Found problems: 3597
1992 Putnam, B4
Let $p(x)$ be a nonzero polynomial of degree less than $1992$ having no nonconstant factor in common with
$x^3 -x$. Let
$$ \frac{d^{1992}}{dx^{1992}} \left( \frac{p(x)}{x^3 -x } \right) =\frac{f(x)}{g(x)}$$
for polynomials $f(x)$ and $g(x).$ Find the smallest possible degree of $f(x)$.
2012 India Regional Mathematical Olympiad, 2
Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$
2013 South africa National Olympiad, 4
Determine all pairs of polynomials $f$ and $g$ with real coefficients such that \[ x^2 \cdot g(x) = f(g(x)). \]
2008 CentroAmerican, 5
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$.
2014 Israel National Olympiad, 7
Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.
2001 Moldova Team Selection Test, 7
Let $(P_n(X))_{n\in\mathbb{N}}$ be a sequence of polynomials defined as: $P_1(X)=X-1, P_2(X)=X^2-X-1, P_n(X)=XP_{n-1}(X)-P_{n-2}(X), \forall n>2$. For every nonnegative integer $n{}$ find all roots of the polynomial $P_n(X)$.
2008 Putnam, A5
Let $ n\ge 3$ be an integer. Let $ f(x)$ and $ g(x)$ be polynomials with real coefficients such that the points $ (f(1),g(1)),(f(2),g(2)),\dots,(f(n),g(n))$ in $ \mathbb{R}^2$ are the vertices of a regular $ n$-gon in counterclockwise order. Prove that at least one of $ f(x)$ and $ g(x)$ has degree greater than or equal to $ n\minus{}1.$
2017 NMTC Junior, 5
(a) Prove that $x^4+3x^3+6x^2+9x+12$ cannot be expressed as product of two polynomials of degree 2 with integers coefficients.
(b) $2n+1$ segments are marked on a line. Each of these segments intersects at least $n$ other segments. Prove that one of these segments intersects all other segments.
1990 Romania Team Selection Test, 3
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
2004 Purple Comet Problems, 3
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]
1993 Greece National Olympiad, 5
Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?
2007 Ukraine Team Selection Test, 8
$ F(x)$ is polynomial with real coefficients. $ F(x) \equal{} x^{4}\plus{}a_{1}x^{3}\plus{}a_{2}x^{2}\plus{}a_{1}x^{1}\plus{}a_{0}$. $ M$ is local maximum and $ m$ is minimum. Prove that $ \frac{3}{10}(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}< M\minus{}m < 3(\frac{a_{1}^{2}}{4}\minus{}\frac{2a_{2}}{3^{2}})^{2}$
2014 Vietnam National Olympiad, 2
Given the polynomial $P(x)=(x^2-7x+6)^{2n}+13$ where $n$ is a positive integer. Prove that $P(x)$ can't be written as a product of $n+1$ non-constant polynomials with integer coefficients.
1994 Greece National Olympiad, 2
Fow which real values of $m$ does the polynomial $x^3+1995x^2-1994x+m$ have all three roots integers?
2001 National Olympiad First Round, 4
How many real solution does the equation $\dfrac{x^{2000}}{2001} + 2\sqrt 3 x^2 - 2\sqrt 5 x + \sqrt 3$ have?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$
1987 Brazil National Olympiad, 1
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
1996 Poland - Second Round, 1
Can every polynomial with integer coefficients be expressed as a sum of cubes
of polynomials with integer coefficients?
[hide]I found the following statement that can be linked to this problem: "It is easy to see that every polynomial in F[x] is sum of cubes if char (F)$\ne$3 and card (F)=2,4"[/hide]
2017 Thailand TSTST, 6
Find all polynomials $f$ with real coefficients such that for all reals $x, y, z$ such that $x+y+z =0$, the following relation holds: $$f(xy) + f(yz) + f(zx) = f(xy + yz + zx).$$
2016 IMO, 4
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
2022 Tuymaada Olympiad, 2
Given are integers $a, b, c$ and an odd prime $p.$ Prove that $p$ divides $x^2 + y^2 + ax + by + c$ for some integers $x$ and $y.$
[i](A. Golovanov )[/i]
2018 Balkan MO Shortlist, A5
Let $f: \mathbb {R} \to \mathbb {R}$ be a concave function and $g: \mathbb {R} \to \mathbb {R}$ be a continuous function . If $$ f (x + y) + f (x-y) -2f (x) = g (x) y^2 $$for all $x, y \in \mathbb {R}, $ prove that $f $ is a second degree polynomial.
2017 Romania National Olympiad, 2
Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that
[b]a)[/b] if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
[b]b)[/b] there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that
$$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$
1999 AMC 12/AHSME, 17
Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$?
$ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad
\textbf{(B)}\ x \plus{} 80 \qquad
\textbf{(C)}\ \minus{}x \plus{} 118 \qquad
\textbf{(D)}\ x \plus{} 118 \qquad
\textbf{(E)}\ 0$
2023 Puerto Rico Team Selection Test, 3
Let $p(x)$ be a polynomial of degree $2022$ such that:
$$p(k) =\frac{1}{k+1}\,\,\, \text{for }\,\,\, k = 0, 1, . . . , 2022$$
Find $p(2023)$.
2015 Balkan MO Shortlist, A6
For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid
$$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$
and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$?
(Serbia)