This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3597

2006 Korea National Olympiad, 1
Tags: polynomial , algebra
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$

2006 Tournament of Towns, 3
Tags: polynomial , algebra
Consider a polynomial $P(x) = x^4+x^3-3x^2+x+2$. Prove that at least one of the coefficients of $(P(x))^k$, ($k$ is any positive integer) is negative. (5)

2006 Romania Team Selection Test, 2
Tags: quadratic , real analysis , complex number , algebra , polynomial , sum of roots , number theory
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]

2018 AMC 12/AHSME, 5
Tags: algebra , polynomial
What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? $ \textbf{(A) }3 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5 \qquad \textbf{(D) }6 \qquad \textbf{(E) }10 \qquad $

1957 Moscow Mathematical Olympiad, 347
Tags: cubic , algebra , polynomial , divisible
a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$. b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.

2021 Alibaba Global Math Competition, 19
Tags: nested radical , algebra , polynomial , sum of roots , roots of unity
Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.

2011 Harvard-MIT Mathematics Tournament, 6
Tags: hmmt , algebra , polynomial
How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?

1995 China Team Selection Test, 2
Tags: polynomial , algebra
$ A$ and $ B$ play the following game with a polynomial of degree at least 4: \[ x^{2n} \plus{} \_x^{2n \minus{} 1} \plus{} \_x^{2n \minus{} 2} \plus{} \ldots \plus{} \_x \plus{} 1 \equal{} 0 \] $ A$ and $ B$ take turns to fill in one of the blanks with a real number until all the blanks are filled up. If the resulting polynomial has no real roots, $ A$ wins. Otherwise, $ B$ wins. If $ A$ begins, which player has a winning strategy?

2015 ISI Entrance Examination, 4
Tags: polynomial , algebra
Let $p(x) = x^7 +x^6 + b_5 x^5 + \cdots +b_0 $ and $q(x) = x^5 + c_4 x^4 + \cdots +c_0$ . If $p(i)=q(i)$ for $i=1,2,3,\cdots,6$ . Show that there exists a negative integer r such that $p(r)=q(r)$ .

2024 India Iran Friendly Math Competition, 3
Tags: algebra , polynomial
Let $n \ge 3$ be an integer. Let $\mathcal{P}$ denote the set of vertices of a regular $n$-gon on the plane. A polynomial $f(x, y)$ of two variables with real coefficients is called $\textit{regular}$ if $$\mathcal{P} = \{(u, v) \in \mathbb{R}^2 \, | \, f(u, v) = 0 \}.$$ Find the smallest possible value of the degree of a regular polynomial. [i]Proposed by Navid Safaei[/i]

2020 USA IMO Team Selection Test, 5
Tags: polynomial , integer polynomial , inequalities , number theory
Find all integers $n \ge 2$ for which there exists an integer $m$ and a polynomial $P(x)$ with integer coefficients satisfying the following three conditions: [list] [*]$m > 1$ and $\gcd(m,n) = 1$; [*]the numbers $P(0)$, $P^2(0)$, $\ldots$, $P^{m-1}(0)$ are not divisible by $n$; and [*]$P^m(0)$ is divisible by $n$. [/list] Here $P^k$ means $P$ applied $k$ times, so $P^1(0) = P(0)$, $P^2(0) = P(P(0))$, etc. [i]Carl Schildkraut[/i]

2010 Postal Coaching, 1
Tags: polynomial , algebra
A polynomial $P (x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 <x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} \] Prove that the maximum value of $|P (x)|$ on the interval $[x_1 , x_n ]$ is attained in the interval $[x_{n-1} , x_n ]$.

2001 All-Russian Olympiad Regional Round, 10.5
Tags: algebra , polynomial , number theory , trinomial
Given integers $a$, $ b$ and $c$, $c\ne b$. It is known that the square trinomials $ax^2 + bx + c$ and $(c-b)x^2 + (c- a)x + (a + b)$ have a common root (not necessarily integer). Prove that $a+b+2c$ is divisible by $3$.

