Olympiad problems

1951 Miklós Schweitzer, 10

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

VMEO II 2005, 7

Tags: algebra , functional equation , function , polynomial

Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$

2017 IOM, 3

Tags: polynomial , algebra

Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.

2009 Harvard-MIT Mathematics Tournament, 8

Tags: vieta , sum of roots , algebra , polynomial

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

1990 ITAMO, 3

Tags: algebra , remainder , polynomial

Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.

1997 Pre-Preparation Course Examination, 1

Tags: greatest common divisor , algebra , polynomial

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

2010 India Regional Mathematical Olympiad, 2

Tags: quadratic , polynomial , algebra

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

2014 Chile TST IMO, 4

Tags: modulo , algebra , polynomial

Let $ f(n) $ be a polynomial with integer coefficients. Prove that if $ f(-1) $, $ f(0) $, and $ f(1) $ are not divisible by 3, then $ f(n) \neq 0 $ for all integers $ n $.

1994 Swedish Mathematical Competition, 5

Tags: algebra , polynomial

The polynomial $x^k + a_1x^{k-1} + a_2x^{k-2} +... + a_k$ has $k$ distinct real roots. Show that $a_1^2 > \frac{2ka_2}{k-1}$.

2010 Indonesia TST, 1

Tags: relatively prime , number theory , algebra , polynomial

find all pairs of relatively prime natural numbers $ (m,n) $ in such a way that there exists non constant polynomial f satisfying \[ gcd(a+b+1, mf(a)+nf(b) > 1 \] for every natural numbers $ a $ and $ b $

2014 ELMO Shortlist, 11

Tags: modular arithmetic , algebra , number theory , polynomial

Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define \[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \] Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$. [i]Proposed by Victor Wang[/i]

2007 Putnam, 4

Tags: algebra , polynomial

Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that \[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\] and $ \text{deg}P<\text{deg}{Q}.$

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)

2019 Iran RMM TST, 2

Tags: chebyshev polynomial , algebra , polynomial

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$.\\ Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $.\\ Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. [i]Proposed by Morteza Saghafian[/i]

2006 Thailand Mathematical Olympiad, 3

Tags: divide , algebra , polynomial

Let $P(x), Q(x)$ and $R(x)$ be polynomials satisfying the equation $2xP(x^3) + Q(-x -x^3) = (1 + x + x^2)R(x)$. Show that $x - 1$ divides $P(x) - Q(x)$.

1960 Putnam, A5

Tags: commutative , algebra , polynomial

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

1969 Putnam, A1

Tags: algebra , polynomial

Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$

2019 CMIMC, 15

Tags: polynomial , algebra , team

Call a polynomial $P$ [i]prime-covering[/i] if for every prime $p$, there exists an integer $n$ for which $p$ divides $P(n)$. Determine the number of ordered triples of integers $(a,b,c)$, with $1\leq a < b < c \leq 25$, for which $P(x)=(x^2-a)(x^2-b)(x^2-c)$ is prime-covering.

1989 IMO Longlists, 8

Tags: polynomial , equation , root , diophantine equation , algebra

Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2022 Bulgarian Autumn Math Competition, Problem 12.1

Tags: polynomial , algebra

Find $A=x^5+y^5+z^5$ if $x+y+z=1$, $x^2+y^2+z^2=2$ and $x^3+y^3+z^3=3$.

1995 Poland - First Round, 9

Tags: polynomial , algebra

A polynomial with integer coefficients when divided by $x^2-12x+11$ gives the remainder $990x-889$. Prove that the polynomial has no integer roots.

PEN H Problems, 16

Tags: diophantine equation , algebra , function , polynomial

Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$, where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$.

2020-IMOC, A6

Tags: polynomial , algebra

$\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.}$ Let $ P (x)$ be a polynomial with real coefficients such that $\deg P \ge 3$ is an odd integer. Let $f : \mathbb{R}\rightarrow\mathbb{Z}$ be a function such that $$\definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)).$$ $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)}$ Prove that the range of $f$ is finite. $\definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)}$ Show that for any positive integer $n$, there exist $P$, $f$ that satisfies the above condition and also that the range of $f$ has cardinality $n$. [i]Proposed by [/i][b][color=#419DAB]ltf0501[/color][/b]. [color=#3D9186]#1735[/color]

2011 HMNT, 10

Tags: polynomial , hmmt , algebra

Let $r_1, r_2, \cdots, r_7$ be the distinct complex roots of the polynomial $P(x) = x^7 - 7$ Let \[K = \prod_{1 \leq i < j \leq 7} (r_i + r_j)\] that is, the product of all the numbers of the form $r_i + r_j$, where $i$ and $j$ are integers for which $1 \leq i < j \leq 7$. Determine the value of $K^2$.

2006 Turkey MO (2nd round), 3

Tags: polynomial , algebra , number theory

Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where \[P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.\]