Found problems: 3597
2003 India IMO Training Camp, 7
$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$
2019 Serbia National Math Olympiad, 2
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .
A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove :
a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.
b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
2000 Moldova Team Selection Test, 3
For each positive integer $ n$, evaluate the sum
\[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]
2014 IberoAmerican, 2
Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:
$xP(x-c) = (x - 2014)P(x)$
2020 BMT Fall, 17
Let $T$ be the answer to question $16$. Compute the number of distinct real roots of the polynomial $x^4 + 6x^3 +\frac{T}{2}x^2 + 6x + 1$.
1985 All Soviet Union Mathematical Olympiad, 400
The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.
2005 MOP Homework, 4
Deos there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$,
$f(x^2y+f(x+y^2))=x^3+y^3+f(xy)$
2014 Iran MO (3rd Round), 5
We say $p(x,y)\in \mathbb{R}\left[x,y\right]$ is [i]good[/i] if for any $y \neq 0$ we have $p(x,y) = p\left(xy,\frac{1}{y}\right)$ . Prove that there are good polynomials $r(x,y) ,s(x,y)\in \mathbb{R}\left[x,y\right]$ such that for any good polynomial $p$ there is a $f(x,y)\in \mathbb{R}\left[x,y\right]$ such that \[f(r(x,y),s(x,y))= p(x,y)\]
[i]Proposed by Mohammad Ahmadi[/i]
2013 All-Russian Olympiad, 2
Peter and Basil together thought of ten quadratic trinomials. Then, Basil began calling consecutive natural numbers starting with some natural number. After each called number, Peter chose one of the ten polynomials at random and plugged in the called number. The results were recorded on the board. They eventually form a sequence. After they finished, their sequence was arithmetic. What is the greatest number of numbers that Basil could have called out?
2002 Mongolian Mathematical Olympiad, Problem 2
Prove that for each $n\in\mathbb N$ the polynomial $(x^2+x)^{2^n}+1$ is irreducible over the polynomials with integer coefficients.
2011 AIME Problems, 15
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
2006 Taiwan National Olympiad, 3
$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.
1988 IMO Shortlist, 2
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
2022 Flanders Math Olympiad, 4
Determine all real polynomials $P$ of degree at most $22$ for which
$$kP (k + 1) - (k + 1)P (k) = k^2 + k + 1$$
for all $k \in \{1, 2, 3, . . . , 21, 22\}$.
2015 Polish MO Finals, 2
Let $P$ be a polynomial with real coefficients. Prove that if for some integer $k$ $P(k)$ isn't integral, then there exist infinitely many integers $m$, for which $P(m)$ isn't integral.
2003 Moldova Team Selection Test, 1
Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form
$ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$,
if it is known that all the roots of them are positive reals.
[i]Proposer[/i]: [b]Baltag Valeriu[/b]
2019 AMC 12/AHSME, 14
For a certain complex number $c$, the polynomial
\[ P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]
has exactly 4 distinct roots. What is $|c|$?
$\textbf{(A) } 2 \qquad \textbf{(B) } \sqrt{6} \qquad \textbf{(C) } 2\sqrt{2} \qquad \textbf{(D) } 3 \qquad \textbf{(E) } \sqrt{10}$
1979 Bundeswettbewerb Mathematik, 4
Prove that the polynomial $P(x) = x^5-x+a$ is irreducible over $\mathbb{Z}$ if $5 \nmid a$.
2021 Durer Math Competition Finals, 8
John found all real numbers $p$ such that in the polynomial $g(x) = (x -1)^2(p + 2x)^2$ , the quadratic term has coefficient $2021$. What is the sum of all of these values $p$?
1951 Polish MO Finals, 4
Determine the coefficients of the equation $$ x^3 - ax^2 + bx - c = 0$$
in such a way that the roots of this equation are the numbers $ a $, $ b $, $ c $.
1967 IMO Shortlist, 2
Find all real solutions of the system of equations:
\[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$
2023 Ukraine National Mathematical Olympiad, 10.6
Let $P(x), Q(x), R(x)$ be polynomials with integer coefficients, such that $P(x) = Q(x)R(x)$. Let's denote by $a$ and $b$ the largest absolute values of coefficients of $P, Q$ correspondingly. Does $b \le 2023a$ always hold?
[i]Proposed by Dmytro Petrovsky[/i]
2002 Vietnam National Olympiad, 1
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?
2009 IMO Shortlist, 6
Let $k$ be a positive integer. Show that if there exists a sequence $a_0,a_1,\ldots$ of integers satisfying the condition \[a_n=\frac{a_{n-1}+n^k}{n}\text{ for all } n\geq 1,\] then $k-2$ is divisible by $3$.
[i]Proposed by Okan Tekman, Turkey[/i]
2018 CCA Math Bonanza, I13
$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\right)$?
[i]2018 CCA Math Bonanza Individual Round #13[/i]