The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? [asy] import cse5;pathpen=black;pointpen=black; size(1.5inch); D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W)); path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle; draw(P,linewidth(2)); fill(P,gray); [/asy] $\textbf{(A) } \dfrac{4\pi}{5} \qquad\textbf{(B) } \dfrac{9\pi}{8} \qquad\textbf{(C) } \dfrac{4\pi}{3} \qquad\textbf{(D) } \dfrac{7\pi}{5} \qquad\textbf{(E) } \dfrac{3\pi}{2} $

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

$(x_1, y_1), (x_2, y_2), (x_3, y_3)$ lie on a straight line and on the curve $y^2 = x^3$. Show that $\frac{x_1}{y_1} + \frac{x_2}{y_2}+\frac{x_3}{y_3} = 0$.

In triangle $ABC$, in which $AB = BC$, on side $AB$ is selected point $D$, and the ciscumcircles of triangles $ADC$ and $BDC$ , $S1$ and $S2$ respectively. The tangent drawn to $S_1$ at point $D$ intersects $S_2$ for second time at point $M$. Prove that $BM \parallel AC$.

Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$. Prove that $FB = FD$. [i]Milan Haiman[/i]

$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that : $ |T| \leq \frac {4}{9}n + \log_2 n + 2$

In $ \triangle ABC$ we have $ AB \equal{} 7$, $ AC \equal{} 8$, and $ BC \equal{} 9$. Point $ D$ is on the circumscribed circle of the triangle so that $ \overline{AD}$ bisects $ \angle BAC$. What is the value of $ AD/CD$? $ \textbf{(A)}\ \frac{9}{8}\qquad \textbf{(B)}\ \frac{5}{3}\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ \frac{17}{7}\qquad \textbf{(E)}\ \frac{5}{2}$

How many ways are there to arrange the $6$ permutations of the tuple $(1, 2, 3)$ in a sequence, such that each pair of adjacent permutations contains at least one entry in common? For example, a valid such sequence is given by $(3, 2, 1) - (2, 3, 1) - (2, 1, 3) - (1, 2, 3) - (1, 3, 2) - (3, 1, 2)$.

Let $k\leq n$ be two positive integers. IMO-nation has $n$ villages, some of which are connected by a road. For any two villages, the distance between them is the minimum number of toads that one needs to travel from one of the villages to the other, if the traveling is impossible, then the distance is set as infinite. Alice, who just arrived IMO-nation, is doing her quarantine in some place, so she does not know the configuration of roads, but she knows $n$ and $k$. She wants to know whether the furthest two villages have finite distance. To do so, for every phone call she dials to the IMO office, she can choose two villages, and ask the office whether the distance between them is larger than, equal to, or smaller than $k$. The office answers faithfully (infinite distance is larger than $k$). Prove that Alice can know whether the furthest two villages have finite distance between them in at most $2n^2/k$ calls. [i]Proposed by usjl and Cheng-Ying Chang[/i]

Let $ z_1,z_2,z_3\in\mathbb{C} $ such that $\text{(i)}\quad \left|z_1\right| = \left|z_2\right| = \left|z_3\right| = 1$ $\text{(ii)}\quad z_1+z_2+z_3\neq 0 $ $\text{(iii)}\quad z_1^2 +z_2^2+z_3^2 =0. $ Show that $ \left| z_1^3+z_2^3+z_3^3\right| = 1. $

Let $a,b,c$ be sidelengths of a triangle and $r,R,s$ be the inradius, the circumradius and the semiperimeter respectively of the same triangle. Prove that: $$\frac{1}{a + b} + \frac{1}{a + c} + \frac{1}{b + c} \leq \frac{r}{16Rs}+\frac{s}{16Rr} + \frac{11}{8s}$$ (Albania)

[b]p22.[/b] Find the unique ordered pair $(m, n)$ of positive integers such that $x = \sqrt[3]{m} -\sqrt[3]{n}$ satisfies $x^6 + 4x^3 - 36x^2 + 4 = 0$. [b]p23.[/b] Jenny plays with a die by placing it flat on the ground and rolling it along any edge for each step. Initially the face with $1$ pip is face up. How many ways are there to roll the dice for $6$ steps and end with the $1$ face up again? [b]p24.[/b] There exists a unique positive five-digit integer with all odd digits that is divisible by $5^5$. Find this integer. PS. You should use hide for answers.

A mouse is on the below grid: \begin{center} \begin{asy} unitsize(2cm); filldraw(circle((0,0),0.07), black); filldraw(circle((0,1),0.07), black); filldraw(circle((1,0),0.07), black); filldraw(circle((0.5,0.5),0.07), black); filldraw(circle((1,1),0.07), black); draw((0,0)--(1,0)); draw((0,0)--(0,1)); draw((1,0)--(1,1)); draw((0,1)--(1,1)); draw((0,1)--(0.5,0.5)); draw((1,0)--(0.5,0.5)); draw((1,1)--(0.5,0.5)); draw((0,0)--(0.5,0.5)); \end{asy} \end{center} The paths connecting each node are the possible paths the mouse can take to walk from a node to another node. Call a ``turn" the action of a walk from one node to another. Given the mouse starts off on an arbitrary node, what is the expected number of turns it takes for the mouse to return to its original node? [i]Lightning 4.2[/i]

At the Berkeley Math Tournament, teams are composed of $6$ students, each of whom pick two distinct subject tests out of $5$ choices. How many different distributions across subjects are possible for a team?

Compare $\tan(\sin x)$ with $\sin(\tan x)$, for $x\in \left]0,\frac{\pi}{2}\right[$.

Prove that \[\sum_{i=-2007}^{2007}\frac{\sqrt{|i+1|}}{(\sqrt2)^{|i|}}>\sum_{i=-2007}^{2007}\frac{\sqrt{|i|}}{(\sqrt2)^{|i|}}\]

Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

Let $n$ and $k$ be positive integers with $1 \leq k \leq n$. A set of cards numbered $1$ to $n$ are arranged randomly in a row from left to right. A person alternates between performing the following moves: [list=a] [*] The leftmost card in the row is moved $k-1$ positions to the right while the cards in positions $2$ through $k$ are each moved one place to the left. [*] The rightmost card in the row is moved $k-1$ positions to the left while the cards in positions $n-k+1$ through $n-1$ are each moved one place to the right. [/list] Determine the probability that after some number of moves the cards end up in order from $1$ to $n$, left to right.

$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)

Let $ T$ be the set of all triplets $ (a,b,c)$ of integers such that $ 1 \leq a < b < c \leq 6$ For each triplet $ (a,b,c)$ in $ T$, take number $ a\cdot b \cdot c$. Add all these numbers corresponding to all the triplets in $ T$. Prove that the answer is divisible by 7.

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

Each day, Jenny ate $ 20\%$ of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, $ 32$ remained. How many jellybeans were in the jar originally? $ \textbf{(A)}\ 40\qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 60\qquad \textbf{(E)}\ 75$

In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$.

Let $n$ be a non-zero natural number, $n \ge 5$. Consider $n$ distinct points in the plane, each colored or white, or black. For each natural $k$ , a move of type $k, 1 \le k <\frac {n} {2}$, means selecting exactly $k$ points and changing their color. Determine the values of $n$ for which, whatever $k$ and regardless of the initial coloring, there is a finite sequence of $k$ type moves, at the end of which all points have the same color.