Find all functions $f:\mathbb{Q}\to \mathbb{Q}$ such that \[ f(x+3f(y))=f(x)+f(y)+2y \quad \forall x,y\in \mathbb{Q}\]

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

$g(x) \colon \int_{10}^{x} \log_{10}(\log_{10}(t^2-1000t+10^{1000})) dt$ (a) Find the domain of $g(x)$ (b) Approximate the value of $g(1000)$ (c) Find $x \in [10, 1000]$ to maximize the slope of $g(x)$ (d) Find $x \in [10, 1000]$ to minimize the slope of $g(x)$ (e) Determine, if it exists, $\lim_{x \to \infty} \frac{\ln(x)}{g(x)}$

Let $ f: \mathbb{N} \rightarrow \mathbb{N} $ be a function from the positive integers to the positive integers for which $ f(1)=1,f(2n)=f(n) $ and $ f(2n+1)=f(n)+f(n+1) $ for all $ n\in \mathbb{N} $. Prove that for any natural number $ n $, the number of odd natural numbers $ m $ such that $ f(m)=n $ is equal to the number of positive integers not greater than $ n $ having no common prime factors with $ n $.

Let a function $f : \Bbb{R}^+ \to \Bbb{R}$ satisfy: (i) $f$ is strictly increasing, (ii) $f(x) > -1/x$ for all $x > 0$, (iii)$ f(x)f (f(x) + 1/x) = 1$ for all $x > 0$. Determine $f(1)$.

Let $\alpha$ be a root of $x^6-x-1$, and call two polynomials $p$ and $q$ with integer coefficients $\textit{equivalent}$ if $p(\alpha)\equiv q(\alpha)\pmod3$. It is known that every such polynomial is equivalent to exactly one of $0,1,x,x^2,\ldots,x^{727}$. Find the largest integer $n<728$ for which there exists a polynomial $p$ such that $p^3-p-x^n$ is equivalent to $0$.

Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

For function $f:\mathbb Z^+ \to \mathbb R$ and coprime positive integers $p,q$ ; define $f_p,f_q$ as $$f_p(x)=f(px)-f(x), f_q(x)=f(qx)-f(x) \space \space (x\in\mathbb Z^+)$$ $f$ satisfies following conditions. $ $ $ $ $(i)$ $ $ for all $r$ that isn't multiple of $pq$, $f(r)=0$ $ $ $ $ $(ii)$ $ $ $\exists m\in \mathbb Z^+$ $ $ $s.t.$ $ $ $\forall x\in \mathbb Z^+, f_p(x+m)=f_p(x)$ and $f_q(x+m)=f_q(x)$ Prove that if $x\equiv y$ $ $ $(mod m)$, then $f(x)=f(y)$ $ $ ($x, y\in \mathbb Z^+$).

Evaluate $\int_0^1\frac{\ln(x+1)}{x^2+1}\,dx.$

With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team. There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week: \begin{align*}\dbinom{16}4&=1820,\\\dbinom{16}5&=4368.\end{align*} Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test: \[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\] Wendy works the solution out correctly. What is her answer?

Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$ [i]Proposed by Mihai Baluna, Romania[/i]

Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.

We call a function $g$ [i]special [/i] if $g(x)=a^{f(x)}$ (for all $x$) where $a$ is a positive integer and $f$ is polynomial with integer coefficients such that $f(n)>0$ for all positive integers $n$. A function is called an [i]exponential polynomial[/i] if it is obtained from the product or sum of special functions. For instance, $2^{x}3^{x^{2}+x-1}+5^{2x}$ is an exponential polynomial. Prove that there does not exist a non-zero exponential polynomial $f(x)$ and a non-constant polynomial $P(x)$ with integer coefficients such that $$P(n)|f(n)$$ for all positive integers $n$.

Let a function $f$ satisfy $f(1) = 1$ and $f(1)+ f(2)+...+ f(n) = n^2f(n)$ for all $n \in N$. Determine $f(1995)$.

Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \] for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.

$f$ is a function defined on integers and satisfies $f(x)+f(x+3)=x^2$ for every integer $x$. If $f(19)=94$, then calculate $f(94)$.

Let $S$ be the domain surrounded by the two curves $C_1:y=ax^2,\ C_2:y=-ax^2+2abx$ for constant positive numbers $a,b$. Let $V_x$ be the volume of the solid formed by the revolution of $S$ about the axis of $x$, $V_y$ be the volume of the solid formed by the revolution of $S$ about the axis of $y$. Find the ratio of $\frac{V_x}{V_y}$.

Let $f : N \to N$ be a function which satisfies $f(x)+ f(x+2) \le 2 f(x+1)$ for any $x \in N$. Prove that there exists a line in the coordinate plane containing infinitely many points of the form $(n, f(n)), n \in N$.

Find the smallest nonnegative integer $n$ for which $\binom{2006}{n}$ is divisible by $7^3$.

For an arbitrary real number $ x$, $ \lfloor x\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \displaystyle m\minus{}\left\lfloor \frac{m}{2005}\right\rfloor\equal{}2005$.

Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

Find number of solutions in non-negative reals to the following equations: \begin{eqnarray*}x_1 + x_n ^2 = 4x_n \\ x_2 + x_1 ^2 = 4x_1 \\ ... \\ x_n + x_{n-1}^2 = 4x_{n-1} \end{eqnarray*}

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

If $a,b,c>0$, what is the smallest possible value of $ \left\lfloor \dfrac {a+b}{c} \right\rfloor + \left\lfloor \dfrac {b+c}{a} \right\rfloor + \left\lfloor \dfrac {c+a}{b} \right\rfloor $? (Note that $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $x$.)