Let $ f(x)\equal{}\sin 3x\plus{}\cos x,\ g(x)\equal{}\cos 3x\plus{}\sin x.$ (1) Evaluate $ \int_0^{2\pi} \{f(x)^2\plus{}g(x)^2\}\ dx$. (2) Find the area of the region bounded by two curves $ y\equal{}f(x)$ and $ y\equal{}g(x)\ (0\leq x\leq \pi).$

For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.

In each lattice-point of an $m \times n$ grid and in the centre of each of the formed unit squares a pawn is placed. a) Find all such grids with exactly $500$ pawns. b) Prove that there are infinitely many positive integers $k$ for which therer is no grid containing exactly $k$ pawns.

Find all the functions $f : C \to R^+$ such that they fulfill simultaneously the following conditions: $$(i) \ \ f(uv) = f(u)f(v) \ \ \forall u, v \in C$$ $$(ii) \ \ f(au) = |a | f(u) \ \ \forall a \in R, u \in C$$ $$(iii) \ \ f(u) + f(v) \le |u| + |v| \ \ \forall u, v \in C$$

Suppose $x^9 = 1$ but $x^3 \ne 1$. Find a polynomial of minimal degree equal to $\frac{1}{1+x}$ .

What is the hundreds digit of $(20!-15!)?$ $\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$

After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$

Prove that it is not possible to write numbers $ 1,2,3,...,25$ on the squares of $ 5$x$ 5$ chessboard such that any neighboring numbers differ by at most $ 4$.

If $p,q\in\mathbb{R}[x]$ satisfy $p(p(x))=q(x)^2$, does it follow that $p(x)=r(x)^2$ for some $r\in\mathbb{R}[x]$?

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

Let $ABC$ be a triangle with $\angle A,\angle B,\angle C <120^{\circ}$, $T$ is its [i]Fermat-Torricelli[/i] point. Consider a point $P$ lying on the same plane with $\triangle ABC$. Prove that: a)$$\dfrac{\overrightarrow {TA}}{TA}+\dfrac{\overrightarrow {TB}}{TB}+\dfrac{\overrightarrow {TC}}{TC}=\overrightarrow {0}.$$ b)$$PA + PB + PC \ge \frac{{\overrightarrow {PA} \overrightarrow {.TA} }}{{TA}} + \frac{{\overrightarrow {PB} .\overrightarrow {TB} }}{{TB}} + \frac{{\overrightarrow {PC} \overrightarrow {.TC} }}{{TC}}.$$ c)$$PA + PB + PC \ge TA + TB + TC$$and the equality occurs iff $P\equiv T$.

An equilateral triangle is given. A point lies on the incircle of this triangle. If the smallest two distances from the point to the sides of the triangle is $1$ and $4$, the sidelength of this equilateral triangle can be expressed as $\tfrac{a\sqrt b}c$ where $(a,c)=1$ and $b$ is not divisible by the square of an integer greater than $1$. Find $a+b+c$.

Let $a_1, a_2,\cdots
,a_n$ be positive integers and let $a$ be an integer greater than $1$ and divisible by the product $a_1a_2\cdots a_n$. Prove that $a^{n+1} + a-1$ is not divisible by the product $(a + a_1 - 1)(a + a_2 - 1) \cdots
(a + a_n - 1)$.

$$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\dots+\frac{1}{2014\cdot2015}=\frac{m}{n},$$ where $\frac{m}{n}$ is irreducible. a) Find $m+n.$ b) Find the remainder of division of $(m+3)^{1444}$ to $n{}$.

In a blackboard, it's written the following expression $ 1-2-2^2-2^3-2^4-2^5-2^6-2^7-2^8-2^9-2^{10}$ We put parenthesis by different ways and then we calculate the result. For example: $ 1-2-\left(2^2-2^3\right)-2^4-\left(2^5-2^6-2^7\right)-2^8-\left( 2^9-2^{10}\right)= 403$ and $ 1-\left(2-2^2 \left(-2^3-2^4 \right)-\left(2^5-2^6-2^7\right)\right)- \left(2^8- 2^9 \right)-2^{10}= -933$ How many different results can we obtain?

Define $E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$ such that $E$ posseses the following properties:\\ $\textbf{(i)}$ If $\int_0^1 f(x)g(x) dx = 0$ for $f\in E$ with $\int_0^1f^2(t)dt \neq 0$, then $g\in E$; \\ $\textbf{(ii)}$ There exists $h\in E$ with $\int_0^1 h^2(t)dt\neq 0$.\\ Prove that $E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}$. \\ [i](Andrei Bâra)[/i]

Let $p , q, r , s$ be four integers such that $s$ is not divisible by $5$. If there is an integer $a$ such that $pa^3 + qa^2+ ra +s$ is divisible be 5, prove that there is an integer $b$ such that $sb^3 + rb^2 + qb + p$ is also divisible by 5.

In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that: [LIST] [*] There are no two numbers written in the first row that are equal to each other.[/*] [*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*] [*] The two numbers written in each column are different and they add up to a rational number.[/*] [/LIST] Determine the maximum quantity of irrational numbers that can be in the chessboard.

Let $x_0 = [a], x_1 = [2a] - [a], x_2 = [3a] - [2a], x_3 = [3a] - [4a],x_4 = [5a] - [4a],x_5 = [6a] - [5a], . . . , $ where $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ .The value of $x_9$ is: (A): $2$ (B): $3$ (C): $4$ (D): $5$ (E): None of the above.

An integer-sided triangle has angles $ p\theta$ and $ q\theta$, where $ p$ and $ q$ are relatively prime integers. Prove that $ \cos\theta$ is irrational.

Prove that: a) There are infinitely many pairs $(x,y)$ of real numbers from the interval $[0,\sqrt{3}]$ which satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$. b) There do not exist any pairs $(x,y)$ of rational numbers from the interval $[0,\sqrt{3}]$ that satisfy the equation $x\sqrt{3-y^2}+y\sqrt{3-x^2}=3$.

Steve is piling $m\geq 1$ indistinguishable stones on the squares of an $n\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform [i]stone moves[/i], defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\leq i, j, k, l\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively. Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?

Alex and Kevin are radish watching. The probability that they will see a radish within the next hour is $\frac{1}{17}$. If the probability that they will see a radish within the next $15$ minutes is $p$, determine $\lfloor 1000p \rfloor$. Assume that the probability of seeing a radish at any given moment is uniform for the entire hour. [i]Proposed by Ephram Chun[/i]

It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_{2013}$

Compute \[ \left\lfloor \dfrac {2005^3}{2003 \cdot 2004} - \dfrac {2003^3}{2004 \cdot 2005} \right\rfloor \]