Olympiad problems

1997 Vietnam Team Selection Test, 3

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

1964 Spain Mathematical Olympiad, 1

Tags: polynomial , algebra

Given the equation $x^2+ax+1=0$, determine: a) The interval of possible values for $a$ where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when $a$ is in the previous interval.

1974 IMO Longlists, 39

Tags: algebra , polynomial , complex number

Let $n$ be a positive integer, $n \geq 2$, and consider the polynomial equation \[x^n - x^{n-2} - x + 2 = 0.\] For each $n,$ determine all complex numbers $x$ that satisfy the equation and have modulus $|x| = 1.$

2010 Germany Team Selection Test, 1

Tags: algebra , polynomial

Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

2017 Saudi Arabia JBMO TST, 1

Tags: algebra , polynomial

Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation

2017 Korea USCM, 2

Tags: algebra , polynomial , linear algebra

Show that any real coefficient polynomial $f(x,y)$ is a linear combination of polynomials of the form $(x+ay)^k$. ($a$ is a real number and $k$ is a non-negative integer.)

1998 National Olympiad First Round, 34

Tags: polynomial , algebra

Let $ a,b,c,d$ be rational numbers with $ a>0$. If for every integer $ n\ge 0$, the number $ an^{3} \plus{}bn^{2} \plus{}cn\plus{}d$ is also integer, then the minimal value of $ a$ will be $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ \frac{1}{6} \qquad\textbf{(D)}\ \text{Cannot be found} \qquad\textbf{(E)}\ \text{None}$

1994 China Team Selection Test, 2

Tags: algebra , integration , gauss , prime number , calculus , polynomial , number theory

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1999 Putnam, 1

Tags: algebra , derivative , polynomial , analytic geometry , graphing lines , calculus

Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all $x$, \[|f(x)|-|g(x)|+h(x)=\begin{cases}-1 & \text{if }x<-1\\3x+2 &\text{if }-1\leq x\leq 0\\-2x+2 & \text{if }x>0.\end{cases}\]

1999 Romania Team Selection Test, 11

Tags: polynomial , absolute value , inequalities , algebra

Let $a,n$ be integer numbers, $p$ a prime number such that $p>|a|+1$. Prove that the polynomial $f(x)=x^n+ax+p$ cannot be represented as a product of two integer polynomials. [i]Laurentiu Panaitopol[/i]

2017 Iran MO (3rd round), 2

Tags: polynomial , algebra , complex number

Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by $$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$ (a) Prove that $$P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) } $$ (b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial $$Q(z)=z^m q(z)+ q^*(z)$$ lie on the unit circle.

2014 Taiwan TST Round 1, 1

Tags: absolute value , polynomial , inequalities , algebra

Let $f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0$ be a polynomial with real coefficients $(n \ge 2)$. Suppose all roots of $f$ are real. Prove that the absolute value of each root is at most $\sqrt{\frac{2(1-n)}n a_{n-2}}$.

2021 Durer Math Competition (First Round), 4

Tags: polynomial , algebra

Find all pairs of polynomials $(p, q)$ with integer coefficients that satisfy the equation $$p(x^2) + q(x^2) = p(x)q(x)$$ such that $p$ is of degree $n$ and has $n$ nonnegative real roots (with multiplicity).

2025 Greece National Olympiad, 1

Tags: root , integer , polynomial , algebra

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$. If $m,n \in \mathbb {Z_+}$, find the values of $m,n$ and their corresponding roots.

2004 Romania Team Selection Test, 7

Tags: linear algebra , matrix , binomial coefficient , polynomial , induction , algebra , number theory

Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have \[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \] Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.

2005 Georgia Team Selection Test, 4

Tags: polynomial , algebra

Find all polynomials with real coefficients, for which the equality \[ P(2P(x)) \equal{} 2P(P(x)) \plus{} 2(P(x))^{2}\] holds for any real number $ x$.

2014 Paenza, 3

Tags: polynomial , vieta , algebra

Find all $(m,n)$ in $\mathbb{N}^2$ such that $m\mid n^2+1$ and $n\mid m^2+1$.

Kvant 2022, M2714

Tags: number theory , polynomial , divisibility

Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$. [i]From the folklore[/i]

2009 Baltic Way, 1

Tags: inequalities , function , polynomial , algebra

A polynomial $p(x)$ of degree $n\ge 2$ has exactly $n$ real roots, counted with multiplicity. We know that the coefficient of $x^n$ is $1$, all the roots are less than or equal to $1$, and $p(2)=3^n$. What values can $p(1)$ take?

1984 IMO Shortlist, 11

Tags: equation , polynomial , diophantine equation , number theory

Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying \[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]

2022 Saudi Arabia BMO + EGMO TST, 1.1

Tags: algebra , polynomial

For each non-constant integer polynomial $P(x)$, let’s define $$M_{P(x)} = \underset{x\in [0,2021]}{\max} |P(x)|.$$ 1. Find the minimum value of $M_{P(x)}$ when deg $P(x) = 1$. 2. Suppose that $P(x) \in Z[x]$ when deg $P(x) = n$ and $2 \le n \le 2022$. Prove that $M_{P(x)} \ge 1011$.

2006 Polish MO Finals, 3

Tags: polynomial , root , quadratic , coefficient , algebra

Find all pairs of integers $a,b$ for which there exists a polynomial $P(x) \in \mathbb{Z}[X]$ such that product $(x^2+ax+b)\cdot P(x)$ is a polynomial of a form \[ x^n+c_{n-1}x^{n-1}+\cdots+c_1x+c_0 \] where each of $c_0,c_1,\ldots,c_{n-1}$ is equal to $1$ or $-1$.

2012 Brazil Team Selection Test, 1

Tags: algebra , geometry , inequalities , polynomial

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

2011 Singapore MO Open, 4

Tags: polynomial , calculus , number theory , algebra

Find all polynomials $P(x)$ with real coefficients such that \[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]

1988 IMO Longlists, 39

Tags: algebra , function , functional equation , polynomial

[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$? [b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of \[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$ [b]iii.)[/b] The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$