Let $a,b,c$ be the roots of the cubic $x^3 + 3x^2 + 5x + 7$. Given that $P$ is a cubic polynomial such that $P(a)=b+c$, $P(b) = c+a$, $P(c) = a+b$, and $P(a+b+c) = -16$, find $P(0)$. [i]Author: Alex Zhu[/i]

Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i]

Let $f$ be a function from the set $Q$ of the rational numbers onto itself such that $f(x+y)=f(x)+f(y)+2547$ for all rational numbers $x,y$. Moreover $f(2004) = 2547$. Determine $f(2547).$ Pierre.

Let $ n\left(A\right)$ be the number of distinct real solutions of the equation $ x^{6} \minus{}2x^{4} \plus{}x^{2} \equal{}A$. When $ A$ takes every value on real numbers, the set of values of $ n\left(A\right)$ is $\textbf{(A)}\ \left\{0,1,2,3,4,5,6\right\} \\ \textbf{(B)}\ \left\{0,2,4,6\right\} \\ \textbf{(C)}\ \left\{0,3,4,6\right\} \\ \textbf{(D)}\ \left\{0,2,3,4,6\right\} \\ \textbf{(E)}\ \left\{0,2,3,4\right\}$

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $ a\in \mathbb{R} $ and $ f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R} $ are the additive functions such that for every $ x\in \mathbb{R} $ we have $ f_1(x)f_2(x) \cdots f_n(x) =ax^n $. Show that there exists $ b\in \mathbb {R} $ and $ i\in {\{1,2,\ldots,n}\} $ such that for every $ x\in \mathbb{R} $ we have $ f_i(x)=bx $.

Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:\[(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i\]

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

a) For each $n \ge 2$, find the maximum constant $c_{n}$ such that $\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$ for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$. b) For each $n \ge 2$, find the maximum constant $d_{n}$ such that $\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$ for all positive reals $a_{1},a_{2},\ldots,a_{n}$ such that $a_{1}a_{2}\cdots a_{n}= 1$.

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

The function f is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1)=2$ and $f(xy)=f(x)f(y)-f(x+y)+1$ for all $x,y \in \mathbb{Q}$. Determine f (with proof)

Given $n \in\mathbb{N}$, find all continuous functions $f : \mathbb{R}\to \mathbb{R}$ such that for all $x\in\mathbb{R},$ \[\sum_{k=0}^{n}\binom{n}{k}f(x^{2^{k}})=0. \]

In a culturing of bacteria, there are two species of them: red and blue bacteria. When two red bacteria meet, they transform into one blue bacterium. When two blue bacteria meet, they transform into four red bacteria. When a red and a blue bacteria meet, they transform into three red bacteria. Find, in function of the amount of blue bacteria and the red bacteria initially in the culturing, all possible amounts of bacteria, and for every possible amount, the possible amounts of red and blue bacteria.

For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because \[4=3+1=2+2=2+1+1=1+1+1.\] Also define $p(0)=1$. Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1. [i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]

Number $a$ is such that $\forall a_1, a_2, a_3, a_4 \in \mathbb{R}$, there are integers $k_1, k_2, k_3, k_4$ such that $\sum_{1 \leq i < j \leq 4} ((a_i - k_i) - (a_j - k_j))^2 \leq a$. Find the minimum of $a$.

Let $ \mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $ f: \mathbb{N} \mapsto \mathbb{N}$ satisfying \[ f(m \plus{} f(n)) \equal{} n \plus{} f(m \plus{} 95) \] for all $ m$ and $ n$ in $ \mathbb{N}.$ What is the value of $ \sum^{19}_{k \equal{} 1} f(k)?$

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by \[F(x)=\sum_{n=1}^{\infty} a_nx^n\] and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.

Find all functions $f: \mathbb{N} \mapsto \mathbb{N}$ so that for any positive integer $n$ and finite sequence of positive integers $a_0, \dots, a_n$, whenever the polynomial $a_0+a_1x+\dots+a_nx^n$ has at least one integer root, so does \[f(a_0)+f(a_1)x+\dots+f(a_n)x^n.\] [i]Proposed by Sutanay Bhattacharya[/i]

Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that $$a(2a+3b+6c)>0.$$

Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following: $$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$ [i]Proposed by Mykhailo Shtandenko[/i]

Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.

Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition $f (xf (y) + f (x)) = 2f (x) + xy$ for any real $x$ and $y$.

Let $F(z)=\frac{z+i}{z-i}$ for all complex numbers $z\not= i,$ and let $z_n=F(z_{n-1})$ for all positive integers $n.$ Given that $z_0=\frac 1{137}+i$ and $z_{2002}=a+bi,$ where $a$ and $b$ are real numbers, find $a+b.$