For integer $ n,\ a_n$ is difined by $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx$. (1) Find $ a_{\minus{}2},\ a_{\minus{}1}$. (2) Find the relation of $ a_n$ and $ a_{n\minus{}2}$. (3) Prove that $ a_{2n}\equal{}b_n\plus{}\pi c_n$ for some rational number $ b_n,\ c_n$, then find $ c_n$ for $ n<0$.

You roll three fair six-sided dice. Given that the highest number you rolled is a $5$, the expected value of the sum of the three dice can be written as $\tfrac ab$ in simplest form. Find $a+b$.

Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

Let $F$ be a closed convex region inside a circle $C$ with center $O$ and radius $1$. Furthermore, assume that from each point of $C$ one can draw two rays tangent to $F$ which form an angle of $60^o$. Prove that $F$ is the disc centered at $O$ with radius $1/2$.

Find all real numbers for which $3^x+4^x=5^x$.

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Find all the roots of $(x^2 + 3x + 2)(x^2 - 7x + 12)(x^2- 2x -1) + 24 = 0$.

Let $x_1, x_2 \dots, x_{2024}$ be non-negative real numbers such that $x_1 \le x_2\cdots \le x_{2024}$, and $x_1^3 + x_2^3 + \dots + x_{2024}^3 = 2024$. Prove that \[\sum_{1 \le i < j \le 2024} (-1)^{i+j} x_i^2 x_j \ge -1012.\] [i]Proposed by Shantanu Nene[/i]

Let $A,B$ be adjacent vertices of a regular $n$-gon ($n\ge5$) with center $O$. A triangle $XYZ$, which is congruent to and initially coincides with $OAB$, moves in the plane in such a way that $Y$ and $Z$ each trace out the whole boundary of the polygon, with $X$ remaining inside the polygon. Find the locus of $X$.

Right triangle $ABC$ has its right angle at $A$. A semicircle with center $O$ is inscribed inside triangle $ABC$ with the diameter along $AB$. Let $D$ be the point where the semicircle is tangent to $BC$. If $AD=4$ and $CO=5$, find $\cos\angle{ABC}$. [asy] import olympiad; pair A, B, C, D, O; A = (1,0); B = origin; C = (1,1); O = incenter(C, B, (1,-1)); draw(A--B--C--cycle); dot(O); draw(arc(O, 0.41421356237,0,180)); D = O+0.41421356237*dir(135); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NE); label("$D$",D,NW); label("$O$",O,S); [/asy] $\text{(A) }\frac{\sqrt{5}}{4}\qquad\text{(B) }\frac{3}{5}\qquad\text{(C) }\frac{12}{25}\qquad\text{(D) }\frac{4}{5}\qquad\text{(E) }\frac{2\sqrt{5}}{5}$

The ruler of Laputa will set up a train network between cities in the state, which satisfies the following conditions: - [i]Uniformity[/i]: From one city to another, by train, possibly through exchanges. - [i]Prohibition N[/i]: There exist no four cities $A, B, C, D$ such that there are direct routes between $A$ and $B, B$ and $C$, and $C$ and $D$, but taking a shortcut is not possible, that is, there are no direct rout between $A$ and $C, B$ and $D$, or $A$ and $D$. In addition, a direct airliner connection will be established exactly between their city pairs, with no direct train connection. Prove that the airline network is not connected when there is more than one city.

Given a square $ABCD$ with side $10$. On sides BC and $AD$ of this square are selected respectively points $E$ and $F$ such that formed a rectangle $ABEF$. Rectangle $KLMN$ is located so that its the vertices $K, L, M$ and $N$ lie one on each segments $CD, DF, FE$ and $EC$, respectively. It turned out that the rectangles $ABEF$ and $KLMN$ are equal with $AB = MN$. Find the length of segment $AL$.

Find all natural numbers $k$ which can be represented as the sum of two relatively prime numbers not equal to $1$.

Let $ f: (1,\infty)\to\mathbb{R}$ be a differentiable function such that \[ f'(x)\equal{}\frac{x^2\minus{}\left(f(x)\right)^2}{x^2\left(\left(f(x)\right)^2\plus{}1\right)}\quad\text{for all }x>1.\] Prove that $ \displaystyle\lim_{x\to\infty}f(x)\equal{}\infty.$

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

An equilateral triangle with side length $3$ is divided into $9$ congruent triangular cells as shown in the figure below. Initially all the cells contain $0$. A [i]move[/i] consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by $1$ simultaneously. Determine all positive integers $n$ such that after performing several such moves one can obtain $9$ consecutive numbers $n,(n+1),\cdots ,(n+8)$ in some order. [asy] size(3cm); pair A=(0,0),D=(1,0),B,C,E,F,G,H,I; G=rotate(60,A)*D; B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A; draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]

Given integers $a,b,c,d,m,n$ such that $ad-bc\ne 0$ and any real $\varepsilon >0$, show that one can find rational numbers $x,y$ such that $0<|ax+by-m|<\varepsilon$ and $0<|cx+dy-n|<\varepsilon$.

Let $E, F$ be two distinct points inside a parallelogram $ABCD$ . Determine the maximum possible number of triangles having the same area with three vertices from points $A, B, C, D, E, F$.

In Abaliba country there are twenty cities and two airline companies, Blue Planes and Red Planes. The flights are planned as follows: $\bullet$ Given any two cities, one and only one of the two companies operates direct flights (in both directions and without stops) between the two cities. Furthermore: $\bullet$There are two cities A and B between which it is not possible to fly (with possible stops) using only Red Planes. Prove that, given any two cities, a passenger can travel from one to the other using only Blue Planes, making at most one stop in a third city.

The satellites $A$ and $B$ circle the Earth in the equatorial plane at altitude $h$. They are separated by distance $2r$, where $r$ is the radius of the Earth. For which $h$ can they be seen in mutually perpendicular directions from some point on the equator?

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \] Holds for all $x \not= y \in \mathbb{R}$

Find all real solutions $(x,y,z)$ of the system of equations $$\begin{cases} x^3 = 2y-1 \\y^3 = 2z-1\\ z^3 = 2x-1\end{cases}$$

Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle? ${{ \textbf{(A)}\ \frac{bm}{h+m}\qquad\textbf{(B)}\ \frac{bh}{2h+2m}\qquad\textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad\textbf{(D)}\ \frac{b(m+h)}{2m+h} }\qquad\textbf{(E)}\ \frac{b(2m+h)}{2(h+m)} } $

Balls of $3$ colours — red, blue and white — are placed in two boxes. If you take out $3$ balls from the first box, there would definitely be a blue one among them. If you take out $4$ balls from the second box, there would definitely be a red one among them. If you take out any $5$ balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.