Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label("$A$",A,W); label("$B$",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi\minus{}\sqrt3 \qquad \textbf{(B)}\ \pi\minus{}\sqrt2 \qquad \textbf{(C)}\ \frac{\pi\plus{}\sqrt2}{2} \qquad \textbf{(D)}\ \frac{\pi\plus{}\sqrt3}{2}$ $ \textbf{(E)}\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}$

The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other cities can be reached from both cities by a single road. Compute the number of ways HMMTLand could have constructed the roads.

Triangle $ABC$ is right angled such that $\angle ACB=90^{\circ}$ and $\frac {AC}{BC} = 2$. Let the line parallel to side $AC$ intersects line segments $AB$ and $BC$ in $M$ and $N$ such that $\frac {CN}{BN} = 2$. Let $O$ be the intersection point of lines $CM$ and $AN$. On segment $ON$ lies point $K$ such that $OM+OK=KN$. Let $T$ be the intersection point of angle bisector of $\angle ABC$ and line from $K$ perpendicular to $AN$. Determine value of $\angle MTB$.

Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying \[m(m + k) = n(n + 1).\]

Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.

There are $900$ students in an International School. There are $59$ international boys and $59$ international girls. The Students are partitioned into $30$ classrooms (each classrooms have equal number of student) and in each of the classrooms, the student will labelled number from $1$ to $30$. The Partition must satisfy at least one follow condition: [list=i] [*] Any Two international boys in same classroom can't have consecutive numbers. [*] For every classroom, the student who is labelled $1$ must be a boy. [/list] Prove that there are $2$ classrooms, each of which has $2$ international boys with their labels difference equal.

Integers $a_1, a_2, \ldots, a_n$ satisfy $$1<a_1<a_2<\ldots < a_n < 2a_1.$$ If $m$ is the number of distinct prime factors of $a_1a_2\cdots a_n$, then prove that $$(a_1a_2\cdots a_n)^{m-1}\geq (n!)^m.$$

Let $x_1$, $x_2$,$x_3$, $y_1$, $y_2$, and $y_3 $ be positive real numbers, such that $x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1$. Prove that $$ x_1y_1 + x_2y_2 + x_3y_3 <1$$

Let $n \geq 2$ be an integer. An $n \times n$ board is initially empty. Each minute, you may perform one of three moves: [list] [*] If there is an L-shaped tromino region of three cells without stones on the board (see figure; rotations not allowed), you may place a stone in each of those cells. [*] If all cells in a column have a stone, you may remove all stones from that column. [*] If all cells in a row have a stone, you may remove all stones from that row. [/list] [asy] unitsize(20); draw((0,0)--(4,0)--(4,4)--(0,4)--(0,0)); fill((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--cycle, grey); draw((0.2,3.8)--(1.8,3.8)--(1.8, 1.8)--(3.8, 1.8)--(3.8, 0.2)--(0.2, 0.2)--(0.2, 3.8), linewidth(2)); draw((0,2)--(4,2)); draw((2,4)--(2,0)); [/asy] For which $n$ is it possible that, after some non-zero number of moves, the board has no stones?

Let $ABCD$ be a convex quadrilateral. Two squares are constructed such that $AB{}$ and $CD{}$ are their diagonals. Show that if these squares have a common vertex inside $ABCD$, then the squares that have $BC{}$ and $AD{}$ as diagonals also have a common vertex inside $ABCD$.

