Found problems: 3597
2004 BAMO, 5
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f (0) = 2004$.
2. If $x$ is irrational, then $f (x)$ is also irrational.
(Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)
2003 South africa National Olympiad, 5
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.
1966 AMC 12/AHSME, 37
Three men, Alpha, Beta, and Gamma, working together, do a job in $6$ hours less time than Alpha alone, in $1$ hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together to do the job. Then $h$ equals:
$\text{(A)}\ \dfrac{5}{2}\qquad
\text{(B)}\ \frac{3}{2}\qquad
\text{(C)}\ \dfrac{4}{3}\qquad
\text{(D)}\ \dfrac{5}{4}\qquad
\text{(E)}\ \dfrac{3}{4}$
1992 Vietnam National Olympiad, 1
Let $ 9 < n_{1} < n_{2} < \ldots < n_{s} < 1992$ be positive integers and \[ P(x) \equal{} 1 \plus{} x^{2} \plus{} x^{9} \plus{} x^{n_{1}} \plus{} \cdots \plus{} x^{n_{s}} \plus{} x^{1992}.\] Prove that if $ x_{0}$ is real root of $ P(x)$ then $ x_{0}\leq\frac {1 \minus{} \sqrt {5}}{2}$.
2013 AIME Problems, 10
There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.
2016 Balkan MO Shortlist, N4
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
[i]Note: A monic polynomial has a leading coefficient equal to 1.[/i]
[i](Greece - Panagiotis Lolas and Silouanos Brazitikos)[/i]
1994 Irish Math Olympiad, 2
Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$.
PEN Q Problems, 10
Suppose that the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are distinct. Show that \[(x-a_{1})(x-a_{2}) \cdots (x-a_{n})-1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.
2014 India National Olympiad, 4
Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.
1980 Putnam, A1
Let $b$ and $c$ be fixed real numbers and let the ten points $(j,y_j )$ for $j=1,2,\ldots,10$ lie on the parabola $y =x^2 +bx+c.$ For $j=1,2,\ldots, 9$ let $I_j$ be the intersection of the tangents to the given parabola at $(j, y_j )$ and $(j+1, y_{j+1}).$ Determine the poynomial function $y=g(x)$ of least degree whose graph passes through all nine points $I_j .$
2005 District Olympiad, 4
Let $f:\mathbb{Q}\rightarrow \mathbb{Q}$ a monotonic bijective function.
a)Prove that there exist a unique continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ such that $F(x)=f(x),\ (\forall)x\in \mathbb{Q}$.
b)Give an example of a non-injective polynomial function $G:\mathbb{R}\rightarrow \mathbb{R}$ such that $G(\mathbb{Q})\subset \mathbb{Q}$ and it's restriction defined on $\mathbb{Q}$ is injective.
2010 Swedish Mathematical Competition, 3
Find all natural numbers $n \ge 1$ such that there is a polynomial $p(x)$ with integer coefficients for which $p (1) = p (2) = 0$ and where $p (n)$ is a prime number .
2006 IMC, 6
The scores of this problem were:
one time 17/20 (by the runner-up)
one time 4/20 (by Andrei Negut)
one time 1/20 (by the winner)
the rest had zero... just to give an idea of the difficulty.
Let $A_{i},B_{i},S_{i}$ ($i=1,2,3$) be invertible real $2\times 2$ matrices such that [list][*]not all $A_{i}$ have a common real eigenvector, [*]$A_{i}=S_{i}^{-1}B_{i}S_{i}$ for $i=1,2,3$, [*]$A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I$.[/list] Prove that there is an invertible $2\times 2$ matrix $S$ such that $A_{i}=S^{-1}B_{i}S$ for all $i=1,2,3$.
2003 Moldova National Olympiad, 12.5
Consider the polynomial $P(x)=X^{2n}-X^{2n-1}+\dots-x+1$, where
$n\in{N^*}$. Find the remainder of the division of polynomial
$P(x^{2n+1})$ by $P(x)$.
1979 Romania Team Selection Tests, 1.
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
[i]Dumitru Bușneag[/i]
2007 All-Russian Olympiad, 5
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
2009 All-Russian Olympiad, 1
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
2022 Indonesia TST, N
Let $n$ be a natural number, with the prime factorisation
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
\[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.
1989 IMO Longlists, 7
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.
2016 Mathematical Talent Reward Programme, SAQ: P 1
Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .
1988 Czech And Slovak Olympiad IIIA, 2
If for the coefficients of equation $x^3+ax^2+bx+c=0$ whose roots are all real, holds, $a^2= 2(b+1)$ then $|a-c|\le 2$. Prove it.
2014 NIMO Problems, 8
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$.
[i]Proposed by Evan Chen[/i]
1982 IMO Longlists, 16
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
2010 Polish MO Finals, 2
Positive rational number $a$ and $b$ satisfy the equality
\[a^3 + 4a^2b = 4a^2 + b^4.\]
Prove that the number $\sqrt{a}-1$ is a square of a rational number.
2017 Greece National Olympiad, 4
Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial
$$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$ where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$.
1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$.
2) Find the minimum possible value of $a_0+a_1+...+a_n$.