The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then the number of possible coverings is even. [i]D. Karpov, C. Gukshin, D. Fon-der-Flaas[/i]

A non-square rectangle is cut into $N$ rectangles of various shapes and sizes. Prove that one can always cut each of these rectangles into two rectangles so that one can construct a square and rectangle, each figure consisting of $N$ pieces. [i](6 points)[/i]

In an acute triangle $ABC,$ let $O,G,H$ be its circumcentre, centroid and orthocenter. Let $D\in BC, E\in CA$ and $OD\perp BC, HE\perp CA.$ Let $F$ be the midpoint of $AB.$ If the triangles $ODC, HEA, GFB$ have the same area, find all the possible values of $\angle C.$

Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. [i]Ray Li.[/i]

Let $ABCD$ be a rectangle with sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $ A$ intersects $BD$ at the point $H$. We denote by $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the length of segment $MN$.

Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$. [b](a)[/b] Prove that $r_{1}+r_{2}=2r$. [b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle? [asy] draw((0,8)--(0,0)--(4,0)--(4,8)--(0,8)--(3.5,8.5)--(3.5,8)); draw((2,-1)--(2,9),dashed);[/asy] $ \text{(A)}\ \frac{1}{3}\qquad\text{(B)}\ \frac{1}{2}\qquad\text{(C)}\ \frac{3}{4}\qquad\text{(D)}\ \frac{4}{5}\qquad\text{(E)}\ \frac{5}{6} $

Consider a $1 \times n$ rectangle and some tiles of size $1 \times 1$ of four different colours. The rectangle is tiled in such a way that no two neighboring square tiles have the same colour. a) Find the number of distinct symmetrical tilings. b) Find the number of tilings such that any consecutive square tiles have distinct colours.

The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is dibided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: $\text{(A)} \ \frac 32 \qquad \text{(B)} \ \frac 53 \qquad \text{(C)} \ \frac 52 \qquad \text{(D)} \ \frac13\sqrt{34} \qquad \text{(E)} \ \frac13\sqrt{68}$

There are $n$ rectangles drawn on the rectangular sheet of paper with the sides of the rectangles parallel to the sheet sides. The rectangles do not have pairwise common interior points. Prove that after cutting out the rectangles the sheet will split into not more than $n+1$ part.

Consider on the coordinate plane all rectangles whose (i) vertices have integer coordinates; (ii) edges are parallel to coordinate axes; (iii) area is $2^k$, where $k = 0,1,2....$ Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?

There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

There are $2024$ rectangles $1 \times n$ for $n=1, 2, \ldots, 2024$. Is it possible to make a square using some of them, such that the side length of the square is greater than $1$?

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] $\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}\text{ cannot be determined}$

A cylinder radius $12$ and a cylinder radius $36$ are held tangent to each other with a tight band. The length of the band is $m\sqrt{k}+n\pi$ where $m$, $k$, and $n$ are positive integers, and $k$ is not divisible by the square of any prime. Find $m + k + n$. [asy] size(150); real t=0.3; void cyl(pair x, real r, real h) { pair xx=(x.x,t*x.y); path B=ellipse(xx,r,t*r), T=ellipse((x.x,t*x.y+h),r,t*r), S=xx+(r,0)--xx+(r,h)--(xx+(-r,h))--xx-(r,0); unfill(S--cycle); draw(S); unfill(B); draw(B); unfill(T); draw(T); } real h=8, R=3,r=1.2; pair X=(0,0), Y=(R+r)*dir(-50); cyl(X,R,h); draw(shift((0,5))*yscale(t)*arc(X,R,180,360)); cyl(Y,r,h); void str (pair x, pair y, real R, real r, real h, real w) { real u=(angle(y-x)+asin((R-r)/(R+r)))*180/pi+270; path P=yscale(t)*(arc(x,R,180,u)--arc(y,r,u,360)); path Q=shift((0,h))*P--shift((0,h+w))*reverse(P)--cycle; fill(Q,grey);draw(Q); } str(X,Y,R,r,3.5,1.5);[/asy]

A $ 100 \times 100$ square floor consisting of $ 10000$ squares is to be tiled by rectangular $ 1 \times 3$ tiles, fitting exactly over three squares of the floor. $ (a)$ If a $ 2 \times 2$ square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles. $ (b)$ If, instead, a $ 2 \times 2$ square is removed from the corner, prove that such a tiling is not possble.

Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.

Positive real numbers, $a_1, .. ,a_6$ satisfy $a_1^2+..+a_6^2=2$. Think six squares that has side length of $a_i$ ($i=1,2,\ldots,6$). Show that the squares can be packed inside a square of length $2$, without overlapping.

On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

A point $K$ is chosen on the side $CD$ of a rectangle $ABCD$. From the vertex $B$, the perpendicular $BH$ is dropped to the segment $AK$. The segments $AK$ and $BH$ divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to $CD$ are equal then the third circle is also equal to them.

The area of polygon $ ABCDEF$, in square units, is [asy]draw((0,0)--(4,0)--(4,9)--(-2,9)--(-2,4)--(0,4)--cycle); label("A",(-2,9),NW); label("B",(4,9),NE); label("C",(4,0),SE); label("D",(0,0),SW); label("E",(0,4),NE); label("F",(-2,4),SW); label("5",(-2,6.5),W); label("4",(2,0),S); label("9",(4,4.5),E); label("6",(1,9),N); label("All angles in this diagram are right.",(0,-3),S);[/asy] \[ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74 \]

Given a rectangle $ABCD$ such that $AB = b > 2a = BC$, let $E$ be the midpoint of $AD$. On a line parallel to $AB$ through point $E$, a point $G$ is chosen such that the area of $GCE$ is $$(GCE)= \frac12 \left(\frac{a^3}{b}+ab\right)$$ Point $H$ is the foot of the perpendicular from $E$ to $GD$ and a point $I$ is taken on the diagonal $AC$ such that the triangles $ACE$ and $AEI$ are similar. The lines $BH$ and $IE$ intersect at $K$ and the lines $CA$ and $EH$ intersect at $J$. Prove that $KJ \perp AB$.

Given a rectangle $ABCD$, points $M$ -- the midpoint of $[AD]$ side, $N$ -- the midpoint of $[BC]$ side. Let us take a point $P$ on the extension of the $[DC]$ segment over the point $D$. Let us denote the intersection point of lines $(PM)$ and $(AC)$ as $Q$. Prove that the $\angle QNM= \angle MNP$