Found problems: 3597
2016 CCA Math Bonanza, I9
Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$).
[i]2016 CCA Math Bonanza Individual #9[/i]
2016 Kyrgyzstan National Olympiad, 5
Given two monic polynomials $P(x)$ and $Q(x)$ with degrees 2016.
$P(x)=Q(x)$ has no real root. [b]Prove that P(x)=Q(x+1) has at least one real root.[/b]
1963 Polish MO Finals, 5
Prove that a fifth-degree polynomial $$ P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6$$ is not the product of two lower-degree polynomials with integer coefficients.
1987 IMO, 3
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.
1971 Canada National Olympiad, 4
Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.
2025 India STEMS Category B, 1
Let $\mathcal{P}$ be the set of all polynomials with coefficients in $\{0, 1\}$. Suppose $a, b$ are non-zero integers such that for every $f \in \mathcal{P}$ with $f(a)\neq 0$, we have $f(a) \mid f(b)$. Prove that $a=b$.
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]
2010 Iran MO (3rd Round), 1
[b]two variable ploynomial[/b]
$P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$.
(for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.)
suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that
$Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$
prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded.
(we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.)
time allowed for this question was 1 hour.
2006 Putnam, A2
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
2023 Thailand TST, 3
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
2014 Saudi Arabia Pre-TST, 2.2
Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.
2013 Iran MO (3rd Round), 4
Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation:
\[x^2 - (n^2 +1)y^2 = n^2.\]
(25 points)
2023 Bulgaria National Olympiad, 3
Let $f(x)$ be a polynomial with positive integer coefficients. For every $n\in\mathbb{N}$, let $a_{1}^{(n)}, a_{2}^{(n)}, \dots , a_{n}^{(n)}$ be fixed positive integers that give pairwise different residues modulo $n$ and let
\[g(n) = \sum\limits_{i=1}^{n} f(a_{i}^{(n)}) = f(a_{1}^{(n)}) + f(a_{2}^{(n)}) + \dots + f(a_{n}^{(n)})\]
Prove that there exists a constant $M$ such that for all integers $m>M$ we have $\gcd(m, g(m))>2023^{2023}$.
1997 Pre-Preparation Course Examination, 1
Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that
\[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]
1965 AMC 12/AHSME, 27
When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is:
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$
2025 Canada Junior National Olympiad, 5
A polynomial $c_dx^d+c_{d-1}x^{d-1}+\dots+c_1x+c_0$ with degree $d$ is [i]reflexive[/i] if there is an integer $n\ge d$ such that $c_i=c_{n-i}$ for every $0\le i\le n$, where $c_i=0$ for $i>d$. Let $\ell\ge 2$ be an integer and $p(x)$ be a polynomial with integer coefficients. Prove that there exist reflexive polynomials $q(x)$, $r(x)$ with integer coefficients such that
\[(1+x+x^2+\dots+x^{\ell-1})p(x)=q(x)+x^\ell r(x)\]
2017 IMC, 5
Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$, and let
$$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$
be a polynomial with complex coefficients such that
$$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$
Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.
2010 Romanian Masters In Mathematics, 6
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]
2018 Mathematical Talent Reward Programme, SAQ: P 5
[list=1]
[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number.
[*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime.
[/list]
2003 Singapore Senior Math Olympiad, 2
For each positive integer $k$, we define the polynomial $S_k(x)=1+x+x^2+x^3+...+x^{k-1}$
Show that $n \choose 1$ $S_1(x) +$ $n \choose 2$ $S_2(x) +$ $n \choose 3$ $S_3(x)+...+$ $n \choose n$ $S_n(x) = 2^{n-1}S_n\left(\frac{1+x}{2}\right)$
for every positive integer $n$ and every real number $x$.
2003 India IMO Training Camp, 6
A zig-zag in the plane consists of two parallel half-lines connected by a line segment. Find $z_n$, the maximum number of regions into which $n$ zig-zags can divide the plane. For example, $z_1=2,z_2=12$(see the diagram). Of these $z_n$ regions how many are bounded? [The zig-zags can be as narrow as you please.] Express your answers as polynomials in $n$ of degree not exceeding $2$.
[asy]
draw((30,0)--(-70,0), Arrow);
draw((30,0)--(-20,-40));
draw((-20,-40)--(80,-40), Arrow);
draw((0,-60)--(-40,20), dashed, Arrow);
draw((0,-60)--(0,15), dashed);
draw((0,15)--(40,-65),dashed, Arrow);
[/asy]
2016 AIME Problems, 11
Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$, and $\left(P(2)\right)^2 = P(3)$. Then $P(\tfrac72)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2013 Baltic Way, 2
Let $k$ and $n$ be positive integers and let $x_1, x_2, \cdots, x_k, y_1, y_2, \cdots, y_n$ be distinct integers. A polynomial $P$ with integer coefficients satisfies
\[P(x_1)=P(x_2)= \cdots = P(x_k)=54\]
\[P(y_1)=P(y_2)= \cdots = P(y_n)=2013.\]
Determine the maximal value of $kn$.
2017 China Team Selection Test, 2
Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.
2001 Tuymaada Olympiad, 3
Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$
[i]Proposed by A. Golovanov[/i]
2000 Vietnam National Olympiad, 3
Consider the polynomial $ P(x) \equal{} x^3 \plus{} 153x^2 \minus{} 111x \plus{} 38$.
(a) Prove that there are at least nine integers $ a$ in the interval $ [1, 3^{2000}]$ for which $ P(a)$ is divisible by $ 3^{2000}$.
(b) Find the number of integers $ a$ in $ [1, 3^{2000}]$ with the property from (a).