Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?

The incircle $\mathcal{C}_1$ of triangle $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. The incircle $\mathcal{C}_2$ of the triangle $ADE$ touches the sides $AB$ and $AC$ at the points $P$ and $Q$, and intersects the circle $\mathcal{C}_1$ at the points $M$ and $n$. Prove that (a) the center of the circle $\mathcal{C}_2$ lies on the circle $\mathcal{C}_1$. (b) the four points $M,N,P,Q$ in appropriate order form a rectangle if and only if twice the radius of $\mathcal{C}_1$ is three times the radius of $\mathcal{C}_2$.

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

A circle with radius $r$ is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$.The area of the rectangle in terms of $r$, is $\textbf{(A) }4r^2\qquad\textbf{(B) }6r^2\qquad\textbf{(C) }8r^2\qquad\textbf{(D) }12r^2\qquad \textbf{(E) }20r^2$

A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

Let $ ABC$ be an acute angled triangle and let $ H$ be its ortho centre. Let $ h_{max}$ denote the largest altitude of the triangle $ ABC$. Prove that: $AH \plus{} BH \plus{} CH\leq2h_{max}$

Let $ABCD$ be a rectangle of center $O$ in the plane $\alpha$, and let $V \notin\alpha$ be a point in space such that $V O \perp \alpha$. Let $A' \in (V A)$, $B'\in (V B)$, $C'\in (V C)$, $D'\in (V D)$ be four points, and let $M$ and $N$ be the midpoints of the segments $A'C'$ and $B'D'$. .Prove that $MN \parallel \alpha$ if and only if $V , A', B', C', D'$ all lie on a sphere.

What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{33}{100} \qquad \textbf{(C)}\ \frac{17}{50} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{18}{25}$

Let $A_1A_2A_3A_4$ be a rectangle and let $S_1,S_2,S_3,S_4$ four circumferences inside of the rectangle such that $S_k$ and $S_{k+1}$ are tangent to each other and tangent to the side $A_kA_{k+1}$ for $k=1,2,3,4$, where $A_5=A_1$ and $S_5=S_1$. Prove that $A_1A_2A_3A_4$ is a square.

The adjacent map is part of a city: the small rectangles are rocks, and the paths in between are streets. Each morning, a student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. For variety, at each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through $C$. [asy] defaultpen(linewidth(0.7)+fontsize(8)); size(250); path p=origin--(5,0)--(5,3)--(0,3)--cycle; path q=(5,19)--(6,19)--(6,20)--(5,20)--cycle; int i,j; for(i=0; i<5; i=i+1) { for(j=0; j<6; j=j+1) { draw(shift(6*i, 4*j)*p); }} clip((4,2)--(25,2)--(25,21)--(4,21)--cycle); fill(q^^shift(18,-16)*q^^shift(18,-12)*q, black); label("A", (6,19), SE); label("B", (23,4), NW); label("C", (23,8), NW); draw((26,11.5)--(30,11.5), Arrows(5)); draw((28,9.5)--(28,13.5), Arrows(5)); label("N", (28,13.5), N); label("W", (26,11.5), W); label("E", (30,11.5), E); label("S", (28,9.5), S);[/asy] $\textbf {(A) } \frac{11}{32} \qquad \textbf {(B) } \frac 12 \qquad \textbf {(C) } \frac 47 \qquad \textbf {(D) } \frac{21}{32} \qquad \textbf {(E) } \frac 34$

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

Let $ ABC$ be an acute-angled triangle; $ AD$ be the bisector of $ \angle BAC$ with $ D$ on $ BC$; and $ BE$ be the altitude from $ B$ on $ AC$. Show that $ \angle CED > 45^\circ .$ [b][weightage 17/100][/b]

Consider a convex $n$-gon $A_1A_2 \dots A_n$. (Note: In a convex polygon, all interior angles are less than $180 \circ$.) Let $h$ be a positive number. Using the sides of the polygon as bases, we draw $n$ rectangles, each of height $h$, so that each rectangle is either entirely inside the $n$-gon or partially overlaps the inside of the $n$-gon. As an example, the left figure below shows a pentagon with a correct configuration of rectangles, while the right figure shows an incorrect configuration of rectangles (since some of the rectangles do not overlap with the pentagon):

Let $f:[0,1]\to\mathbb{R}$ be a continuous function and let $\{a_n\}_n$, $\{b_n\}_n$ be sequences of reals such that \[ \lim_{n\to\infty} \int^1_0 | f(x) - a_nx - b_n | dx = 0 . \] Prove that: a) The sequences $\{a_n\}_n$, $\{b_n\}_n$ are convergent; b) The function $f$ is linear.

The cells of a $11\times 11$ chess-board are colored in 3 colors. Prove that there exists on the board a $m\times n$ rectangle such that the four cells interior to the rectangle and containing the four vertices of the rectangle have the same color. [i]Ioan Tomescu[/i]

There are several equal (possibly overlapping) square-shaped napkins on a rectangular table, with sides parallel to the sides of the table. Prove that it is possible to nail some of them to the table in such a way that every napkin is nailed exactly once.

A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.

Let $S$ be the set of ordered pairs $(x, y)$ such that $0<x\le 1$, $0<y\le 1$, and $\left[\log_2{\left(\frac 1x\right)}\right]$ and $\left[\log_5{\left(\frac 1y\right)}\right]$ are both even. Given that the area of the graph of $S$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. The notation $[z]$ denotes the greatest integer that is less than or equal to $z$.

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ ad $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares? [asy] pair A,B,C,D,E,F,G,H,I,J; A = (0.5,2); B = (1.5,2); C = (1.5,1); D = (0.5,1); E = (0,1); F = (0,0); G = (1,0); H = (1,1); I = (2,1); J = (2,0); draw(A--B); draw(C--B); draw(D--A); draw(F--E); draw(I--J); draw(J--F); draw(G--H); draw(A--J); filldraw(A--B--C--I--J--cycle,grey); draw(E--I); dot("$A$", A, NW); dot("$B$", B, NE); dot("$C$", C, NE); dot("$D$", D, NW); dot("$E$", E, NW); dot("$F$", F, SW); dot("$G$", G, S); dot("$H$", H, N); dot("$I$", I, NE); dot("$J$", J, SE);[/asy] $\textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac7{24} \qquad \textbf{(C)}\ \frac13 \qquad \textbf{(D)}\ \frac38 \qquad \textbf{(E)}\ \frac5{12}$

(a) A regular $4k$-gon is cut into parallelograms. Prove that among these there are at least $k$ rectangles. (b) Find the total area of the rectangles in (a) if the lengths of the sides of the $4k$-gon equal $a$. (VV Proizvolov, Moscow)

$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle. (M Panov)

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); draw(scale(4)*unitsquare); draw((0,3)--(4,3)); draw((1,3)--(1,4)); draw((2,3)--(2,4)); draw((3,3)--(3,4));[/asy]$ \textbf{(A)}\ \frac {5}{4} \qquad \textbf{(B)}\ \frac {4}{3} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$

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} $

Find all rectangles that can be cut into $13$ equal squares.