Let $f : [0,1] \to [0,1]$ be a continuous, strictly increasing function such that $f(0) = 0$ and $f(1) = 1$. Prove that $$f\left(\frac{1}{10}\right) + f\left(\frac{2}{10}\right) +...+f\left(\frac{9}{10}\right) +f^{-1}\left(\frac{1}{10}\right) +...+f^{-1}\left(\frac{9}{10}\right) \le \frac{99}{10}$$

From a given triangle, cut out the rectangle with the largest area.

[b]a) [/b]Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? [b]b)[/b] Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j\plus{}1$ where $ j$ is a natural number.

A city has $4$ horizontal and $n\geq3$ vertical boulevards which intersect at $4n$ crossroads. The crossroads divide every horizontal boulevard into $n-1$ streets and every vertical boulevard into $3$ streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city doesn't have a closed path(this means that starting from any street and going only through open crossroads without turning back you can't return to the same street). $a)$Prove that exactly $n$ crossroads are closed. $b)$Prove that if from any street you can go to any other street and none of the $4$ corner crossroads are closed then exactly $3$ crossroads on the border are closed(A crossroad is on the border if it lies either on the first or fourth horizontal boulevard, or on the first or the n-th vertical boulevard).

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

We color numbers $1,2,3,...,20$ in two colors, blue and yellow, such that both colors are used (not all numbers are colored in one color). Determine number of ways we can color those numbers, such that product of all blue numbers and product of all yellow numbers have greatest common divisor $1$.

Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

There are $n \ge 2$ numbers on the blackboard: $1, 2,..., n$. It is permitted to erase two of those numbers $x,y$ and write $2x - y$ instead. Find all values of $n$ such that it is possible to leave number $0$ on the blackboard after $n - 1$ such procedures.

A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? $ \textbf {(A) } \frac{b}{1080t} \qquad \textbf {(B) } \frac{30t}{b} \qquad \textbf {(C) } \frac{30b}{t}\qquad \textbf {(D) } \frac{10t}{b} \qquad \textbf {(E) } \frac{10b}{t}$

Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.

Let $ABC$ be a triangle and let $P$ be a point in the interior of the side $BC$. Let $I_1$ and $I_2$ be the incenters of the triangles $AP B$ and $AP C$, respectively. Let $X$ be the closest point to $A$ on the line $AP$ such that $XI_1$ is perpendicular to $XI_2$. Prove that the distance $AX$ is independent of the choice of $P$.

Masaru randomly paints $50\%$ of the area of a square. What is the probability that at least $60\%$ of the left side of the square is painted? [asy] size(2cm); defaultpen(fontsize(7)); draw((0,0)--(4,0)--(4,4)--(0,4)--cycle,linewidth(1.5)); fill(circle((0.3,0.3),0.3),paleblue); fill(circle((1,1),0.5),palered); fill(circle((0.8,2),0.8),purple); fill(circle((3,3),0.6),orange); fill(circle((3.4,1.5),0.6),mediumgreen); fill(circle((1.4,1.6),0.4),yellow); fill(circle((2.5,2.8),0.4),cyan); fill(circle((1,3.5),0.5),red); fill((3,0)--(4,0)--(4,0.4)--(3.5,0.8)--cycle,magenta); fill((2,4)--(3,4)--(3.1,3.8)--(2.7,3.5)--(2.4,3.1)--cycle,olive); draw((2,0)--(2,4),dashed); [/asy] $\textbf{(A) } 25\%\qquad\textbf{(B) } 30\%\qquad\textbf{(C) } 35\%\qquad\textbf{(D) } 40\%\qquad\textbf{(E) } 45\%$

Find all integer values of $x$ for which the value of the expression \[x^2+6x+33\] is a perfect square.

In a group of $2021$ people, $1400$ of them are $\emph{saboteurs}$. Sherlock wants to find one saboteur. There are some missions that each needs exactly $3$ people to be done. A mission fails if at least one of the three participants in that mission is a saboteur! In each round, Sherlock chooses $3$ people, sends them to a mission and sees whether it fails or not. What is the minimum number of rounds he needs to accomplish his goal?

In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.

Each square of an $n \times m$ board is assigned a pair of coordinates $(x,y)$ with $1 \le x \le m$ and $1 \le y \le n$. Let $p$ and $q$ be positive integers. A pawn can be moved from the square $(x,y)$ to $(x',y')$ if and only if $|x - x'| = p$ and $|y- y'| = q$. There is a pawn on each square. We want to move each pawn at the same time so that no two pawns are moved onto the same square. In how many ways can this be done?

There are $n$ holes in a circle. The holes are numbered $1,2,3$ and so on to $n$. In the beginning, there is a peg in every hole except for hole $1$. A peg can jump in either direction over one adjacent peg to an empty hole immediately on the other side. After a peg moves, the peg it jumped over is removed. The puzzle will be solved if all pegs disappear except for one. For example, if $n=4$ the puzzle can be solved in two jumps: peg $3$ jumps peg $4$ to hole $1$, then peg $2$ jumps the peg in $1$ to hole $4$. (See illustration below, in which black circles indicate pegs and white circles are holes.) [center][img]http://i.imgur.com/4ggOa8m.png[/img][/center] [list=a] [*]Can the puzzle be solved for $n=5$? [*]Can the puzzle be solved for $n=2014$? [/list] In each part (a) and (b) either describe a sequence of moves to solve the puzzle or explain why it is impossible to solve the puzzle.