Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.

A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$ $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$ $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$ $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of $f(2023)$.

For which positive integers $n$, $m$ does there exist a polynomial of degree $n$, all coefficients of which are powers of $m$ with integer exponents, having $n$ rational roots, counting multiplicities? [i]Proposed by Fedor Petrov[/i]

Let $p = 2^{16}+1$ be an odd prime. Define $H_n = 1+ \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$. Compute the remainder when \[ (p-1)! \sum_{n = 1}^{p-1} H_n \cdot 4^n \cdot \binom{2p-2n}{p-n} \] is divided by $p$. [i]Proposed by Yang Liu[/i]

Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

The nonzero real numbers $a, b, c$ satisfy the following system: $$\begin{cases} a+ab=c\\ b+bc=a\\ c+ca=b \end{cases}$$ Find all possible values of the $abc$.

We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$ We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$. Prove that for every $n>0$, $y_n$ is the square of an odd integer

Let $ABC$ be an isosceles right-angled triangle of area 1. Find the length of the shortest segment that divides the triangle into 2 parts of equal area.

Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that: \[ 45 < x_{1000} < 45.1. \]

Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]

We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.

Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds: [list] [*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$; [*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$. [/list]

Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$.

Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n. We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.

Let $x_1,x_2, ..., x_n$ be non-negative real numbers. Solve the system of equations: $$x_k+x_{k+1}=x^2_{k+2}\,\,,\,\,\, (k =1,2,...,n),$$ where $x_{n+1} = x_1$, $x_{n+2} = x_2$.

[b]p1.[/b] In a given tetrahedron the sum of the measures of the three face angles at each of the vertices is $180$ degrees. Prove that all faces of the tetrahedron are congruent triangles. [img]https://cdn.artofproblemsolving.com/attachments/c/c/40f03324fd19f6a5e0a5e541153a2b38faac79.png[/img] [b]p2.[/b] The digital sum $D(n)$ of a positive integer $n$ is defined recursively by: $D(n) = n$ if $1 \le n \le 9$ $D(n) = D(a_0 + a_1 + ... + a_m)$ if $n>9$ where $a_0 , a_1 ,..,a_m$ are all the digits of $n$ expressed in base ten. (For example, $D(959) = D(26) = D(8) = 8$.) Prove that $D(n \times 1234)= D(n)$ fcr all positive integers $n$ . [b]p3.[/b] A right triangle has area $A$ and perimeter $P$ . Find the largest possible value for the positive constant $k$ such that for every such triangle, $P^2 \ge kA$ . [b]p4.[/b] In the accompanying diagram, $\overline{AB}$ is tangent at $A$ to a circle of radius $1$ centered at $O$ . The segment $\overline{AP}$ is equal in length to the arc $AB$ . Let $C$ be the point of intersection of the lines $AO$ and $PB$ . Determine the length of segment $\overline{AC}$ in terms of $a$ , where $a$ is the measure of $\angle AOB$ in radians. [img]https://cdn.artofproblemsolving.com/attachments/e/0/596e269a89a896365b405af7bc6ca47a1f7c57.png[/img] [b]p5.[/b] Let $a_1 = a > 0$ and $a_2 = b >a$. Consider the sequence $\{a_1,a_2,a_3,...\}$ of positive numbers defined by: $a_3=\sqrt{a_1a_2}$, $a_4=\sqrt{a_2a_3}$, $...$ and in general, $a_n=\sqrt{a_{n-2}a_{n-1}}$, for $n\ge 3$ . Develop a formula $a_n$ expressing in terms of $a$, $b$ and $n$ , and determine $\lim_{n \to \infty} a_n$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the minimal value of $\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}$, if $x$, $y$ and $z$ are nonnegative real numbers such that $x+y+z=4$

$a_{n}$ is a sequence of positive integers such that, for every $n\geq 1$, $0<a_{n+1}-a_{n}<\sqrt{a_{n}}$. Prove that for every $x,y\in{R}$ such that $0<x<y<1$ $x< \frac{a_{k}}{a_{m}}<y$ we can find such $k,m\in{Z^{+}}$.

Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy \[xP(x - a) = (x - b)P(x).\]

Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.

If $n \ge 2$ is an integer, prove the equality $$\prod_{k=1}^n \tan \frac{\pi}{3}\left(1+\frac{3^k}{3^n-1}\right)=\prod_{k=1}^n \cot \frac{\pi}{3}\left(1-\frac{3^k}{3^n-1}\right)$$

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that $$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$ for any $x;y\in\mathbb{Q}$

Let be given an open interval $(\alpha; \beta)$ with $\alpha - \beta = \frac{1}{27}$. Determine the maximum number of irreducible fractions $\frac{a}{b}$ in $(\alpha; \beta)$ with $1 \leq b \leq 2007$?