$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .

What is the largest integer that is a divisor of \[ (n\plus{}1)(n\plus{}3)(n\plus{}5)(n\plus{}7)(n\plus{}9) \]for all positive even integers $ n$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 165$

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]

Given a non-zero real number $a$. Find all functions $f : R \to R$, such that $$f(f(x + y)) = f(x + y) + f(x)f(y) + axy$$ for all $x, y \in R$.

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5 $

A [i]tile[/i] $T$ is a union of finitely many pairwise disjoint arcs of a unit circle $K$. The [i]size[/i] of $T$, denoted by $|T|$, is the sum of the lengths of the arcs $T$ consists of, divided by $2\pi$. A [i]copy[/i] of $T$ is a tile $T'$ obtained by rotating $T$ about the centre of $K$ through some angle. Given a positive real number $\varepsilon < 1$, does there exist an infinite sequence of tiles $T_1,T_2,\ldots,T_n,\ldots$ satisfying the following two conditions simultaneously: 1) $|T_n| > 1 - \varepsilon$ for all $n$; 2) The union of all $T_n'$ (as $n$ runs through the positive integers) is a proper subset of $K$ for any choice of the copies $T_1'$, $T_2'$, $\ldots$, $T_n', \ldots$? [hide=Note] In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.[/hide]

Find all possible real values of $a$ for which the system of equations \[\{\begin{array}{cc}x +y +z=0\\\text{ } \\ xy+yz+azx=0\end{array}\] has exactly one solution.

For certain $a,b,c$ holds: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ Prove that for all odd $n$ holds, $$\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}=\frac{1}{a^n+b^n+c^n}.$$

a sequence $(a_n)$ $n$ $\geq 1$ is defined by the following equations; $a_1=1$, $a_2=2$ ,$a_3=1$, $a_{2n-1}$$a_{2n}$=$a_2$$a_{2n-3}$+$(a_2a_{2n-3}+a_4a_{2n-5}.....+a_{2n-2}a_1)$ for $n$ $\geq 2$ $na_{2n}$$a_{2n+1}$=$a_2$$a_{2n-2}$+$(a_2a_{2n-2}+a_4a_{2n-4}.....+a_{2n-2}a_2)$ for $n$ $\geq 2$ find $a_{2020}$

For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that \[ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) \]

A sequence $ (x_n)$ is given as follows: $ x_0,x_1$ are arbitrary positive real numbers, and $ x_{n\plus{}2}\equal{}\frac{1\plus{}x_{n\plus{}1}}{x_n}$ for $ n \ge 0$. Find $ x_{1998}$.

Prove that if all roots of a monic cubic polynomial have modulus $ 1, $ then, the two middle coefficients have the same modulus.

Let $n$ be a positive integer. A finite set of integers is called $n$-divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$-divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$, or $\{1, 5, 7\}$ and $\{3, 4, 6\}$. Find all the integers $n > 0$ for which there exists a $n$-divided set. Proposed by [i]Martin Rakovsky, France[/i]

Let $a$, $b$, and $c$ be the solutions of the equation $$x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.$$ Compute $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$, \[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\] ($\mathbb{R}^+$ denotes the set of positive real numbers.) [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

Find all arcs $\theta$ such that $\frac{1}{\sin ^2 \theta}, \frac{1}{\cos ^2 \theta} $ are integer numbers and roots of equation $$x^2-ax+a=0.$$

Find all triplets $(a,b,c)$ of complex numbers such that the equation \[x^4-ax^3-bx+c=0\] has $a,b,c$ as roots.

The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$, $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$. Prove that every $p_n$ is different from $5$.

What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?

Show that the polynomial $x^6 -x^5 +x^4 -x^3 +x^2 -x+\frac34$ has no real zeroes.

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

Let $a$ be a real number such that $\left(a + \frac{1}{a}\right)^2=11$. For which $n\in N$ is $a^n + \frac{1}{a^n}$ an integer? Does this depend on the exact value of $a$?