Found problems: 3597
2018 Pan-African Shortlist, C2
Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that:
[list]
[*] Adamu has a winning strategy if $n$ is odd.
[*] Afaafa has a winning strategy if $n$ is even.
[/list]
2023 Israel TST, P3
Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?
2022 Vietnam National Olympiad, 1
Consider 2 non-constant polynomials $P(x),Q(x)$, with nonnegative coefficients. The coefficients of $P(x)$ is not larger than $2021$ and $Q(x)$ has at least one coefficient larger than $2021$. Assume that $P(2022)=Q(2022)$ and $P(x),Q(x)$ has a root $\frac p q \ne 0 (p,q\in \mathbb Z,(p,q)=1)$. Prove that $|p|+n|q|\le Q(n)-P(n)$ for all $n=1,2,...,2021$
2007 Harvard-MIT Mathematics Tournament, 9
The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]
2020 German National Olympiad, 3
Show that the equation
\[x(x+1)(x+2)\dots (x+2020)-1=0\]
has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies
\[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]
2019 USEMO, 2
Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials)
such that
[list]
[*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$;
[*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does.
[/list]
[i]Carl Schildkraut[/i]
1991 Brazil National Olympiad, 3
Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \]
a)For which $k$ can we write $a_n$ as a polynomial in $n$?
b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).
2016 China National Olympiad, 3
Let $p$ be an odd prime and $a_1, a_2,...,a_p$ be integers. Prove that the following two conditions are equivalent:
1) There exists a polynomial $P(x)$ with degree $\leq \frac{p-1}{2}$ such that $P(i) \equiv a_i \pmod p$ for all $1 \leq i \leq p$
2) For any natural $d \leq \frac{p-1}{2}$,
$$ \sum_{i=1}^p (a_{i+d} - a_i )^2 \equiv 0 \pmod p$$
where indices are taken $\pmod p$
2010 District Olympiad, 2
Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
2002 Moldova Team Selection Test, 1
Prove that for every positive integer n, there exists a polynomial p(x) with integer coefficients such that p(1), p(2),..., p(n-1), p(n) are distinct powers of 2.
1993 Kurschak Competition, 3
Let $n$ be a fixed positive integer. Compute over $\mathbb{R}$ the minimum of the following polynomial:
\[f(x)=\sum_{t=0}^{2n}(2n+1-t)x^t.\]
2019 Taiwan TST Round 3, 2
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
1966 Miklós Schweitzer, 8
Prove that in Euclidean ring $ R$ the quotient and remainder are always uniquely determined if and only if $ R$ is a polynomial ring over some field and the value of the norm is a strictly monotone function of the degree of the polynomial. (To be precise, there are two trivial cases: $ R$ can also be a field or the null ring.)
[i]E. Fried[/i]
1978 IMO Longlists, 3
Find all numbers $\alpha$ for which the equation
\[x^2 - 2x[x] + x -\alpha = 0\]
has two nonnegative roots. ($[x]$ denotes the largest integer less than or equal to x.)
2015 Iran Team Selection Test, 1
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that
$$P(x)^3+Q(x)^3=x^{12}+1.$$
1995 IMO Shortlist, 3
For an integer $x \geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \ldots$ defined by $x_0 = 1$ and \[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \] for $n \geq 0$. Find all $n$ such that $x_n = 1995$.
2020 Jozsef Wildt International Math Competition, W31
Let $P$ be a real polynomial with degree $n\ge1$ such that
$$P(0),P(1),P(4),P(9),\ldots,P(n^2)$$
are in $\mathbb Z$. Prove that $\forall a\in\mathbb Z,P(a^2)\in\mathbb Z$.
[i]Proposed by Moubinool Omarjee[/i]
2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 1
Let $A=\left(
\begin{array}{ccc}
1 & 1& 0 \\
0 & 1& 0 \\
0 &0 & 2
\end{array}
\right),\ B=\left(
\begin{array}{ccc}
a & 1& 0 \\
b & 2& c \\
0 &0 & a+1
\end{array}
\right)\ (a,\ b,\ c\in{\mathbb{C}}).$
(1) Find the condition for $a,\ b,\ c$ such that ${\text{rank} (AB-BA})\leq 1.$
(2) Under the condition of (1), find the condition for $a,\ b,\ c$ such that $B$ is diagonalizable.
1990 IMO Longlists, 95
Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.
2024 Indonesia MO, 7
Suppose $P(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$ where $a_0, a_1, \ldots, a_{n-1}$ are reals for $n\geq 1$ (monic $n$th-degree polynomial with real coefficients). If the inequality
\[ 3(P(x)+P(y)) \geq P(x+y) \] holds for all reals $x,y$, determine the minimum possible value of $P(2024)$.
2005 Greece National Olympiad, 1
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
2008 China Girls Math Olympiad, 2
Let $ \varphi(x) \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} d$ be a polynomial with real coefficients. Given that $ \varphi(x)$ has three positive real roots and that $ \varphi(0) < 0$, prove that
\[ 2b^3 \plus{} 9a^2d \minus{} 7abc \leq 0.
\]
1990 IMO Longlists, 92
Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$. In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$, satisfying the following conditions:
[b][i]i)[/i][/b] Any two lines are non-parallel.
[b][i]ii)[/i][/b] Any three lines are non-concurrent.
[b][i]iii)[/i][/b] Only $A_1$ does not lies on any line $L_k$, and there are exactly $k + 1$ points $A_j$'s that lie on line $L_k$ $(k = 1, 2, \ldots, n).$
Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$
PEN L Problems, 13
The sequence $\{x_{n}\}_{n \ge 1}$ is defined by \[x_{1}=x_{2}=1, \; x_{n+2}= 14x_{n+1}-x_{n}-4.\] Prove that $x_{n}$ is always a perfect square.
1996 Spain Mathematical Olympiad, 3
Consider the functions $ f(x) = ax^{2} + bx + c $ , $ g(x) = cx^{2} + bx + a $, where a, b, c are real numbers. Given that $ |f(-1)| \leq 1 $, $ |f(0)| \leq 1 $, $ |f(1)| \leq 1 $, prove that $ |f(x)| \leq \frac{5}{4} $ and $ |g(x)| \leq 2 $ for $ -1 \leq x \leq 1 $.