Olympiad problems

1991 Baltic Way, 4

A polynomial $p$ with integer coefficients is such that $p(-n) < p(n) < n$ for some integer $n$. Prove that $p(-n) < -n$.

2024 Saint Petersburg Mathematical Olympiad, 6

Tags: number theory , polynomial , algebra

Polynomial $P(x)$ with integer coefficients is given. For some positive integer $n$ numbers $P(0),P(1),\dots,P(2^n+1)$ are all divisible by $2^{2^n}$. Prove that values of $P(x)$ in all integer points are divisible by $2^{2^n}$.

2017 Hong Kong TST, 3

Tags: algebra , polynomial

Let $f(x)$ be a monic cubic polynomial with $f(0)=-64$, and all roots of $f(x)$ are non-negative real numbers. What is the largest possible value of $f(-1)$? (A polynomial is monic if its leading coefficient is 1.)

1996 IMO Shortlist, 7

Tags: function , algebra , polynomial , functional equation , periodic function

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and \[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\] Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).

2024 Vietnam National Olympiad, 5

Tags: algebra , polynomial

For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots?

2013 Saudi Arabia BMO TST, 5

Tags: polynomial , root , algebra

Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots.

2007 Moldova Team Selection Test, 2

Tags: modular arithmetic , algebra , polynomial , number theory

Consider $p$ a prime number and $p$ consecutive positive integers $m_{1}, m_{2}, \ldots, m_{p}$. Choose a permutation $\sigma$ of $1, 2, \ldots, p$. Show that there exist two different numbers $k,l \in \{1,2, \ldots, p\}$ such that $m_{k}m_{\sigma(k)}-m_{l}m_{\sigma(l)}$ is divisible by $p$.

1954 Moscow Mathematical Olympiad, 267

Tags: divide , polynomial , divisible , algebra

Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

1997 Bulgaria National Olympiad, 1

Tags: polynomial , floor function , algebra

Consider the polynomial $P_n(x) = \binom {n}{2}+\binom {n}{5}x+\binom {n}{8}x^2 + \cdots + \binom {n}{3k+2}x^{3k}$ where $n \ge 2$ is a natural number and $k = \left\lfloor \frac{n-2}{3} \right \rfloor$ [b](a)[/b] Prove that $P_{n+3}(x)=3P_{n+2}(x)-3P_{n+1}(x)+(x+1)P_n(x)$ [b](b)[/b] Find all integer numbers $a$ such that $P_n(a^3)$ is divisible by $3^{ \lfloor \frac{n-1}{2} \rfloor}$ for all $n \ge 2$

2024 pOMA, 5

Tags: algebra , polynomial

Prove that there do not exist positive integers $a,b,c$ such that the polynomial \[ P(x) = x^3 - 2^ax^2 + 3^bx - 6^c \] has three integer roots.

1997 Irish Math Olympiad, 3

Tags: polynomial , algebra

Find all polynomials $ p(x)$ satisfying the equation: $ (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x)$ for all $ x$.

1980 IMO, 19

Tags: diophantine equation , algebra , equation , polynomial , number theory

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

2024 India IMOTC, 10

Tags: algebra , polynomial

Let $r>0$ be a real number. We call a monic polynomial with complex coefficients $r$-[i]good[/i] if all of its roots have absolute value at most $r$. We call a monic polynomial with complex coefficients [i]primordial[/i] if all of its coefficients have absolute value at most $1$. a) Prove that any $1$-good polynomial has a primordial multiple. b) If $r>1$, prove that there exists an $r$-good polynomial that does not have a primordial multiple. [i]Proposed by Pranjal Srivastava[/i]

2014 ELMO Shortlist, 7

Tags: polynomial , algebra

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

2005 IberoAmerican Olympiad For University Students, 2

Tags: linear algebra , matrix , polynomial , algebra

Let $A,B,C$ be real square matrices of order $n$ such that $A^3=-I$, $BA^2+BA=C^6+C+I$ and $C$ is symmetric. Is it possible that $n=2005$?

2008 Tournament Of Towns, 3

Tags: algebra , polynomial , inequalities , sum , root

A polynomial $x^n + a_1x^{n-1} + a_2x^{n-2} +... + a_{n-2}x^2 + a_{n-1}x + a_n$ has $n$ distinct real roots $x_1, x_2,...,x_n$, where $n > 1$. The polynomial $nx^{n-1}+ (n - 1)a_1x^{n-2} + (n - 2)a_2x^{n-3} + ...+ 2a_{n-2}x + a_{n-1}$ has roots $y_1, y_2,..., y_{n_1}$. Prove that $\frac{x^2_1+ x^2_2+ ...+ x^2_n}{n}>\frac{y^2_1 + y^2_2 + ...+ y^2_{n-1}}{n - 1}$

1992 USAMO, 5

Tags: algebra , polynomial , induction , complex number

Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]

2012 Iran Team Selection Test, 3

Tags: algebra , polynomial , function , vector , induction , combinatorics

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them. Find maximum of $\mid S\mid$. [i]Proposed by Erfan Salavati[/i]

2005 Tournament of Towns, 1

Tags: analytic geometry , polynomial , algebra

On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis. [i](4 points)[/i]

2006 Irish Math Olympiad, 5

Tags: function , algebra , polynomial

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

VI Soros Olympiad 1999 - 2000 (Russia), 10.6

Tags: algebra , polynomial

A natural number $n$ is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it $a_0$, $a_1$, $a_2$, $...$, $a_{2n-1}$ the polynomial $x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0$ has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)

2017 Vietnamese Southern Summer School contest, Problem 2

Tags: polynomial , root , equation , algebra

Let $P,Q$ be the polynomials: $$x^3-4x^2+39x-46, x^3+3x^2+4x-3,$$ respectively. 1. Prove that each of $P, Q$ has an unique real root. Let them be $\alpha,\beta$, respectively. 2. Prove that $\{ \alpha\}>\{ \beta\} ^2$, where $\{ x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.

2004 IMC, 6

Tags: function , polynomial , calculus , derivative , integration , logarithm , algebra

For every complex number $z$ different from 0 and 1 we define the following function \[ f(z) := \sum \frac 1 { \log^4 z } \] where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$. b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have \[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]

2017 Saudi Arabia Pre-TST + Training Tests, 4

Tags: arithmetic sequence , algebra , polynomial

Does there exist an integer $n \ge 3$ and an arithmetic sequence $a_0, a_1, ... , a_n$ such that the polynomial $a_nx^n +... + a_1x + a_0$ has $n$ roots which also form an arithmetic sequence?

2004 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , limit , ratio , algebra , polynomial , logarithm , absolute value

Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.