A uniform round tabletop of diameter $\text{4.0 m}$ and mass $\text{50.0 kg}$ rests on massless, evenly spaced legs of length $\text{1.0 m}$ and spacing $\text{3.0 m}$. A carpenter sits on the edge of the table. What is the maximum mass of the carpenter such that the table remains upright? Assume that the force exerted by the carpenter on the table is vertical and at the edge of the table. [asy] size(10cm); import graph; xaxis(-3.6,3.6); //The legs draw((-1.4,1.4)--(-1.4,0)); draw((-1.6,1.4)--(-1.6,0)); draw((1.4,1.4)--(1.4,0)); draw((1.6,1.4)--(1.6,0)); //The tabletop draw((-2.5,1.6)--(-2.5,1.4)--(2.5,1.4)--(2.5,1.6)--cycle); path a = ((-3.2,0.1)--(-3.2,1.5)); draw(a,Arrows); label("1.0m",a,E); draw((-2.6,1.6)--(-3.6,1.6),dashed+linewidth(0.4)); path c = (-1.4,-0.5)--(1.4,-0.5); draw(c,Arrows); label("3.0m",c,2*S); draw((-1.5,-0.1)--(-1.5,-0.8),dashed+linewidth(0.4)); draw((1.5,-0.1)--(1.5,-0.8),dashed+linewidth(0.4)); draw((-2.5,-0.1)--(-2.5,-1.8),dashed+linewidth(0.4)); draw((2.5,-0.1)--(2.5,-1.8),dashed+linewidth(0.4)); path d = (-2.4,-1.5)--(2.4,-1.5); draw(d,Arrows); label("4.0m",d,2*S); //The Man pair A = (2.4,1.7); draw(circle((2.5,3.7),0.5),linewidth(1.6)); draw((2.4,1.7)--(2.5,3.2),linewidth(1.6)); draw((2.15,2.8)--(2.85,2.8),linewidth(1.6)); draw(A--A+(0.6,0)--A+(0.4,-0.8)--A+(0.6,-1.2),linewidth(1.6)); draw(A--(A+(0.64,-0.12)--A+(0.73,-0.9)--A+(1,-0.9)),linewidth(1.6)); draw((2.15,2.8)--(2.15,2.4)--(2.9,2),linewidth(1.6)); draw((2.85,2.8)--(3,2.3)--(3.5,2.1),linewidth(1.6)); [/asy] (a) $\text{67 kg}$ (b) $\text{75 kg}$ (c) $\text{81 kg}$ (d) $\text{150 kg}$ (e) $\text{350 kg}$

For positive integer $n$, let $f(n)$ be the largest integer $k$ such that $k!\leq n$, let $g(n)=n-(f(n))!$, and for $j\geq 1$ let $$g^j(n)=\underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.$$ Find the smallest positive integer $n$ such that $g^{j}(n)> 0$ for all $j<30$ and $g^{30}(n)=0$. [i]Proposed by Connor Gordon[/i]

Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions : 1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$ 2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.

How many four-digit positive integers have exactly one digit equal to $1$ and exactly one digit equal to $3$?

A function $f$ is defined for all real numbers and satisfies \[f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)\] for all real $x$. If $x = 0$ is a root of $f(x) = 0$, what is the least number of roots $f(x) = 0$ must have in the interval $-1000 \le x \le 1000$?

A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimetres and its base has radius $30$ centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

Six distinct integers are picked at random from $\{1,2,3,\ldots,10\}$. What is the probability that, among those selected, the second smallest is $3$? $ \textbf{(A)}\ \frac{1}{60}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{3}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \text{none of these} $

A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in the lab by creating a copy $I'$ of each imon $I$. During this procedure, the two copies $I'$ and $J'$ become entangled if and only if the original imons $I$ and $J$ are entangled, and each copy $I'$ becomes entangled with its original imon $I$; no other entanglements occur or disappear at this moment. Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

Find all positive integers $k$ such that $k^k +1$ is divisible by $30$. Justify your answer.

Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.

Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$. [hide] Stronger one. Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]

Write down $f(n)$ for the greatest odd divisor of $n \in N_0$. (a) Determine $f (n + 1) + f (n + 2) + ... + f(2n)$. (b) Determine $f(1) + f(2) + f(3) + ... + f(2n)$.

Compute the value of \[\int_{-\pi}^\pi\frac{e^{x^2}-e^{-x^2}}{e^{x^2}-x\sqrt{2}}|x|dx.\]

Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $. Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.

Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment? [i]Proposed by Timothy Qian[/i]