Let $n$ be a positive integer, $h$ a real number and $f(x)$ a polynomial (whole rational function) with real coefficients of degree n, which has no real zeros. Prove that then also the polynomial $$F(x) = f(x) + h f'(x) + h^2 f''(x) +... + h^n f^{(n)}(x)$$ has no real zeros.

Solve : $y(x+y)^2=9 $ ; $y(x^3-y^3)=7$

Is there a real number $\alpha$ such that $\cos\alpha$ is irrational but $\cos 2\alpha$, $\cos 3\alpha$, $\cos 4\alpha$, $\cos 5\alpha$ are all rational? (Author: V. Senderov)

Prove that every rational number $x$ in the interval $(0, 1)$ can be written as a finite sum of different fractions of the type $\frac{1}{k(k + 1)}$ , that is, different elements in the sequence $\frac12$ , $\frac{1}{6}$ , $\frac{1}{12}$,$...$.

A continuous function $f : R \to R$ satisfies the conditions $f(1000) = 999$ and $f(x)f(f(x)) = 1$ for all real $x$. Determine $f(500)$.

Calculate the following $$P=(4\sin^2{0} -3)(4\sin^2\frac{\pi}{2^{2005}} -3)(4\sin^2\frac{2\pi}{2^{2005}} -3)(4\sin^2\frac{3\pi}{2^{2005}} -3)...$$ $$...\,\,\,\,(4\sin^2\frac{(2^{2004}-1)\pi}{2^{2005}} -3)(4\sin^2\frac{\pi}{2} -3)$$

Find all functions $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(x^2 - y^2) = (x+y)(f(x) - f(y))$ for all real numbers $x$ and $y$.

A train is going up a hill with vertical velocity given as a function of $t$ by $\frac{1}{1 - t^4}$ , where $t$ is between $[0, 1)$. Determine its height as a function of $t$.

The given numbers are real numbers $ q,t \in \langle \frac{1}{2}; 1) $, $ t \in (0; 1 \rangle $. Prove that there is an increasing sequence of natural numbers $ {n_k} $ ($ k = 1,2, \ldots $) such that $$ t = \lim_{N\to \infty} \sum_{j=1}^N q^{n_j}.$$

Show that for every real $x \ge \frac12$ there is an integer $n$ such that $|x - n^2| \le \sqrt{x-\frac{1}{4}}$.

Let positive integers $M$, $m$, $n$ be such that $1\leq m \leq n$, $1\leq M \leq \dfrac {m(m+1)} {2}$, and let $A \subseteq \{1,2,\ldots,n\}$ with $|A|=m$. Prove there exists a subset $B\subseteq A$ with $$0 \leq \sum_{b\in B} b - M \leq n-m.$$ ([i]Dan Schwarz[/i])

Given A, non-inverted matrices of order n with real elements, $n\ge 2$ and given ${{A}^{*}}$adjoin matrix A. Prove that $tr({{A}^{*}})\ne -1$ if and only if the matrix ${{I}_{n}}+{{A}^{*}}$ is invertible.

Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

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

Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy $$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$

$a, b$ are odd numbers that satisfy $(a-b)^2 \le 8\sqrt {ab}$. For $n=ab$, show that the equation $$x^2-2([\sqrt n]+1)x+n=0$$ has two integral solutions. $[r]$ denotes the biggest integer, not strictly bigger than $r$.

Find all real solutions: $$\begin{cases}a^3=2024bc \\ b^3=2024cd \\ c^3=2024da \\ d^3=2024ab \end{cases}$$

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]

Find all functions $f : \mathbb R \to \mathbb R$ such that \[f(x)+f\left(1-\frac{1}{x}\right)=x,\] holds for all real $x$.

Let $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0}$, where $a_{0},\ldots,a_{n}$ are integers, $a_{n}>0$, $n\geq 2$. Prove that there exists a positive integer $m$ such that $P(m!)$ is a composite number.

