[asy] import cse5; size(180); real a=4, b=3; pathpen=black; pair A=(a,0), B=(0,b), C=(0,0); D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle); pair X=IP(B--A,(0,0)--(b,a)); D(CP((X+C)/2,C)); D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0)))); //Credit to chezbgone2 for the diagram[/asy] In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is $\textbf{(A) }4.75\qquad\textbf{(B) }4.8\qquad\textbf{(C) }5\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$

A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.

Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number? $\text{(A)}\ 24 \qquad\text{(B)}\ 27\qquad\text{(C)}\ 39\qquad\text{(D)}\ 40 \qquad\text{(E)}\ 42$

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited. In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i]. Determine the number of different MEMOrable rectangles. [i]Remark: Rectangles are different unless they consist of exactly the same squares.[/i]

The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a$, $y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is: $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ \text{infinitely many}$

A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then: $\textbf{(A)}\ x=\dfrac{1}{2}h \qquad \textbf{(B)}\ x=\dfrac{bh}{b+h} \qquad \textbf{(C)}\ x=\dfrac{bh}{2h+b} \qquad \textbf{(D)}\ x=\sqrt{\dfrac{hb}{2}} \qquad \textbf{(E)}\ x=\dfrac{1}{2}b$

It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that: 1: if 2 squares are adjacent then one of them is marked. 2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked. Find the minimun number of squares we most mark.

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$

Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $

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.

assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.

Square with side 100 was cut by 99 horizontal and 99 vertical lines into 10000 rectangles (not necessarily with integer sides). How many rectangles in this square with area not exceeding 1 at least can be?

Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.

Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$? Remark: Two cells are called connected if they have a common edge.

An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.