For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?

A $ 9 \times 12$ rectangle is partitioned into unit squares. The centers of all the unit squares, except for the four corner squares and eight squares sharing a common side with one of them, are coloured red. Is it possible to label these red centres $ C_1,C_2,\ldots ,C_{96}$ in such way that the following to conditions are both fulfilled i) the distances $C_1C_2,\ldots ,C_{95}C_{96}, C_{96}C_{1}$ are all equal to $ \sqrt {13}$, ii) the closed broken line $ C_1C_2\ldots C_{96}C_1$ has a centre of symmetry? [i]Bulgaria[/i]

Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times 1, 1 \times 2, 1 \times 3$, or $3 \times 1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.

A rectangular sheet of paper whose dimensions are $12 \times 18$ is folded along a diagonal, creating the $M$-shaped region drawn in the picture (see below). Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/4/7/d82cde3e91ab83fa14cd6cefe9bba28462dde1.png[/img]

Let $ABCD$ be a rectangle. Points $E$ and $F$ are on diagonal $AC$ such that $F$ lies between $A$ and $E$; and $E$ lies between $C$ and $F$. The circumcircle of triangle $BEF$ intersects $AB$ and $BC$ at $G$ and $H$ respectively, and the circumcircle of triangle $DEF$ intersects $AD$ and $CD$ at $I$ and $J$ respectively. Prove that the lines $GJ, IH$ and $AC$ concur at a point.

Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure). (M. Evdokimov) [img]https://cdn.artofproblemsolving.com/attachments/f/f/9c3e345f33cabbbd83f65d7240aac29a163b19.png[/img]

A rectangle with a diagonal of length $ x$ is twice as long as it is wide. What is the area of the rectangle? $ \textbf{(A)}\ \frac14x^2 \qquad \textbf{(B)}\ \frac25x^2 \qquad \textbf{(C)}\ \frac12x^2 \qquad \textbf{(D)}\ x^2 \qquad \textbf{(E)}\ \frac32x^2$

Tiles I, II, III and IV are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$? [asy] size(400); defaultpen(linewidth(0.8)); path p=origin--(8,0)--(8,6)--(0,6)--cycle; draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p); draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p); label("8", (4,6+10), S); label("6", (4+8.5,6+10), S); label("7", (4,6), S); label("2", (4+8.5,6), S); label("I", (4,6+10), N); label("II", (4+8.5,6+10), N); label("III", (4,6), N); label("IV", (4+8.5,6), N); label("3", (0,3+10), E); label("4", (0+8.5,3+10), E); label("1", (0,3), E); label("9", (0+8.5,3), E); label("7", (4,10), N); label("2", (4+8.5,10), N); label("0", (4,0), N); label("6", (4+8.5,0), N); label("9", (8,3+10), W); label("3", (8+8.5,3+10), W); label("5", (8,3), W); label("1", (8+8.5,3), W); label("A", (24,10), N); label("B", (32,10), N); label("C", (24,4), N); label("D", (32,4), N); [/asy] $\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}\text{ cannot be determined}$

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.

The nine horizontal and nine vertical lines on an $8\times8$ checkerboard form $r$ rectangles, of which $s$ are squares. The number $s/r$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Rectangles $1\times20$, $1\times 19$, ..., $1\times 1$ were cut out of $20\times20$ table. Prove that at least 85 dominoes(1×2 rectangle) can be removed from the remainder. Proposed by S. Berlov

The numbers $p,q>0$ are given. Construct a rectangle $ABCD$ with $AE=p,AF=q$ where $E,F$ are midpoints of $BC,CD,$ respectively. Discuss conditions of solvability.

In rectangle $ ABCD$, $ AB\equal{}100$. Let $ E$ be the midpoint of $ \overline{AD}$. Given that line $ AC$ and line $ BE$ are perpendicular, find the greatest integer less than $ AD$.

The shaded region formed by the two intersecting perpendicular rectangles, in square units, is [asy] fill((0,0)--(6,0)--(6,-3.5)--(9,-3.5)--(9,0)--(10,0)--(10,2)--(9,2)--(9,4.5)--(6,4.5)--(6,2)--(0,2)--cycle,black); label("2",(0,.9),W); label("3",(7.3,4.5),N); draw((0,-3.3)--(0,-5.3),linewidth(1)); draw((0,-4.3)--(3.7,-4.3),linewidth(1)); label("10",(4.7,-3.7),S); draw((5.7,-4.3)--(10,-4.3),linewidth(1)); draw((10,-3.3)--(10,-5.3),linewidth(1)); draw((11,4.5)--(13,4.5),linewidth(1)); draw((12,4.5)--(12,2),linewidth(1)); label("8",(11.3,1),E); draw((12,0)--(12,-3.5),linewidth(1)); draw((11,-3.5)--(13,-3.5),linewidth(1));[/asy] $ \text{(A)}\ 23\qquad\text{(B)}\ 38\qquad\text{(C)}\ 44\qquad\text{(D)}\ 46\qquad\text{(E)}\ \text{unable to be determined from the information given} $