Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$

Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$. [b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar. [b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.

(1) Determine if there exists an infinite sequence $\{a_n\}$ with positive integer terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$. (2) Determine if there exists an infinite sequence $\{a_n\}$ with positive irrational terms, such that $a^2_{n+1}\ge 2a_na_{n+2}$ for any positive integer $n$.

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

Prove that the polynomial \[ f(x) \equal{} \frac {x^n \plus{} x^m \minus{} 2}{x^{\gcd(m,n)} \minus{} 1}\] is irreducible over $ \mathbb{Q}$ for all integers $ n > m > 0$.

Numbers $1,\ldots,2025$ are written in a circle in increasing order. For every three consecutive numbers $i,j,k$ we consider the polynomial $(x-i)(x-j)(x-k)$. Let $s(x)$ be the sum of all $2025$ these polynomials. Prove that $s(x)$ has an integral root. [i]A. Voidelevich[/i]

Real function $f$ [b]generates[/b] real function $g$ if there exists a natural $k$ such that $f^k=g$ and we show this by $f \rightarrow g$. In this question we are trying to find some properties for relation $\rightarrow$, for example it's trivial that if $f \rightarrow g$ and $g \rightarrow h$ then $f \rightarrow h$.(transitivity) (a) Give an example of two real functions $f,g$ such that $f\not = g$ ,$f\rightarrow g$ and $g\rightarrow f$. (b) Prove that for each real function $f$ there exists a finite number of real functions $g$ such that $f \rightarrow g$ and $g \rightarrow f$. (c) Does there exist a real function $g$ such that no function generates it, except for $g$ itself? (d) Does there exist a real function which generates both $x^3$ and $x^5$? (e) Prove that if a function generates two polynomials of degree 1 $P,Q$ then there exists a polynomial $R$ of degree 1 which generates $P$ and $Q$. Time allowed for this problem was 75 minutes.

Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.

Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$. (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$. (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$. (c) Show, in (b) that $s=0$ implies $a_2=0$.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Find all polynomials $P$ in two variables with real coefficients satisfying the identity $$P(x,y)P(z,t)=P(xz-yt,xt+yz).$$

A polynomial $p(x)$ has remainder three when divided by $x-1$ and remainder five when divided by $x-3$. The remainder when $p(x)$ is divided by $(x-1)(x-3)$ is $\textbf{(A) }x-2\qquad\textbf{(B) }x+2\qquad\textbf{(C) }2\qquad\textbf{(D) }8\qquad \textbf{(E) }15$

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

Show that $\cos \frac{\pi}{7}$ is irrational.

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]

Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 4x^2 -3x + 2$. The sum $|x_1| + |x_2| + |x_3|$ is (A): $4$ (B): $6$ (C): $8$ (D): $10$ (E): None of the above.

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

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

Given $n\ge 2$ a natural number, $(K,+,\cdot )$ a body with commutative property that $\underbrace{1+...+}_{m}1\ne 0,m=2,...,n,f\in K[X]$ a polynomial of degree $n$ and $G$ a subgroup of the additive group $(K,+,\cdot )$, $G\ne K.$Show that there is $a\in K$ so$f(a)\notin G$.

The square polynomial $x^2+ax+b+1$ has natural roots. Prove that $(a^2+b^2)$ is a composite number.

Do there exist two polynomials $P$ and $Q$ with integer coefficient such that i) both $P$ and $Q$ have a coefficient with absolute value bigger than $2021$, ii) all coefficients of $P \cdot Q$ by absolute value are at most $1$.

You are given $n$ different primes $p_1, p_2,..., p_n$. Consider the polynomial $$x^n + a_1x^{n -1} + a_2x^{n - 2} + ...+ a_{n - 1}x + a_n$$, where $a_i$ is the product of the first $i$ given prime numbers. For what $n$ can it have an integer root?

Let $f\in\mathbb{Z}[X]$ be an irreducible polynomial over the ring of integer polynomials, such that $|f(0)|$ is not a perfect square. Prove that if the leading coefficient of $f$ is 1 (the coefficient of the term having the highest degree in $f$) then $f(X^2)$ is also irreducible in the ring of integer polynomials. [i]Mihai Piticari[/i]