Found problems: 3597
1987 Vietnam National Olympiad, 2
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.
2012 Germany Team Selection Test, 1
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
2006 Princeton University Math Competition, 3
Let $r_1, \dots , r_5$ be the roots of the polynomial $x^5+5x^4-79x^3+64x^2+60x+144$. What is $r^2_1+\dots+r^2_5$?
1967 IMO Longlists, 47
Prove the following inequality:
\[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1}
x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in
\mathbb{N}.$
2006 Iran MO (3rd Round), 6
$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$.
a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial.
b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?
1995 Austrian-Polish Competition, 4
Determine all polynomials $P(x)$ with real coefficients such that
$P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.
2016 Kosovo National Mathematical Olympiad, 2
Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .
2016 CentroAmerican, 3
The polynomial $Q(x)=x^3-21x+35$ has three different real roots. Find real numbers $a$ and $b$ such that the polynomial $x^2+ax+b$ cyclically permutes the roots of $Q$, that is, if $r$, $s$ and $t$ are the roots of $Q$ (in some order) then $P(r)=s$, $P(s)=t$ and $P(t)=r$.
1989 AIME Problems, 8
Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that
\[ \begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\,\,\,\,\,\,\,\, \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\,\,\,\,\, \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\ \end{array} \] Find the value of \[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\]
2009 Bulgaria National Olympiad, 4
Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$, for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\]
for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$).
2020 AIME Problems, 14
Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1$. Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b$. Find the sum of all possible values of $(a+b)^2$.
2017 China Team Selection Test, 4
Find out all the integer pairs $(m,n)$ such that there exist two monic polynomials $P(x)$ and $Q(x)$ ,with $\deg{P}=m$ and $\deg{Q}=n$,satisfy that $$P(Q(t))\not=Q(P(t))$$ holds for any real number $t$.
2020 Dutch IMO TST, 2
Determine all polynomials $P (x)$ with real coefficients that apply $P (x^2) + 2P (x) = P (x)^2 + 2$.
1977 USAMO, 1
Determine all pairs of positive integers $ (m,n)$ such that
$ (1\plus{}x^n\plus{}x^{2n}\plus{}\cdots\plus{}x^{mn})$ is divisible by $ (1\plus{}x\plus{}x^2\plus{}\cdots\plus{}x^{m})$.
1995 China Team Selection Test, 2
$ A$ and $ B$ play the following game with a polynomial of degree at least 4:
\[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0
\]
$ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?
2021 Azerbaijan Senior NMO, 5
Define $P(x)=((x-a_1)(x-a_2)...(x-a_n))^2 +1$, where $a_1,a_2...,a_n\in\mathbb{Z}$ and $n\in\mathbb{N^+}$. Prove that $P(x)$ couldn't be expressed as product of two non-constant polynomials with integer coefficients.
1969 IMO Shortlist, 24
$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$
2014 Contests, 3
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)
2012 ELMO Shortlist, 9
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
2007 China Team Selection Test, 1
When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.
1969 AMC 12/AHSME, 32
Let a sequence $\{u_n\}$ be defined by $u_1=5$ and the relation $u_{n+1}-u_n=3+4(n-1)$, $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, the algebraic sum of its coefficients is:
$\textbf{(A) }3\qquad
\textbf{(B) }4\qquad
\textbf{(C) }5\qquad
\textbf{(D) }6\qquad
\textbf{(E) }11$
2012 Putnam, 6
Let $p$ be an odd prime number such that $p\equiv 2\pmod{3}.$ Define a permutation $\pi$ of the residue classes modulo $p$ by $\pi(x)\equiv x^3\pmod{p}.$ Show that $\pi$ is an even permutation if and only if $p\equiv 3\pmod{4}.$
2006 Bulgaria Team Selection Test, 2
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]
2000 Moldova National Olympiad, Problem 4
Find all polynomials $P(x)$ with real coefficients that satisfy the relation
$$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$
1963 Putnam, B1
For what integers $a$ does $x^2 -x+a$ divide $x^{13}+ x +90$ ?