If $ a_0,a_1,\dots,a_{50}$ are the coefficients of the polynomial \[ \left(1\plus{}x\plus{}x^2\right)^{25}\] show that $ a_0\plus{}a_2\plus{}a_4\plus{}\cdots\plus{}a_{50}$ is even.

Let $ a_i,b_i$ $ (i\equal{}1,2,...,n)$ be real numbers such that the $ a_i$ are distinct, and suppose that there is a real number $ \alpha$ such that the product $ (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n)$ is equal to $ \alpha$ for each $ i$. Prove that there is a real number $ \beta$ such that $ (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j)$ is equal to $ \beta$ for each $ j$.

It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .

Two players alternately replace the stars in the expression \[*x^{2000}+*x^{1999}+...+*x+1 \] by real numbers. The player who makes the last move loses if the resulting polynomial has a real root $t$ with $|t| < 1$, and wins otherwise. Give a winning strategy for one of the players.

Given is the equation: \[x^2+mx+2022=0\] a) Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{natural}$ numbers b)Find all the values of the parameter $m$, such that the two solutions of the equation $x_1, x_2$ are $\textbf{integer}$ numbers

Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$. [I]Proposed by Italy[/i]

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

Let $r_1, r_2, \cdots, r_7$ be the distinct complex roots of the polynomial $P(x) = x^7 - 7$ Let \[K = \prod_{1 \leq i < j \leq 7} (r_i + r_j)\] that is, the product of all the numbers of the form $r_i + r_j$, where $i$ and $j$ are integers for which $1 \leq i < j \leq 7$. Determine the value of $K^2$.

Find all polynomials $P(x)$ with real coefficients such that \[P(x)P(x + 1) = P(x^2), \quad \forall x \in \mathbb R.\]

Let $P(x),Q(x)$ be monic polynomials with integer coeeficients. Let $a_n=n!+n$ for all natural numbers $n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for all positive integer $n$ then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n\neq0$. \\ [i]Hint (given in question): Try applying division algorithm for polynomials [/i]

Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}\left(\omega^{a_p} - \omega^{a_q}\right).\] Given that $S$ is a positive integer, compute the number of positive divisors of $S$.

Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.

We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$: \[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\] Find the sum of all $a_{2023}$'s possible values.

We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.

The monic polynomial $P(x) = x^n + a_{n-1}x^{n-1} +...+ a_0$ of degree $n > 1$ has $n$ distinct negative roots. Prove that $a_1P(1) > 2n^2a_o$

$23$ frat brothers are sitting in a circle. One, call him Alex, starts with a gallon of water. On the first turn, Alex gives each person in the circle some rational fraction of his water. On each subsequent turn, every person with water uses the same scheme as Alex did to distribute his water, but in relation to themselves. For instance, suppose Alex gave $\frac{1}{2}$ and $\frac{1}{6}$ of his water to his left and right neighbors respectively on the first turn and kept $\frac{1}{3}$ for himself. On each subsequent turn everyone gives $\frac{1}{2}$ and $\frac{1}{6}$ of the water they started the turn with to their left and right neighbors, respectively, and keep the final third for themselves. After $23$ turns, Alex again has a gallon of water. What possibilities are there for the scheme he used in the first turn? (Note: you may find it useful to know that $1+x+x^2+\cdot +x^{23}$ has no polynomial factors with rational coefficients)

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Let $ x,y,z$ be positive real numbers, show that $ \frac {xy}{z} \plus{} \frac {yz}{x} \plus{} \frac {zx}{y} > 2\sqrt [3]{x^3 \plus{} y^3 \plus{} z^3}.$

Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$. i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$. ii) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.

Let $x$ and $y$ be real numbers satisfying $x^4y^5+y^4x^5=810$ and $x^3y^6+y^3x^6=945$. Evaluate $2x^3+(xy)^3+2y^3$.

Find all couples of polynomials $(P,Q)$ with real coefficients, such that for infinitely many $x\in\mathbb R$ the condition \[ \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)}\] Holds. [i] Nikolai Nikolov, Oleg Mushkarov[/i]

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

Let $P(x)$ be a polynomial with integer coefficients. We will denote the set of all prime numbers by $\mathbb P$. Show that the set $\mathbb S := \{p\in\mathbb P : \exists\text{ }n \text{ s.t. }p\mid P(n)\}$ is finite if and only if $P(x)$ is a non-zero constant polynomial.

For how many positive integers $n$ is $n^3-8n^2+20n-13$ a prime number? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{more than 4}$

If $ a$ and $b$ are positive numbers, prove that the equation $$\frac{1}{x}+\frac{1}{x - a}+\frac{1}{x+ b}= 0$$ has two rea] roots, one between $ a/3$ and $2a/3$, and one between $-2b/3$ and $-b/3$.