Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.

Find all functions $f:R \rightarrow R$, which satisfy the equality for any $x,y \in R$: $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$,

Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.

For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds: For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which \[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\] a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$. b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.

Let $a,b,c$ be positive reals such that $abc=1$. Prove the inequality $$\frac{4a-1}{(2b+1)^2} + \frac{4b-1}{(2c+1)^2} + \frac{4c-1}{(2a+1)^2}\geqslant 1.$$

For which real values of $x,y$ the expression$\frac{2-\left(\dfrac{x+y}{3}-1\right)^2}{\left(\dfrac{x-3}{2}+\dfrac{2y-x}{3}\right)^2+4}$ becomes maximum? Which is that maximum value?

$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define $A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.

Does there exist a functional equation[sup]1[/sup] that has a solution and the range of any of its solutions is the set of integers? [sup]1[/sup][size=75]A [i]functional equation[/i] has the form $\mbox{\footnotesize \(E = 0\)}$, where $\mbox{\footnotesize \(E\)}$ is a function form. The set of function forms is the smallest set $\mbox{\footnotesize \(\mathcal{F}\)}$ which contains the variables $\mbox{\footnotesize \(x_1, x_2, \dots\)}$, the real numbers $\mbox{\footnotesize \(r \in \mathbb{R}\)}$, and for which $\mbox{\footnotesize \(E, E_1, E_2 \in \mathcal{F}\)}$ implies $\mbox{\footnotesize \(E_1+E_2 \in \mathcal{F}\)}$, $\mbox{\footnotesize \(E_1 \cdot E_2 \in \mathcal{F}\)}$, and $\mbox{\footnotesize \(f(E) \in \mathcal{F}\)}$, where $\mbox{\footnotesize \(f\)}$ is a fixed function symbol. The solution of the functional equation $\mbox{\footnotesize \(E = 0\)}$ is a function $\mbox{\footnotesize \(f: \mathbb{R} \to \mathbb{R}\)}$ such that $\mbox{\footnotesize \(E = 0\)}$ holds for all values of the variables. E.g. $\mbox{\footnotesize \(f\big(x_1 + f(\sqrt{2} \cdot x_2 \cdot x_2)\big) + (-\pi) + (-1) \cdot x_1 \cdot x_1 \cdot x_2 = 0\)}$ is a functional equation.[/size]

Prove that for all integers $n > 1$, $$\frac{1}{n}+\frac{1}{n + 1}+ ...+\frac{1}{n^2- 1}+\frac{1}{n^2} > 1$$

Let $\overline{a_{n}a_{n-1}\ldots a_{1}a_{0}}$ be the decimal representation of a prime positive integer such that $n>1$ and $a_{n}>1$. Prove that the polynomial $P(x)=a_{n}x^{n}+\ldots +a_{1}x+a_{0}$ cannot be written as a product of two non-constant integer polynomials.

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? $\textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} $

Determine all monic polynomials $p(x)$ having real coefficients and satisfying the following two conditions: $\bullet$ $p(x)$ is nonconstant, and all of its roots are distinct reals $\bullet$ If $a $and $b$ are roots of $p(x)$ then $a + b + ab$ is also a root of $p(x)$.

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.) [i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f: R \to R$ such that $f (xy) \le yf (x) + f (y)$, for all $x, y\in R$.

Find all functions $f : R \to R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$ for all reals $x_1, x_2, . . . , x_{2550}$.

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.

Find all real solutions $(x,y,z)$ of the system $$\begin{cases}\dfrac{4x^2}{1+4x^2}= y\\ \\\dfrac{4y^2}{1+4y^2}= z\\ \\ \dfrac{4z^2}{1+4z^2}= x \end{cases}$$

Let $ P(x)$ be the real polynomial function, $ P(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d.$ Prove that if $ |P(x)| \leq 1$ for all $ x$ such that $ |x| \leq 1,$ then \[ |a| \plus{} |b| \plus{} |c| \plus{} |d| \leq 7.\]

Is it possible to simultaneously take away on eight three-ton vehicles $50$ stones, the weight of which is respectively equal to $416, 418, 420, .., 512, 514$ kg?

Solve the equation $$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$