Found problems: 3597
2011 QEDMO 9th, 2
Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.
2007 Bosnia Herzegovina Team Selection Test, 4
Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?
2010 IberoAmerican Olympiad For University Students, 7
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
1984 IMO Longlists, 40
Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.
2012 ELMO Problems, 3
Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant.
[i]Victor Wang.[/i]
1972 Putnam, B4
Show that for $n > 1$ we can find a polynomial $P(a, b, c)$ with integer coefficients such that
$$P(x^{n},x^{n+1},x+x^{n+2})=x.$$
2014 239 Open Mathematical Olympiad, 2
The fourth-degree polynomial $P(x)$ is such that the equation $P(x)=x$ has $4$ roots, and any equation of the form $P(x)=c$ has no more two roots. Prove that the equation $P(x)=-x$ too has no more than two roots.
2019 IMEO, 4
Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well.
[i]Proposed by Valentio Iverson (Indonesia)[/i]
2017 AMC 12/AHSME, 21
A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have?
$\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$
2003 AMC 10, 5
Let $ d$ and $ e$ denote the solutions of $ 2x^2\plus{}3x\minus{}5\equal{}0$. What is the value of $ (d\minus{}1)(e\minus{}1)$?
$ \textbf{(A)}\ \minus{}\frac{5}{2} \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6$
2020 Iran Team Selection Test, 1
We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.
[i]Proposed by Masud Shafaie[/i]
2014 Contests, 1
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
1998 Baltic Way, 8
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that
\[ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) \]
for every real number $x$ and every positive integer $n$.
2010 Contests, 1
Show that a sequence $(a_n)$ of $+1$ and $-1$ is periodic with period a power of $2$ if and only if $a_n=(-1)^{P(n)}$, where $P$ is an integer-valued polynomial with rational coefficients.
1993 Irish Math Olympiad, 4
Let $ f(x)\equal{}x^n\plus{}a_{n\minus{}1} x^{n\minus{}1}\plus{}...\plus{}a_0$ $ (n \ge 1)$ be a polynomial with real coefficients such that $ |f(0)|\equal{}f(1)$ and each root $ \alpha$ of $ f$ is real and lies in the interval $ [0,1]$. Prove that the product of the roots does not exceed $ \frac{1}{2^n}$.
ICMC 7, 3
Let $N{}$ be a fixed positive integer, $S{}$ be the set $\{1, 2,\ldots , N\}$ and $\mathcal{F}$ be the set of functions $f:S\to S$ such that $f(i)\geqslant i$ for all $i\in S.$ For each $f\in\mathcal{F}$ let $P_f$ be the unique polynomial of degree less than $N{}$ satisfying $P_f(i) = f(i)$ for all $i\in S.$ If $f{}$ is chosen uniformly at random from $\mathcal{F}$ determine the expected value of $P_f'(0)$ where\[P_f'(0)=\frac{\mathrm{d}P_f(x)}{\mathrm{d}x}\bigg\vert_{x=0}.\][i]Proposed by Ishan Nath[/i]
1974 Putnam, A6
Given $n$, let $k = k(n)$ be the minimal degree of any monic integral polynomial
$$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$
such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$. In particular, find the value of $k(n)$ corresponding to $n = 10^6.$
2014 France Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1996 AIME Problems, 7
Two of the squares of a $ 7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane of the board. How many inequivalent color schemes are possible?
2005 Croatia National Olympiad, 2
Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.
2007 Iran MO (3rd Round), 5
Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system:
\[ \left\{\begin{array}{c}f(x,y)\equal{}0\\ g(x,y)\equal{}0\end{array}\right.\]
has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.
2018 BMT Spring, 3
If $f$ is a polynomial, and $f(-2)=3$, $f(-1)=-3=f(1)$, $f(2)=6$, and $f(3)=5$, then what is the minimum possible degree of $f$?
1968 IMO Shortlist, 6
If $a_i \ (i = 1, 2, \ldots, n)$ are distinct non-zero real numbers, prove that the equation
\[\frac{a_1}{a_1-x} + \frac{a_2}{a_2-x}+\cdots+\frac{a_n}{a_n-x} = n\]
has at least $n - 1$ real roots.
2014 Belarus Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1984 IMO Shortlist, 2
Prove:
(a) There are infinitely many triples of positive integers $m, n, p$ such that $4mn - m- n = p^2 - 1.$
(b) There are no positive integers $m, n, p$ such that $4mn - m- n = p^2.$