2001 AMC 12/AHSME, 19
Tags: algebra , polynomial , product of roots
The polynomial $ P(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $ y$-intercept of the graph of $ y \equal{} P(x)$ is 2, what is $ b$? $ \textbf{(A)} \ \minus{} 11 \qquad \textbf{(B)} \ \minus{} 10 \qquad \textbf{(C)} \ \minus{} 9 \qquad \textbf{(D)} \ 1 \qquad \textbf{(E)} \ 5$

1988 Swedish Mathematical Competition, 4
Tags: polynomial , algebra , analysis
A polynomial $P(x)$ of degree $3$ has three distinct real roots. Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.

1970 Swedish Mathematical Competition, 3
Tags: algebra , polynomial , integer polynomial
A polynomial with integer coefficients takes the value $5$ at five distinct integers. Show that it does not take the value $9$ at any integer.

2021 All-Russian Olympiad, 6
Tags: algebra , polynomial
Given is a polynomial $P(x)$ of degree $n>1$ with real coefficients. The equation $P(P(P(x)))=P(x)$ has $n^3$ distinct real roots. Prove that these roots could be split into two groups with equal arithmetic mean.

2012 Bosnia Herzegovina Team Selection Test, 4
Tags: function , induction , algebra , polynomial , modular arithmetic , number theory
Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

2017 European Mathematical Cup, 4
Tags: algebra , polynomial
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. [i]Remark:[/i] For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$. Proposed by Adrian Beker.

2004 AMC 12/AHSME, 23
Tags: algebra , polynomial , vieta
The polynomial $ x^3\minus{}2004x^2\plus{}mx\plus{}n$ has integer coefficients and three distinct positive zeros. Exactly one of these is an integer, and it is the sum of the other two. How many values of $ n$ are possible? $ \textbf{(A)}\ 250,\!000 \qquad \textbf{(B)}\ 250,\!250 \qquad \textbf{(C)}\ 250,\!500 \qquad \textbf{(D)}\ 250,\!750 \qquad \textbf{(E)}\ 251,\!000$

1987 Vietnam National Olympiad, 2
Tags: algorithm , polynomial , modular arithmetic , algebra
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.

2000 IMO Shortlist, 7
Tags: coefficient , algebra , polynomial , modular arithmetic
For a polynomial $ P$ of degree 2000 with distinct real coefficients let $ M(P)$ be the set of all polynomials that can be produced from $ P$ by permutation of its coefficients. A polynomial $ P$ will be called [b]$ n$-independent[/b] if $ P(n) \equal{} 0$ and we can get from any $ Q \in M(P)$ a polynomial $ Q_1$ such that $ Q_1(n) \equal{} 0$ by interchanging at most one pair of coefficients of $ Q.$ Find all integers $ n$ for which $ n$-independent polynomials exist.

1970 Putnam, B2
Tags: algebra , polynomial
The time-varying temperature of a certain body is given by a polynomial in the time of degree at most three. Show that the average temperature of the body between $9$ am and $3$ pm can always be found by taking the average of the temperatures at two fixed times, which are independent of the polynomial. Also, show that these two times are $10\colon \! 16$ am and $1\colon \!44$ pm to the nearest minute.

PEN Q Problems, 3
Tags: polynomial
Let $n \ge 2$ be an integer. Prove that if $k^2 + k + n$ is prime for all integers $k$ such that $0 \leq k \leq \sqrt{\frac{n}{3}}$, then $k^2 +k + n$ is prime for all integers $k$ such that $0 \leq k \leq n - 2$.

2008 Greece Team Selection Test, 1
Tags: algebra , polynomial , divisibility
Find all possible values of $a\in \mathbb{R}$ and $n\in \mathbb{N^*}$ such that $f(x)=(x-1)^n+(x-2)^{2n+1}+(1-x^2)^{2n+1}+a$ is divisible by $\phi (x)=x^2-x+1$