Olympiad problems

1999 Harvard-MIT Mathematics Tournament, 8

Tags: polynomial , algebra

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1$, $f(2)=-4$, $f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

1984 Bulgaria National Olympiad, Problem 4

Tags: algebra , polynomial

Let $a,b,a_2,\ldots,a_{n-2}$ be real numbers with $ab\ne0$ such that all the roots of the equation $$ax^n-ax^{n-1}+a_2x^{n-2}+\ldots+a_{n-2}x^2-n^2bx+b=0$$are positive and real. Prove that these roots are all equal.

1993 Irish Math Olympiad, 1

Tags: algebra , polynomial

The following is known about the reals $ \alpha$ and $ \beta$ $ \alpha^{3}-3\alpha^{2}+5\alpha-17=0$ and $ \beta^{3}-3\beta^{2}+5\beta+11=0$ Determine $ \alpha+\beta$

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 $q+1$.

2012 China Team Selection Test, 3

Tags: geometric transformation , algebra , geometry , polynomial , inequalities

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

2021 JHMT HS, 5

Tags: polynomial , algebra

A function $f$ with domain $A$ and range $B$ is called [i]injective[/i] if every input in $A$ maps to a unique output in $B$ (equivalently, if $x, y \in A$ and $x \neq y$, then $f(x) \neq f(y)$). With $\mathbb{C}$ denoting the set of complex numbers, let $P$ be an injective polynomial with domain and range $\mathbb{C}$. Suppose further that $P(0) = 2021$ and that when $P$ is written in standard form, all coefficients of $P$ are integers. Compute the smallest possible positive integer value of $P(10)/P(1)$.

2006 Iran MO (3rd Round), 2

Tags: analytic geometry , algebra , polynomial

A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.

2022 Abelkonkurransen Finale, 4b

Tags: algebra , polynomial

Do there exist $2022$ polynomials with real coefficients, each of degree equal to $2021$, so that the $2021 \cdot 2022 + 1$ coefficients in their product are equal?

2013 Iran MO (3rd Round), 5

Tags: polynomial , algebra , complex number , trigonometry

Prove that there is no polynomial $P \in \mathbb C[x]$ such that set $\left \{ P(z) \; | \; \left | z \right | =1 \right \}$ in complex plane forms a polygon. In other words, a complex polynomial can't map the unit circle to a polygon. (30 points)

1994 Tournament Of Towns, (411) 2

Tags: algebra , polynomial

The sequence of positive integers $a_1$, $a_2$,$...$ is such that for each $n = 1$,$2$, $...$ the quadratic equation $$a_{n+2}x^2 + a_{n+1}x+ a_n = 0$$ has a real root. Can the sequence consist of (a) $10 $ terms, (b) an infinite number of terms? (A Shapovalov)

2020 Latvia Baltic Way TST, 4

Tags: polynomial , quadratic , algebra

Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.

KoMaL A Problems 2020/2021, A. 789

Tags: polynomial , algebra , absolute value

Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$. [i]Submitted by Navid Safaei, Tehran, Iran[/i]

1966 Polish MO Finals, 2

Tags: algebra , polynomial

Prove that when $ f, m, n $, are any non-negative integers, then the polynomial $$ P(x) = x^{3k+2} + x^{3m+1} + x^{3n}$$ is divisible by the polynomial $ x^2 + x + 1 $.

2012 All-Russian Olympiad, 1

Tags: algebra , polynomial

Given is the polynomial $P(x)$ and the numbers $a_1,a_2,a_3,b_1,b_2,b_3$ such that $a_1a_2a_3\not=0$. Suppose that for every $x$, we have \[P(a_1x+b_1)+P(a_2x+b_2)=P(a_3x+b_3)\] Prove that the polynomial $P(x)$ has at least one real root.

1989 IMO Longlists, 51

Tags: polynomial , algebra

Let $ f(x) \equal{} \prod^n_{k\equal{}1} (x \minus{} a_k) \minus{} 2,$ where $ n \geq 3$ and $ a_1, a_2, \ldots,$ an are distinct integers. Suppose that $ f(x) \equal{} g(x)h(x),$ where $ g(x), h(x)$ are both nonconstant polynomials with integer coefficients. Prove that $ n \equal{} 3.$

2012 European Mathematical Cup, 3

Tags: system of equations , algebra , polynomial , vieta

Are there positive real numbers $x$, $y$ and $z$ such that $ x^4 + y^4 + z^4 = 13\text{,} $ $ x^3y^3z + y^3z^3x + z^3x^3y = 6\sqrt{3} \text{,} $ $ x^3yz + y^3zx + z^3xy = 5\sqrt{3} \text{?} $ [i]Proposed by Matko Ljulj.[/i]

2007 Stanford Mathematics Tournament, 11

Tags: polynomial , algebra

The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.

2024 Princeton University Math Competition, A1 / B3

Tags: algebra , polynomial

Consider polynomial $f(x)=ax^3+bx^2+cx+d$ where $a, b, c, d$ are nonnegative integers satisfying $ab+bc+cd+ad=20$. Find the sum of all distinct possible values of $f(1)$.

1981 IMO Shortlist, 6

Tags: algebra , polynomial , functional equation

Let $P(z)$ and $Q(z)$ be complex-variable polynomials, with degree not less than $1$. Let \[P_k = \{z \in \mathbb C | P(z) = k \}, Q_k = \{ z \in \mathbb C | Q(z) = k \}.\] Let also $P_0 = Q_0$ and $P_1 = Q_1$. Prove that $P(z) \equiv Q(z).$

2022 Moldova EGMO TST, 4

Tags: polynomial , algebra , modular arithmetic , abstract algebra

Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

2010 Thailand Mathematical Olympiad, 9

Tags: algebra , polynomial

Let $a, b, c$ be real numbers so that all roots of the equation $2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0$ are real. Find the smallest real root of the equation above.

1996 All-Russian Olympiad, 4

Tags: polynomial , algebra

Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$), $a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs. [i]O. Musin[/i]

1987 Romania Team Selection Test, 10

Tags: modular arithmetic , number theory , algebra , function , induction , polynomial

Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$. [i]Laurentiu Panaitopol[/i]

2014 Iran Team Selection Test, 3

Tags: function , algebra , polynomial

let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending. and for all $x\in \mathbb{R}$ $p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ . prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$ $f(q(p(x))+h(x))=f(x)^{2}+1$

2008 Mathcenter Contest, 2

Tags: polynomial , algebra

Find all polynomials $P(x)$ which have the properties: 1) $P(x)$ is not a constant polynomial and is a mononic polynomial. 2) $P(x)$ has all real roots and no duplicate roots. 3) If $P(a)=0$ then $P(a|a|)=0$ [i](nooonui)[/i]