The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building. [asy] unitsize(5mm); defaultpen(linewidth(.8pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<= 6; ++j) { draw((0,i)--(9,i)); draw((1.5*j,0)--(1.5*j,5)); }} dot((.75, .5)); label("$\ast$",(8.25,4.5)); dot((11, 3)); label("$\ast$",(11,1.75)); label("room with locked external door",(18,1.9)); label("starting position",(15.3,3)); [/asy]

In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates. [b]PS: The rays can be lines and the problem is still true.[/b]

Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10); draw(A--B--C--D--E--F--G--A); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, S); label("$D$", D, NE); label("$E$", E, SE); label("$F$", F, N); label("$G$", G, SE); label("$5$", (11,7.5)); label("$5$", (11,-7.5)); label("$20$", (C+D)/2, N); label("$20$", (F+E)/2, S); label("$10$", (31,0)); [/asy] $ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $

Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]

A [i]tetromino[/i] consists of four squares connected along their edges. There are five possible tetromino shapes, I, O, L, T, S, shown below, which can be rotated or flipped over. Three tetrominos are used to completely cover a $3\times 4$ rectangle. At least one of the titles is an S tile. What are the other two tiles? [img]https://i.imgur.com/9Nxq4y6.png[/img] $\textbf{(A) } \text{I and L} \qquad\textbf{(B) }\text{I and T} \qquad\textbf{(C) }\text{L and L}\qquad\textbf{(D) }\text{L and S} \qquad\textbf{(E) }\text{O and T}$\\

Let $ABC$ be an acute-angled triangle with $AB<AC$, and let $O,H$ be its circumcentre and orthocentre respectively. Points $Z,Y$ lie on segments $AB,AC$ respectively, such that \[\angle ZOB=\angle YOC = 90^{\circ}.\] The perpendicular line from $H$ to line $YZ$ meets lines $BO$ and $CO$ at $Q,R$ respectively. Let the tangents to the circumcircle of $\triangle AYZ$ at points $Y$ and $Z$ meet at point $T$. Prove that $Q, R, O, T$ are concyclic. [i]Proposed by Kazi Aryan Amin and K.V. Sudharshan[/i]

A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years?

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.

Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1\plus{}ax_2\equal{}0$ $ x_2\plus{}a^2x_3\equal{}0$ … $ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$ … $ x_n\plus{}a^nx_1\equal{}0$

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

Let $ABC$ be an acute triangle. Let $J$ be the $A$-excenter. The $A$-excircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The line $DE$ intersects $CJ$ and $BJ$ at $P$ and $Q$, respectively. $M$ is the midpoint of $AD$. Prove that $PM=QM$.

For every non-negative integer $k$ let $S(k)$ denote the sum of decimal digits of $k$. Let $P(x)$ and $Q(x)$ be polynomials with non-negative integer coecients such that $S(P(n)) = S(Q(n))$ for all non-negative integers $n$. Prove that there exists an integer $t$ such that $P(x) - 10^tQ(x)$ is a constant polynomial.

Suppose $n$ is odd and each square of an $n \times n$ grid is arbitrarily filled with either by $1$ or by $-1$. Let $r_j$ and $c_k$ denote the product of all numbers in $j$-th row and $k$-th column respectively, $1 \le j, k \le n$. Prove that $$\sum_{j=1}^{n} r_j+ \sum_{k=1}^{n} c_k\ne 0$$

Find the value of $\left\lfloor \frac{1}{6}\right\rfloor+\left\lfloor\frac{4}{6}\right\rfloor+\left\lfloor\frac{9}{6}\right\rfloor+\dots+\left\lfloor\frac{1296}{6}\right\rfloor$. [i]Proposed by Zachary Perry[/i]

$(a_{n})$ is sequence with positive integer. $a_{1}>10$ $ a_{n}=a_{n-1}+GCD(n,a_{n-1})$, n>1 For some i $a_{i}=2i$. Prove that these numbers are infinite in this sequence.

There is a pile of $1000$ matches. Two players each take turns and can take $1$ to $5$ matches. It is also allowed at most $10$ times during the whole game to take $6$ matches, for example $7$ exceptional moves can be done by the first player and $3$ moves by the second and then no more exceptional moves are allowed. Whoever takes the last match wins. Determine which player has a winning strategy.

Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? $\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

Let $n$ be a natural number. Prove that, \[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \] is even.

There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students. [i]Proposed by Yifan Kang[/i]

Each integer should be colored by one of three colors, including red. Each number which can be represent as a sum of two numbers of different colors should be red. Each color should be used. Is this coloring possible?