Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. [i] Proposed by usjl[/i]

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Let $a, b$ and $c$ be positive real numbers for which $a + b + c = 1$. Prove that $$\frac{a^2}{b^3 + c^4 + 1}+\frac{b^2}{c^3 + a^4 + 1}+\frac{c^2}{a^3 + b^4 + 1} > \frac{1}{5}$$

The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.

Consider the positive reals $ x$, $ y$ and $ z$. Prove that: a) $ \arctan(x) \plus{} \arctan(y) < \frac {\pi}{2}$ iff $ xy < 1$. b) $ \arctan(x) \plus{} \arctan(y) \plus{} \arctan(z) < \pi$ iff $ xyz < x \plus{} y \plus{} z$.

Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

Let $a$ and $b$ satisfy $a \ge b >0, a + b = 1$. i) Prove that if $m$ and $n$ are positive integers with $m < n$, then $a^m - a^n \ge b^m- b^n > 0$. ii) For each positive integer $n$, consider a quadratic function $f_n(x) = x^2 - b^nx- a^n$. Show that $f(x)$ has two roots that are in between $-1$ and $1$.

Write the number $\frac{1997}{1998}$ as a sum of different numbers, inverse to naturals.

A rectangle has an area of $16$ and a perimeter of $18$; determine the length of the diagonal of the rectangle. [i]2015 CCA Math Bonanza Individual Round #8[/i]

Let $a^3 - a- 1 = 0$. Find the exact value of the expression $$\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.$$

Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial. [i]Ashwin Sah[/i]

Let $a,b,c\geq 0$ and $a+b+c=1.$ Prove that$$\frac{a}{2a+1}+\frac{b}{3b+1}+\frac{c}{6c+1}\leq \frac{1}{2}.$$ [size=50](Marian Dinca)[/size]

Real numbers $x$ and $y$ satisfy the system of equations $$x^3 + 3x^2 = -3y - 1$$ $$y^3 + 3y^2 = -3x - 1.$$ What is the greatest possible value of $x$?

Let $a,b,c$ be three different positive integers. Show that the sequence \[a+b+c,ab+bc+ca,3abc\] could be neither an arithmetic nor geometric progression. [i]Fajar Yuliawan, Bandung[/i]

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by [i]Warren Bei[/i]

Find all function $f: \mathbb{R}^+ \rightarrow \mathbb{R}$,such that the inequality $$f(x) + f\left(\frac{y}{x}\right) \leqslant \frac{x^3}{y^2} + \frac{y}{x^3}$$ holds for all positive reals $x,y$ and for every positive real $x$, there exist positive reals $y$, such that the equality holds.

Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.

Consider the regular $1987$-gon $A_1A_2 . . . A_{1987}$ with center $O$. Show that the sum of vectors belonging to any proper subset of $M = \{OA_j | j = 1, 2, . . . , 1987\}$ is nonzero.

Suppose that $f : \mathbb R^+ \to \mathbb R^+$ is a decreasing function such that \[f(x+y)+f(f(x)+f(y))=f(f(x+f(y))+f(y+f(x)), \quad \forall x,y \in \mathbb R^+.\] Prove that $f(x) = f^{-1}(x).$

Find all polynomials $P(x)$ with real coefficients such that for all real $x$ holds the equality $$(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .$$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

Let $a,b,c$ be reals and \[f(a, b, c) = \left| \frac{ |b-a|}{|ab|} +\frac{b+a}{ab} -\frac 2c \right| +\frac{ |b-a|}{|ab|} +\frac{b+a}{ab} +\frac 2c\] Prove that $f(a, b, c) = 4 \max \{\frac 1a, \frac 1b,\frac 1c \}.$

Let $ n \ge 2$ and $x_1$, $x_2$, $...$, $x_n$ be real numbers from the segment $[1,\sqrt2]$. Prove that holds the inequality $$\frac{\sqrt{x_1^2-1}}{x_2}+\frac{\sqrt{x_2^2-1}}{x_3}+...+\frac{\sqrt{x_n^2-1}}{x_1} \le \frac{\sqrt2}{2} n.$$

Find all polynomials $P$ with real coefficients satisfying $P(x^2)=P(x)\cdot P(x+2)$ for all real numbers $x.$