Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

$a,b,c$ are distinct real roots of $x^3-3x+1=0$. $a^8+b^8+c^8$ is $ \textbf{(A)}\ 156 \qquad \textbf{(B)}\ 171 \qquad \textbf{(C)}\ 180 \qquad \textbf{(D)}\ 186 \qquad \textbf{(E)}\ 201$

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

Find, with justification, all positive real numbers $a,b,c$ satisfying the system of equations: $$\begin{cases} a\sqrt{b}=a+c \\ b\sqrt{c}=b+a \\ c\sqrt{a}=c+b \end{cases}$$

Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

Determine the smallest and the greatest possible values of the expression $$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$. [i]Proposed by Walther Janous, Austria [/i]

The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

[b]p1.[/b] (a) Show that for every set of three integers, we can find two of them whose average is also an integer. (b) Show that for every set of $5$ integers, there is a subset of three of them whose average is an integer. [b]p2.[/b] Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two different quadratic polynomials such that $f(7) + f(11) = g(7) + g(11)$. (a) Show that $f(9) = g(9)$. (b) Show that $x = 9$ is the only value of $x$ where $f(x) = g(x)$. [b]p3.[/b] Consider a rectangle $ABCD$ and points $E$ and $F$ on the sides $BC$ and $CD$, respectively, such that the areas of the triangles $ABE$, $ECF$, and $ADF$ are $11$, $3$, and $40$, respectively. Compute the area of rectangle $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/f/0/2b0bb188a4157894b85deb32d73ab0077cd0b7.png[/img] [b]p4.[/b] How many ways are there to put markers on a $8 \times 8$ checkerboard, with at most one marker per square, such that each of the $8$ rows and each of the $8$ columns contain an odd number of markers? [b]p5.[/b] A robot places a red hat or a blue hat on each person in a room. Each person can see the colors of the hats of everyone in the room except for his own. Each person tries to guess the color of his hat. No communication is allowed between people and each person guesses at the same time (so no timing information can be used, for example). The only information a person has is the color of each other person’s hat. Before receiving the hats, the people are allowed to get together and decide on their strategies. One way to think of this is that each of the $n$ people makes a list of all the possible combinations he could see (there are $2^{n-1}$ such combinations). Next to each combination, he writes what his guess will be for the color of his own hat. When the hats are placed, he looks for the combination on his list and makes the guess that is listed there. (a) Prove that if there are exactly two people in the room, then there is a strategy that guarantees that always at least one person gets the right answer for his hat color. (b) Prove that if you have a group of $2008$ people, then there is a strategy that guarantees that always at least $1004$ people will make a correct guess. (c) Prove that if there are $2009$ people, then there is no strategy that guarantees that always at least $1005$ people will make a correct guess. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two. How many three-element subsets of the set of integers $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$ are arithmetic and harmonic? (Remark: The arithmetic mean $A(a,b)$ and the harmonic mean $H(a,b)$ are defined as \[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\] respectively, where $H(a,b)$ is not defined for some $a$, $b$.)

Suppose that $x, y$, and $z$ are positive real numbers satisfying $$\begin{cases} x^2 + xy + y^2 = 64 \\ y^2 + yz + z^2 = 49 \\ z^2 + zx + x^2 = 57 \end{cases}$$ Then $\sqrt[3]{xyz}$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that \[yf(x+1)=f(x+y-f(x))+f(x)f(f(y))\] for all $x,y \in \mathbb{R}$.

Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.

For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?

One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.

Find the smallest positive interger number $n$ such that there exists $n$ real numbers $a_1,a_2,\ldots,a_n$ satisfied three conditions as follow: a. $a_1+a_2+\cdots+a_n>0$; b. $a_1^3+a_2^3+\cdots+a_n^3<0$; c. $a_1^5+a_2^5+\cdots+a_n^5>0$.

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x\plus{}af(y)\leq y\plus{}f(f(x)) \] for all $ x,y\in\mathbb{R}$

Let ${\bf R}$ denote the set of all real numbers. Find all functions $f$ from ${\bf R}$ to ${\bf R}$ satisfying: (i) there are only finitely many $s$ in ${\bf R}$ such that $f(s)=0$, and (ii) $f(x^4+y)=x^3f(x)+f(f(y))$ for all $x,y$ in ${\bf R}$.

Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.

To every natural number $k, k \geq 2$, there corresponds a sequence $a_n(k)$ according to the following rule: \[a_0 = k, \qquad a_n = \tau(a_{n-1}) \quad \forall n \geq 1,\] in which $\tau(a)$ is the number of different divisors of $a$. Find all $k$ for which the sequence $a_n(k)$ does not contain the square of an integer.

If the function $f$ is defined by \[ f(x)=\frac{cx}{2x+3} , ~~~x\neq -\frac 32 , \] satisfies $x=f(f(x))$ for all real numbers $x$ except $-\frac 32$, then $c$ is $\text{(A)} \ -3 \qquad \text{(B)} \ - \frac{3}{2} \qquad \text{(C)} \ \frac{3}{2} \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{not uniquely determined}$

The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} \equal{} 1, a_{2} \equal{} 3$, and $ a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.

A finite set $\mathcal{S}$ of positive integers is called cardinal if $\mathcal{S}$ contains the integer $|\mathcal{S}|$ where $|\mathcal{S}|$ denotes the number of distinct elements in $\mathcal{S}$. Let $f$ be a function from the set of positive integers to itself such that for any cardinal set $\mathcal{S}$, the set $f(\mathcal{S})$ is also cardinal. Here $f(\mathcal{S})$ denotes the set of all integers that can be expressed as $f(a)$ where $a \in \mathcal{S}$. Find all possible values of $f(2024)$ $\quad$ Proposed by Sutanay Bhattacharya

Solve the following equation in $\mathbb{R}$: $$\left(x-\frac{1}{x}\right)^\frac{1}{2}+\left(1-\frac{1}{x}\right)^\frac{1}{2}=x.$$