Let $a,b,c \in \mathbb{R}$. Prove that $$\sum_{cyc} (a^3-b^3)^2+3\sum_{cyc}(a^2-b^2)^2+6(a-b)(b-c)(c-a)(ab+ bc+ca) \ge 0.$$ [i](LightLucifer)[/i]

Find the sum of all real numbers $x$ such that $x^2 = 5x + 6\sqrt{x} - 3$.

[b]p1.[/b] The final Big $10$ standings for the $1996$ Women's Softball season were 1. Michigan 2. Minnesota З. Iowa 4. Indiana 5. Michigan State 6. Purdue 7. Northwestern 8. Ohio State 9. Penn State 10. Wisconsin (Illinois does not participate in Women's Softball.) When you compare the $1996$ final standings (above) to the final standings for the $1999$ season, you find that the following pairs of teams changed order relative to each other from $1996$ to $1999$ (there are no ties, and no other pairs changed places): (Iowa, Michigan State) (Indiana, Penn State) (Purdue, Wisconsin) (Iowa, Penn State) (Indiana, Wisconsin) (Northwestern, Penn State) (Indiana, Michigan State) (Michigan State, Penn State) (Northwestern, Wisconsin) (Indiana, Purdue) (Purdue, Northwestern) (Ohio State, Penn State) (Indiana, Northwestern) (Purdue, Penn State) (Ohio State, Penn State) (Indiana, Ohio State) Determine as much as you can about the final Big $10$ standings for the $1999$ Women's Softball season. If you cannot determine the standings, explain why you do not have enough information. You must justify your answer. [b]p2.[/b] a) Take as a given that any expression of the form $A \sin t + B \cos t$ ($A>0$) can be put in the form $C \sin (t + D)$, where $C>0$ and $-\pi /2 <D <\pi /2 $. Determine $C$ and $D$ in terms of $A$ and $B$. b) For the values of $C$ and $D$ found in part a), prove that $A \sin t + B \cos t = C \sin (t + D)$. c) Find the maximum value of $3 \sin t +2 \cos t$. [b]pЗ.[/b] А $6$-bу-$6$ checkerboard is completelу filled with $18$ dominoes (blocks of size $1$-bу-$2$). Prove that some horizontal or vertical line cuts the board in two parts but does not cut anу of the dominoes. [b]p4.[/b] a) The midpoints of the sides of a regular hexagon are the vertices of a new hexagon. What is the ratio of the area of the new hexagon to the area of the original hexagon? Justify your answer and simplify as much as possible. b) The midpoints of the sides of a regular $n$-gon ($n >2$) are the vertices of a new $n$-gon. What is the ratio of the area of the new $n$-gon to that of the old? Justify your answer and simplify as much as possible. [b]p5. [/b] You run a boarding house that has $90$ rooms. You have $100$ guests registered, but on any given night only $90$ of these guests actually stay in the boarding house. Each evening a different random set of $90$ guests will show up. You don't know which $90$ it will be, but they all arrive for dinner before you have to assign rooms for the night. You want to give out keys to your guests so that for any set of $90$ guests, you can assign each to a private room without any switching of keys. a) You could give every guest a key to every room. But this requires $9000$ keys. Find a way to hand out fewer than $9000$ keys so that each guest will have a key to a private room. b) What is the smallest number of keys necessary so that each guest will have a key to a private room? Describe how you would distribute these keys and assign the rooms. Prove that this number of keys is as small as possible. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1, a_2, \ldots, a_{2024}$ be a permutation of $1, 2, \ldots, 2024$. Find the minimum possible value of\[\sum_{i=1} ^{2023} \Big[(a_i+a_{i+1})\Big(\frac{1}{a_i}+\frac{1}{a_{i+1}}\Big)+\frac{1}{a_ia_{i+1}}\Big]\] [i]Proposed by Md. Ashraful Islam Fahim[/i]

Let $p(x)$ be a polynomial with rational coefficients, of degree at least $2$. Suppose that a sequence $(r_{n})$ of rational numbers satisfies $r_{n}= p(r_{n+1})$ for every $n\geq 1$. Prove that the sequence $(r_{n})$ is periodic.