$ 2. There exist one student from $B$ that knows all students from $C$ 3. There exist one student from $C$ that knows all students from $A$

Let $\tau(n)$ be the number of positive divisors of $n$. Let $\tau_1(n)$ be the number of positive divisors of $n$ which have remainders $1$ when divided by $3$. Find all positive integral values of the fraction $\frac{\tau(10n)}{\tau_1(10n)}$.

Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC =5$. If $D$ is the projection from $B$ onto $AC$, $E$ is the projection from $D$ onto $BC$, and $F$ is the projection from $E$ onto $AC$, compute the length of the segment $DF$. [i]2016 CCA Math Bonanza Individual #5[/i]

Let $ABC$ be an equilateral triangle of sidelength 2 and let $\omega$ be its incircle. a) Show that for every point $P$ on $\omega$ the sum of the squares of its distances to $A$, $B$, $C$ is 5. b) Show that for every point $P$ on $\omega$ it is possible to construct a triangle of sidelengths $AP$, $BP$, $CP$. Also, the area of such triangle is $\frac{\sqrt{3}}{4}$.

The incircle of the triangle $\triangle{ABC}$ has center at $O$ and it is tangent to the sides $BC$, $AC$ and $AB$ at the points $X$, $Y$ and $Z$, respectively. The lines $BO$ and $CO$ intersect the line $YZ$ at the points $P$ and $Q$, respectively. Show that if the segments $XP$ and $XQ$ has the same length, then the triangle $\triangle ABC$ is isosceles.

Show that real numbers, $p, q, r$ satisfy the condition $p^4(q-r)^2 + 2p^2(q+r) + 1 = p^4$ if and only if the quadratic equations $x^2 + px + q = 0$ and $y^2 - py + r = 0$ have real roots (not necessarily distinct) which can be labeled by $x_1,x_2$ and $y_1,y_2$, respectively, in such a way that $x_1y_1 - x_2y_2 = 1$.

The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?

Arnav is placing three rectangles into a $3 \times 3$ grid of unit squares. He has a $1\times 3$ rectangle, a $1\times 2$ rectangle, and a $1\times 1$ rectangle. He must place the rectangles onto the grid such that the edges of the rectangles align with the gridlines of the grid. If he is allowed to rotate the rectangles, how many ways can he place the three rectangles into the grid, without overlap? [i]Proposed by William Yue[/i]