Olympiad problems

2023 India Regional Mathematical Olympiad, 3

Tags: algebra , polynomial

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:\\ \[f(a)=bc , f(b)=ac, f(c)=ab\] Dertermine $f(a+b+c)$ in terms of $a,b,c$.

2007 Tournament Of Towns, 4

Tags: quadratic , polynomial , algebra

Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?

1953 Putnam, B5

Tags: root , polynomial

Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$, if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$

V Soros Olympiad 1998 - 99 (Russia), 9.8

Tags: polynomial , algebra

Calculate $f(\sqrt[3]{2}-1) $, where $$f(x) = x^{1999} + 3x^{1998} + 4x^{1997} + 2x^{1996} + 4x^{1995} + 2x^{1994} + ...$$ $$... + 4x^3 + 2x^2 + 3x+ 1.$$

1994 Polish MO Finals, 1

Tags: rational root theorem , polynomial , algebra

Find all triples $(x,y,z)$ of positive rationals such that $x + y + z$, $\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$ and $xyz$ are all integers.

2020 Balkan MO Shortlist, A4

Tags: algebra , polynomial

Let $P(x) = x^3 + ax^2 + bx + 1$ be a polynomial with real coefficients and three real roots $\rho_1$, $\rho_2$, $\rho_3$ such that $|\rho_1| < |\rho_2| < |\rho_3|$. Let $A$ be the point where the graph of $P(x)$ intersects $yy'$ and the point $B(\rho_1, 0)$, $C(\rho_2, 0)$, $D(\rho_3, 0)$. If the circumcircle of $\vartriangle ABD$ intersects $yy'$ for a second time at $E$, find the minimum value of the length of the segment $EC$ and the polynomials for which this is attained. [i]Brazitikos Silouanos, Greece[/i]

2014 BMT Spring, 4

Tags: polynomial , algebra

The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$.

2022 Latvia Baltic Way TST, P4

Tags: polynomial , algebra

Let $p(x)$ be a polynomial with real coefficients and $n$ be a positive integer. Prove that there exists a non-zero polynomial $q(x)$ with real coefficients such that the polynomial $p(x)\cdot q(x)$ has non-zero coefficients only by the powers which are multiples of $n$.

2005 Iran MO (3rd Round), 3

Tags: polynomial , algebra , number theory

$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that \[p(x)^2|q(x)^2+q(x).r(x)+r(x)^2\]

2022 International Zhautykov Olympiad, 1

Tags: algebra , polynomial

Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities $$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$ Prove that the degrees of the three polynomials are all even.

2011 AMC 12/AHSME, 20

Tags: algebra , polynomial

Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50 < f(7) < 60$, $70 < f(8) < 80$, and $5000k < f(100) < 5000(k+1)$ for some integer $k$. What is $k$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5 $

2006 Pre-Preparation Course Examination, 2

Tags: algebra , function , combinatorics , polynomial

If $f(x)$ is the generating function of the sequence $a_1,a_2,\ldots$ and if $f(x)=\frac{r(x)}{s(x)}$ holds such that $r(x)$ and $s(x)$ are polynomials show that $a_n$ has a homogenous recurrence.

2024 IFYM, Sozopol, 2

Tags: algebra , polynomial

Let $m,n$ and $a$ be positive integers. Lumis has $m$ cards, each with the number $n$ written on it, and an infinite number of cards with each of the symbols addition, subtraction, multiplication, division, opening, and closing brackets. Umbra has composed an arithmetic expression with them, whose value is a positive integer less than $\displaystyle\frac{n}{2^m}$. Prove that if $n$ is replaced everywhere by $a$, then the resulting expression will have the same value as before or will be undefined due to division by zero.

2006 Junior Balkan Team Selection Tests - Moldova, 3

Tags: polynomial , algebra

Determine all 2nd degree polynomials with integer coefficients of the form $P(X)=aX^{2}+bX+c$, that satisfy: $P(a)=b$, $P(b)=a$, with $a\neq b$.

1967 IMO Shortlist, 2

Tags: polynomial , algebra , system of equations

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

2023 VN Math Olympiad For High School Students, Problem 8

Tags: polynomial , algebra

Prove that: for all positive integers $n\ge 2,$ the polynomial$$(x^2-1)^2(x^2-1)^2...(x^2-2023)^2+1$$ is irreducible in $\mathbb{Q}[x].$

1999 Brazil Team Selection Test, Problem 2

Tags: polynomial , algebra

If $a,b,c,d$ are Distinct Real no. such that $a = \sqrt{4+\sqrt{5+a}}$ $b = \sqrt{4-\sqrt{5+b}}$ $c = \sqrt{4+\sqrt{5-c}}$ $d = \sqrt{4-\sqrt{5-d}}$ Then $abcd = $

2019 CCA Math Bonanza, I12

Tags: algebra , polynomial

Let $f\left(x,y\right)=x^2\left(\left(x+2y\right)^2-y^2+x-1\right)$. If $f\left(a,b+c\right)=f\left(b,c+a\right)=f\left(c,a+b\right)$ for distinct numbers $a,b,c$, what are all possible values of $a+b+c$? [i]2019 CCA Math Bonanza Individual Round #12[/i]

1969 IMO Shortlist, 54

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

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2015 Germany Team Selection Test, 1

Tags: real number , integer , root , algebra , polynomial

Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties: - For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$. - There is a real number $\xi$ with $P(\xi)=0$.

2006 China Team Selection Test, 3

Tags: pigeonhole principle , polynomial , algebra

$k$ and $n$ are positive integers that are greater than $1$. $N$ is the set of positive integers. $A_1, A_2, \cdots A_k$ are pairwise not-intersecting subsets of $N$ and $A_1 \cup A_2 \cup \cdots \cup A_k = N$. Prove that for some $i \in \{ 1,2,\cdots,k \}$, there exsits infinity many non-factorable n-th degree polynomials so that coefficients of one polynomial are pairwise distinct and all the coeficients are in $A_i$.

2015 IMC, 10

Tags: polynomial , inequality

Let $n$ be a positive integer, and let $p(x)$ be a polynomial of degree $n$ with integer coefficients. Prove that $$ \max_{0\le x\le1} \big|p(x)\big| > \frac1{e^n}. $$ Proposed by Géza Kós, Eötvös University, Budapest

2018 Moscow Mathematical Olympiad, 3

Tags: algebra , polynomial

Are there such natural $n$, that exist polynomial of degree $n$ and with $n$ different real roots, and a) $P(x)P(x+1)=P(x^2)$ b) $P(x)P(x+1)=P(x^2+1)$

2016 Regional Olympiad of Mexico Southeast, 3

Tags: algebra , polynomial

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

2009 Brazil Team Selection Test, 3

Tags: polynomial , inequalities , algebra , monic polynomial

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$. (a) Prove that $P(x) > -4$ for all real $x$. (b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.