Found problems: 3597
2024 Indonesia TST, A
Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then
$P(a^2+a) \geq a.P(a+1)$
2012 India Regional Mathematical Olympiad, 6
Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.
2019 Romania National Olympiad, 4
Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that
$\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct.
$\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$
2013 Putnam, 3
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
2015 Harvard-MIT Mathematics Tournament, 8
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]
2023 Iran MO (3rd Round), 1
Find all integers $n > 4$ st for every two subsets $A,B$ of $\{0,1,....,n-1\}$ , there exists a polynomial $f$ with integer coefficients st either $f(A) = B$ or $f(B) = A$ where the equations are considered mod n.
We say two subsets are equal mod n if they produce the same set of reminders mod n. and the set $f(X)$ is the set of reminders of $f(x)$ where $x \in X$ mod n.
1984 Vietnam National Olympiad, 2
Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy
\[xP(x - a) = (x - b)P(x).\]
2014 Contests, 1
Let $a$, $b$, $c$, $d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.
2020 BMT Fall, 3
The graph of the degree $2021$ polynomial $P(x)$, which has real coefficients and leading coefficient $1$, meets the $x$-axis at the points $(1,0),\, (2,0),\,(3,0),\dots,\, (2020,0)$ and nowhere else. The mean of all possible values of $P(2021)$ can be written in the form $a!/b$, where $a$ and $b$ are positive integers and $a$ is as small as possible. Compute $a+b$.
2014 Iran MO (3rd Round), 8
The polynomials $k_n(x_1, \ldots, x_n)$, where $n$ is a non-negative integer, satisfy the following conditions
\[k_0=1\]
\[k_1(x_1)=x_1\]
\[k_n(x_1, \ldots, x_n) = x_nk_{n-1}(x_1, \ldots , x_{n-1}) + (x_n^2+x_{n-1}^2)k_{n-2}(x_1,\ldots,x_{n-2})\]
Prove that for each non-negative $n$ we have $k_n(x_1,\ldots,x_n)=k_n(x_n,\ldots,x_1)$.
2014 Contests, 1
Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.
2006 Stanford Mathematics Tournament, 10
Find the smallest positive $m$ for which there are at least 11 even and 11 odd positive integers $n$ so that $\tfrac{n^3+m}{n+2}$ is an integer.
2019 LIMIT Category A, Problem 1
Let $p(x)$ be a polynomial of degree $4$ with leading coefficient $1$. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$ and $p(4)=4$. Then $p(5)=$?
$\textbf{(A)}~5$
$\textbf{(B)}~\frac{25}6$
$\textbf{(C)}~29$
$\textbf{(D)}~35$
2015 VJIMC, 4
[b]Problem 4 [/b]
Let $m$ be a positive integer and let $p$ be a prime divisor of $m$. Suppose that the complex polynomial
$a_0 + a_1x + \ldots + a_nx^n$ with $n < \frac{p}{p-1}\varphi(m)$ and $a_n \neq 0$ is divisible by the cyclotomic polynomial $\phi_m(x)$. Prove that there are at least $p$ nonzero coefficients $a_i\ .$
The cyclotomic polynomial $\phi_m(x)$ is the monic polynomial whose roots are the $m$-th primitive complex
roots of unity. Euler’s totient function $\varphi(m)$ denotes the number of positive integers less than or equal to $m$
which are coprime to $m$.
2019 Serbia National MO, 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 $\left[ 0, 2-\frac{1}{2^n} \right ]$.
1997 Tournament Of Towns, (545) 6
Prove that if $F(x)$ and $G(x)$ are polynomials with coefficients $0$ and $1$ such that $$F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}$$
holds for some $n > 1$, then one of them can be represented in the form
$$ (1 +x + x^2 +...+ x^{k-1}) T(x)$$
for some $k > 1$ where $T(x)$ is a polynomial with coefficients $0$ and $1$.
(V Senderov, M Vialiy)
2017 Australian MO, 1
Determine all polynomial $P(x)\in \mathbb{R}[x]$ satisfying the following two conditions:
(a) $P(2017)=2016$ and
(b) $(P(x)+1)^2=P(x^2+1)$ for all real number $x$.
2011 Postal Coaching, 3
Let $P (x)$ be a polynomial with integer coefficients. Given that for some integer $a$ and some positive integer $n$, where
\[\underbrace{P(P(\ldots P}_{\text{n times}}(a)\ldots)) = a,\]
is it true that $P (P (a)) = a$?
2011 Purple Comet Problems, 9
There are integers $m$ and $n$ so that $9 +\sqrt{11}$ is a root of the polynomial $x^2 + mx + n.$ Find $m + n.$
2017 China Team Selection Test, 5
Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $ are not rational number. there are integers $ a, b,c$ such that $u=av^2+bv+c$. Prove that $b^2 -2b -4ac - 7$ is a square number .
2004 Vietnam National Olympiad, 2
Let $x$, $y$, $z$ be positive reals satisfying $\left(x+y+z\right)^{3}=32xyz$
Find the minimum and the maximum of $P=\frac{x^{4}+y^{4}+z^{4}}{\left(x+y+z\right)^{4}}$
2013 NIMO Summer Contest, 4
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \][i]Proposed by Evan Chen[/i]
2016 India Regional Mathematical Olympiad, 8
At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.
2005 Gheorghe Vranceanu, 2
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
1984 Vietnam National Olympiad, 1
$(a)$ Find a polynomial with integer coefficients of the smallest degree having $\sqrt{2} + \sqrt[3]{3}$ as a root.
$(b)$ Solve $1 +\sqrt{1 + x^2}(\sqrt{(1 + x)^3}-\sqrt{(1- x)^3}) = 2\sqrt{1 - x^2}$.