Olympiad problems

2016 CMIMC, 8

Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$.

2024 Thailand TSTST, 12

Tags: algebra , polynomial

We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

2023 Indonesia TST, N

Tags: number theory , polynomial

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

1977 Poland - Second Round, 5

Tags: algebra , polynomial

Let the polynomials $ w_n $ be given by the formulas: $$ w_1(x) = x^2 - 1, \quad w_{n+1}(x) = w_n(x)^2 - 1, \quad (n = 1, 2, \ldots)$$ and let $a$ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have?

2016 Switzerland Team Selection Test, Problem 2

Tags: algebra , polynomial , function

Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$

2010 Contests, 2

Tags: polynomial , quadratic , algebra

Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

2010 Kyrgyzstan National Olympiad, 5

Tags: polynomial , function , algebra

Let $k$ be a constant number larger than $1$. Find all polynomials $P(x)$ such that $P({x^k}) = {\left( {P(x)} \right)^k}$ for all real $x$.

2007 Hungary-Israel Binational, 3

Tags: polynomial , algebra

Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.

2014 Contests, 901

Tags: calculus , integration , algebra , polynomial , calculus computations

Given the polynomials $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$, find the real numbers $p,\ q,\ r,\ s,\ t$ such that $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$ and $\int_0^1 P(x)dx=-\frac{52}{5}.$

2021 Belarusian National Olympiad, 11.3

Tags: algebra , polynomial

A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root. Find all possible values of $P(0)$

2017 Saudi Arabia BMO TST, 2

Tags: algebra , integer polynomial , polynomial

Polynomial P(x) with integer coefficient is called [i]cube-presented[/i] if it can be represented as sum of several cube of polynomials with integer coefficients. Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$. a) Is $3x^2$ a cube-represented polynomial? b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?

2010 Brazil National Olympiad, 2

Tags: polynomial , ceiling function , number theory , prime number , algebra

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

1966 Miklós Schweitzer, 4

Tags: algebra , polynomial , abstract algebra , vector , calculus , integration , ring theory

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

1989 USAMO, 3

Tags: algebra , polynomial , inequalities

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

2011 NIMO Problems, 9

Tags: algebra , polynomial , vieta

The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$. [i]Proposed by Eugene Chen [/i]

2016 Mathematical Talent Reward Programme, SAQ: P 1

Tags: polynomial , algebra

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 $x$ .

1995 All-Russian Olympiad, 3

Tags: quadratic , polynomial , symmetry , function , complex number , algebra

Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$? [i]S. Tokarev[/i]

1994 Tournament Of Towns, (415) 3

Tags: algebra , polynomial

At least one of the coefficients of a polynomial $P(x)$ is negative. Can all of the coefficients of all of its powers $(P(x))^n$, $n > 1$, be positive? (0 Kryzhanovskij)

2006 Italy TST, 3

Tags: polynomial , complex number , algebra

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

2006 Italy TST, 3

Tags: polynomial , complex number , algebra

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

2016 IFYM, Sozopol, 2

Tags: polynomial , prime number , number theory , algebra

We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.

2021 Canadian Mathematical Olympiad Qualification, 1

Tags: polynomial , functional equation , algebra

Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.

2010 Contests, 4

Tags: polynomial , algebra

Find all polynomials $P(x)$ with real coefficients such that \[(x-2010)P(x+67)=xP(x) \] for every integer $x$.

2024 Israel TST, P3

Tags: algebra , polynomial , number theory , modular arithmetic , prime number

Let $n$ be a positive integer and $p$ be a prime number of the form $8k+5$. A polynomial $Q$ of degree at most $2023$ and nonnegative integer coefficients less than or equal to $n$ will be called "cool" if \[p\mid Q(2)\cdot Q(3) \cdot \ldots \cdot Q(p-2)-1.\] Prove that the number of cool polynomials is even.

2025 India STEMS Category A, 2

Tags: algebra , polynomial

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]