Let $ABCD$ be a square with side length $4$. Consider points $P$ and $Q$ on segments $AB$ and $BC$, respectively, with $BP=3$ and $BQ=1$. Let $R$ be the intersection of $AQ$ and $DP$. If $BR^2$ can be expressed in the form $\frac{m}{n}$ for coprime positive integers $m,n$, compute $m+n$. [i]Proposed by Brandon Wang[/i]

find all the function $f,g:R\rightarrow R$ such that (1)for every $x,y\in R$ we have $f(xg(y+1))+y=xf(y)+f(x+g(y))$ (2)$f(0)+g(0)=0$

The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in each subgrid. [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle, linewidth(2)); draw((0,1)--(9,1)); draw((0,2)--(9,2)); draw((0,3)--(9,3), linewidth(2)); draw((0,4)--(9,4)); draw((0,5)--(9,5)); draw((0,6)--(9,6), linewidth(2)); draw((0,7)--(9,7)); draw((0,8)--(9,8)); draw((1,0)--(1,9)); draw((2,0)--(2,9)); draw((3,0)--(3,9), linewidth(2)); draw((4,0)--(4,9)); draw((5,0)--(5,9)); draw((6,0)--(6,9), linewidth(2)); draw((7,0)--(7,9)); draw((8,0)--(8,9)); [/asy] Now consider an $n^2 \times n^2$ grid divided into $n^2$ subgrids of size $n \times n$. Find the number of ways in which you can select $n^2$ cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.

The squares of an $8 \times 8$ board are colored black and white in such a way that every row and every column contains exactly four black squares. Prove that the number of pairs of neighboring white squares is the same as the number of pairs of neighboring black squares. (Two squares are neighboring if they have a side in common.)

Determine the number of ordered triples $(a, b, c)$, with $0 \le a, b, c \le 10$ for which there exists $(x, y)$ such that $ax^2 + by^2 \equiv c$ (mod $11$)

Let $r>0$ be a real number. We call a monic polynomial with complex coefficients $r$-[i]good[/i] if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients [i]primordial[/i] if all of its coefficients have absolute value at most $1$. a) Prove that any $1$-good polynomial has a primordial multiple. b) If $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. [i]Proposed by Pranjal Srivastava[/i]

How many solutions does the equation $\tan{(2x)} = \cos{(\tfrac{x}{2})}$ have on the interval $[0, 2\pi]?$ $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$

For the infinite series $1-\frac12-\frac14+\frac18-\frac{1}{16}-\frac{1}{32}+\frac{1}{64}-\frac{1}{128}-\cdots$ let $S$ be the (limiting) sum. Then $S$ equals: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac27\qquad\textbf{(C)}\ \frac67\qquad\textbf{(D)}\ \frac{9}{32}\qquad\textbf{(E)}\ \frac{27}{32} $

There are only two letters in the Mumu tribe alphabet: M and $U$. The word in the Mumu language is any sequence of letters $M$ and $U$, in which next to each letter $M$ there is a letter $U$ (for example, $UUU$ and $UMMUM$ are words and $MMU$ is not). Let $f(m,u)$ denote the number of words in the Mumu language which have $m$ times the letter $M$ and $u$ times the letter $U$. Prove that $f (m, u) - f (2u - m + 1, u) = f (m, u - 1) - f (2u - m + 1, u - 1)$ for any $u \ge 2,3 \le m \le 2u$.

A train traveling at $ 80$ mph begins to cross a $ 1$ mile long bridge. At this moment, a man begins to walk from the front of the train to the back of the train at a speed of $5$ mph. The man reaches the back of the train as soon as the train is completely off the bridge. What is the length of the train (as a fraction of a mile)?

Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$, $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$?

Suppose $x,y\in\mathbb{Z}$ satisfy $$y^4+4y^3+28y+8x^3+6y^2+32x+1=(x^2-y^2)(x^2+y^2+24).$$ Find the sum of all possible values of $|xy|$.

p1. Four kite-shaped shapes as shown below ($a > b$, $a$ and $b$ are natural numbers less than $10$) arranged in such a way so that it forms a square with holes in the middle. The square hole in the middle has a perimeter of $16$ units of length. What is the possible perimeter of the outermost square formed if it is also known that $a$ and $b$ are numbers coprime? [img]https://cdn.artofproblemsolving.com/attachments/4/1/fa95f5f557aa0ca5afb9584d5cee74743dcb10.png[/img] p2. If $a = 3^p$, $b = 3^q$, $c = 3^r$, and $d = 3^s$ and if $p, q, r$, and $s$ are natural numbers, what is the smallest value of $p\cdot q\cdot r\cdot s$ that satisfies $a^2 + b^3 + c^5 = d^7$ 3. Ucok intends to compose a key code (password) consisting of 8 numbers and meet the following conditions: i. The numbers used are $1, 2, 3, 4, 5, 6, 7, 8$, and $9$. ii. The first number used is at least $1$, the second number is at least $2$, third digit-at least $3$, and so on. iii. The same number can be used multiple times. a) How many different passwords can Ucok compose? b) How many different passwords can Ucok make, if provision (iii) is replaced with: no numbers may be used more than once. p 4. For any integer $a, b$, and $c$ applies $a\times (b + c) = (a\times b) + (a\times c)$. a) Look for examples that show that $a + (b\times c)\ne (a + b)\times (a + c)$. b) Is it always true that $a + (b\times c) = (a + b)\times(a + c)$? Justify your answer. p5. The results of a survey of $N$ people with the question whether they maintain dogs, birds, or cats at home are as follows: $50$ people keep birds, $61$ people don't have dogs, $13$ people don't keep a cat, and there are at least $74$ people who keep the most a little two kinds of animals in the house. What is the maximum value and minimum of possible value of $N$ ?

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.

A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? [asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9)); [/asy] $\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $

The nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$. Can the product $xy$ be a negative number? [i]Proposed by N. Agakhanov[/i]

We want to place $2012$ pockets, including variously colored balls, into $k$ boxes such that [b]i)[/b] For any box, all pockets in this box must include a ball with the same color or [b]ii)[/b] For any box, all pockets in this box must include a ball having a color which is not included in any other pocket in this box Find the smallest value of $k$ for which we can always do this placement whatever the number of balls in the pockets and whatever the colors of balls.

Let the integer $n \ge 2$ , and $x_1,x_2,\cdots,x_n $ be real numbers such that $\sum_{k=1}^nx_k$ be integer . $d_k=\underset{m\in {Z}}{\min}\left|x_k-m\right| $, $1\leq k\leq n$ .Find the maximum value of $\sum_{k=1}^nd_k$.

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3,\hdots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$.

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.

If $0 < x < \frac{\pi}{2}$ and $\frac{\sin x}{1 + \cos x} = \frac{1}{3}$, what is $\frac{\sin 2x}{1 + \cos 2x}$?

Jay has a $24\times 24$ grid of lights, all of which are initially off. Each of the $48$ rows and columns has a switch that toggles all the lights in that row and column, respectively, i.e. it switches lights that are on to off and lights that are off to on. Jay toggles each of the $48$ rows and columns exactly once, such that after each toggle he waits for one minute before the next toggle. Each light uses no energy while off and 1 kiloJoule of energy per minute while on. To express his creativity, Jay chooses to toggle the rows and columns in a random order. Compute the expected value of the total amount of energy in kiloJoules which has been expended by all the lights after all $48$ toggles. [i]Proposed by James Lin

Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.