A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.

Billy and Bobby are located at points $A$ and $B$, respectively. They each walk directly toward the other point at a constant rate; once the opposite point is reached, they immediately turn around and walk back at the same rate. The first time they meet, they are located 3 units from point $A$; the second time they meet, they are located 10 units from point $B$. Find all possible values for the distance between $A$ and $B$. [i]Proposed by Isabella Grabski[/i]

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

An arbitary triangle is partitioned to some triangles homothetic with itself. The ratio of homothety of the triangles can be positive or negative. Prove that sum of all homothety ratios equals to $1$. Time allowed for this problem was 45 minutes.

After a cyclist has gone $ \frac{2}{3}$ of his route, he gets a flat tire. Finishing on foot, he spends twice as long walking as he did riding. How many times as fast does he ride as walk?

Can we partition the positive integers in two sets such that none of the sets contains an infinite arithmetic progression of nonzero ratio ?

Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer. [i]Radu Gologan[/i]

Prove that for a positive number $ r>1, $ there is a nondecreasing sequence of positive numbers $ \left( a_v\right)_{v\ge 1} $ such that $$ r=\lim_{n\to\infty }\sum_{i=1}^n \frac{a_i}{a_{i+1}} . $$

A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained . (a) What is the ratio into which the diagonals of this quadrilateral divide each other? (b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ . (A Shapovalov)

Let $ABC$ be a triangle. Let $D, E$ be a points on the segment $BC$ such that $BD =DE = EC$. Let $F$ be the mid-point of $AC$. Let $BF$ intersect $AD$ in $P$ and $AE$ in $Q$ respectively. Determine $BP:PQ$.

Acute angle triangle is given such that $ BC $ is the longest side. Let $ E $ and $ G $ be the intersection points from the altitude from $ A $ to $ BC $ with the circumscribed circle of triangle $ ABC $ and $ BC $ respectively. Let the center $ O $ of this circle is positioned on the perpendicular line from $ A $ to $ BE $. Let $ EM $ be perpendicular to $ AC $ and $ EF $ be perpendicular to $ AB $. Prove that the area of $ FBEG $ is greater than the area of $ MFE $.

Suppose line $\ell$ and four points $A,B,C,D$ lies on $\ell$. Suppose that circles $\omega_1 , \omega_2$ passes through $A,B$ and circles $\omega'_1 , \omega'_2$ passes through $C,D$. If $\omega_1 \perp \omega'_1$ and $\omega_2 \perp \omega'_2$ then prove that lines $O_1O'_2 , O_2O'_1 , \ell $ are concurrent where $O_1,O_2,O'_1,O'_2$ are center of $\omega_1 , \omega_2 , \omega'_1 , \omega'_2$.

Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .

[b]Q2.[/b] Let be given a parallegogram $ABCD$ with the area of $12 \ \text{cm}^2$. The line through $A$ and the midpoint $M$ of $BC$ mects $BD$ at $N.$ Compute the area of the quadrilateral $MNDC.$ \[(A) \; 4 \text{cm}^2; \qquad (B) \; 5 \text{cm}^2; \qquad (C ) \; 6 \text{cm}^2; \qquad (D) \; 7 \text{cm}^2; \qquad (E) \; \text{None of the above.}\]

(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.

Triangle $\triangle ABC$ in the figure has area $10$. Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$2$", (2,0), S); label("$3$", (7,0), S);[/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ \frac{5}{3}\sqrt{10}\qquad\textbf{(E)}\ \text{not uniquely determined}$

The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ . $$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$ Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$

Alice creates the graphs $y=|x-a|$ and $y=c-|x-b|$ , where $a,b,c\in\mathbb{R^+}$. She observes that these two graphs and $x$ axis divides the positive side of the plane ($x,y>0$) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral. [hide=There might be a translation error] In the original statement,it says $XOY$ plane,instead of positive side of the plane. I think these 2 are the same,but I might be wrong [/hide]

The midpoint of each side of a convex pentagon is connected by a segment with the centroid of the triangle formed by the remaining three vertices of the pentagon. Prove that these five segments have a common point.

In scalene $\triangle ABC$, $I$ is the incenter, $I_a$ is the $A$-excenter, $D$ is the midpoint of arc $BC$ of the circumcircle of $ABC$ not containing $A$, and $M$ is the midpoint of side $BC$. Extend ray $IM$ past $M$ to point $P$ such that $IM = MP$. Let $Q$ be the intersection of $DP$ and $MI_a$, and $R$ be the point on the line $MI_a$ such that $AR\parallel DP$. Given that $\frac{AI_a}{AI}=9$, the ratio $\frac{QM} {RI_a}$ can be expressed in the form $\frac{m}{n}$ for two relatively prime positive integers $m,n$. Compute $m+n$. [i]Ray Li.[/i] [hide="Clarifications"][list=1][*]"Arc $BC$ of the circumcircle" means "the arc with endpoints $B$ and $C$ not containing $A$".[/list][/hide]

Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$. (Netherlands)

The number $2.5081081081081\ldots$ can be written as $\frac{m}{n}$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.

Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000}\\ &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where $A_k = \binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\ldots,1000$. For which $k$ is $A_k$ the largest?