Olympiad problems

2010 ISI B.Math Entrance Exam, 9

Tags: algebra , polynomial

Let $f(x)$ be a polynomial with integer co-efficients. Assume that $3$ divides the value $f(n)$ for each integer $n$. Prove that when $f(x)$ is divided by $x^3-x$ , the remainder is of the form $3r(x)$ where $r(x)$ is a polynomial with integer coefficients.

1934 Eotvos Mathematical Competition, 2

Tags: cyclic , geometric inequalities , polynomial , geometry

Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?

2011 All-Russian Olympiad, 1

Tags: quadratic , polynomial , algebra , function

Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$.

2017 Estonia Team Selection Test, 11

Tags: polynomial , number theory , functional equation , sum of digits

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2021 Thailand Online MO, P5

Tags: polynomial , algebra

Prove that there exists a polynomial $P(x)$ with real coefficients and degree greater than 1 such that both of the following conditions are true $\cdot$ $P(a)+P(b)+P(c)\ge 2021$ holds for all nonnegative real numbers $a,b,c$ such that $a+b+c=3$ $\cdot$ There are infinitely many triples $(a,b,c)$ of nonnegative real numbers such that $a+b+c=3$ and $P(a)+P(b)+P(c)= 2021$

1986 AMC 12/AHSME, 23

Tags: polynomial , algebra

Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $

2003 National Olympiad First Round, 4

Tags: algebra , polynomial

How many pairs of positive integers $(a,b)$ are there such that the roots of polynomial $x^2-ax-b$ are not greater than $5$? $ \textbf{(A)}\ 40 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 65 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ \text{None of the preceding} $

2021 South Africa National Olympiad, 5

Tags: polynomial , algebra , system of equations

Determine all polynomials $a(x)$, $b(x)$, $c(x)$, $d(x)$ with real coefficients satisfying the simultaneous equations \begin{align*} b(x) c(x) + a(x) d(x) & = 0 \\ a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1. \end{align*}

1987 Polish MO Finals, 3

Tags: sequence , sum of digits , algebra , number theory , polynomial

$w(x)$ is a polynomial with integer coefficients. Let $p_n$ be the sum of the digits of the number $w(n)$. Show that some value must occur infinitely often in the sequence $p_1, p_2, p_3, ...$ .

1982 IMO Longlists, 14

Tags: parametric equation , diophantine equation , geometric progression , polynomial , algebra

Determine all real values of the parameter $a$ for which the equation \[16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0\] has exactly four distinct real roots that form a geometric progression.

2005 Putnam, B5

Tags: derivative , algebra , system of equations , polynomial , calculus

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

2019 Switzerland Team Selection Test, 4

Tags: prime number , number theory , polynomial , algebra

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

2008 Germany Team Selection Test, 3

Tags: inequalities , polynomial , algebra

Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that \[ 2 y f(x \plus{} y) \plus{} (x \minus{} y)(f(x) \plus{} f(y)) \geq 0. \]

2005 Austrian-Polish Competition, 2

Tags: polynomial , ratio , algebra

Determine all polynomials $P$ with integer coefficients satisfying \[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]

1981 Austrian-Polish Competition, 5

Tags: algebra , polynomial , rational

Let $P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4$ be a polynomial with rational coefficients. Show that if $P(x)$ has exactly one real root $\xi$, then $\xi$ is a rational number.

2004 USAMTS Problems, 3

Tags: algebra , polynomial

Find, with proof, a polynomial $f(x,y,z)$ in three variables, with integer coefficients, such that for all $a,b,c$ the sign of $f(a,b,c)$ (that is, positive, negative, or zero) is the same as the sign of $a+b\sqrt[3]{2}+c\sqrt[3]{4}$.

2009 Putnam, B2

Tags: algebra , integration , calculus , function , polynomial , limit

A game involves jumping to the right on the real number line. If $ a$ and $ b$ are real numbers and $ b>a,$ the cost of jumping from $ a$ to $ b$ is $ b^3\minus{}ab^2.$ For what real numbers $ c$ can one travel from $ 0$ to $ 1$ in a finite number of jumps with total cost exactly $ c?$

2010 Brazil Team Selection Test, 2

Tags: number theory , size , polynomial

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

1984 IMO, 2

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

Find one pair of positive integers $a,b$ such that $ab(a+b)$ is not divisible by $7$, but $(a+b)^7-a^7-b^7$ is divisible by $7^7$.

2007 AMC 12/AHSME, 18

Tags: 3d geometry , quadratic , algebra , polynomial , geometry

The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$ $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$

2012 Belarus Team Selection Test, 1

Tags: sum , min , irrational , cubic , cubic equation , polynomial , algebra

A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be [i]irrational [/i] if it has three pairwise distinct real irrational roots $a_1,a_2, a_3$ Find all irrational cubic trinomials for which the value of $|a_1| + [a_2| + |a_3|$ is the minimal possible. (E. Barabanov)

2000 Romania Team Selection Test, 2

Tags: polynomial , algebra

Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$. [i]Marius Cavachi[/i]

1969 IMO Longlists, 66

Tags: algebra , inequality , taylor series , polynomial

$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$ $(b)$ Formulate and prove the analogous result for polynomials of third degree.

2007 Hungary-Israel Binational, 3

Tags: polynomial , algebra

Let $ t \ge 3$ be a given real number and assume that the polynomial $ f(x)$ satisfies $|f(k)\minus{}t^k|<1$, for $ k\equal{}0,1,2,\ldots ,n$. Prove that the degree of $f(x)$ is at least $n$.

Russian TST 2020, P1

Tags: number theory , polynomial

Let $P(x)$ be a polynomial taking integer values at integer inputs. Are there infinitely many natural numbers that are not representable in the form $P(k)-2^n$ where $n{}$ and $k{}$ are non-negative integers? [i]Proposed by F. Petrov[/i]