Olympiad problems

2010 Harvard-MIT Mathematics Tournament, 3

Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeroes of $p^\prime (x)$ are also zeroes of $p(x)$. Find $p$. Note: monic means that the leading coefficient is $1$.

1989 India National Olympiad, 2

Tags: algebra , polynomial , vieta

Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.

1992 IMO Longlists, 82

Tags: algebra , polynomial , product

Let $f(x) = x^m + a_1x^{m-1} + \cdots+ a_{m-1}x + a_m$ and $g(x) = x^n + b_1x^{n-1} + \cdots + b_{n-1}x + b_n$ be two polynomials with real coefficients such that for each real number $x, f(x)$ is the square of an integer if and only if so is $g(x)$. Prove that if $n +m > 0$, then there exists a polynomial $h(x)$ with real coefficients such that $f(x) \cdot g(x) = (h(x))^2.$ [hide="Remark."]Remark. The original problem stated $g(x) = x^n + b_1x^{n-1} + \cdots + {\color{red}{ b_{n-1}}} + b_n$, but I think the right form of the problem is what I wrote.[/hide]

2011 Saudi Arabia BMO TST, 1

Tags: polynomial , algebra

Find all polynomials $P$ with real coefficients such that for all $x, y ,z \in R$, $$P(x)+P(y)+P(z)+P(x+y+z)=P(x+y)+P(y+z)+P(z+x)$$

1981 Canada National Olympiad, 4

Tags: function , algebra , polynomial

$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.

2006 Taiwan National Olympiad, 3

Tags: polynomial , function , algebra

$f(x)=x^3-6x^2+17x$. If $f(a)=16, f(b)=20$, find $a+b$.

2020 Serbian Mathematical Olympiad, Problem 1

Tags: polynomial , algebra

Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.

2008 Hong kong National Olympiad, 1

Tags: induction , function , algebra , polynomial

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

1978 IMO Longlists, 31

Tags: polynomial , algebra

Let the polynomials \[P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,\] \[Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,\] be given satisfying the identity $P(x)^2 = (x^2 - 1)Q(x)^2 + 1$. Prove the identity \[P'(x) = nQ(x).\]

2008 IberoAmerican Olympiad For University Students, 2

Tags: geometry , algebra , polynomial , ratio

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

2010 Princeton University Math Competition, 8

Tags: trigonometry , symmetry , polynomial , algebra , geometric series

The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

2002 India IMO Training Camp, 17

Tags: algebra , trigonometry , polynomial , arithmetic sequence

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

2000 National Olympiad First Round, 2

Tags: quadratic , algebra , polynomial

Discriminant of a second degree polynomial with integer coefficients cannot be $ \textbf{(A)}\ 23 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 25 \qquad\textbf{(D)}\ 28 \qquad\textbf{(E)}\ 33 $

1978 Putnam, B3

Tags: polynomial , sequence , recurrence

The sequence $(Q_{n}(x))$ of polynomials is defined by $$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$ and for $m \geq 1 $ by $$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$ $$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$ Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$

2014 NIMO Problems, 5

Tags: polynomial , algebra , vieta , search

Let $r$, $s$, $t$ be the roots of the polynomial $x^3+2x^2+x-7$. Then \[ \left(1+\frac{1}{(r+2)^2}\right)\left(1+\frac{1}{(s+2)^2}\right)\left(1+\frac{1}{(t+2)^2}\right)=\frac{m}{n} \] for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Justin Stevens[/i]

2012 Indonesia TST, 1

Tags: least common multiple , algebra , polynomial , number theory

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

2005 Taiwan TST Round 2, 1

Tags: polynomial , algebra , quadratic

Prove that for any quadratic polynomial $f(x)=x^2+px+q$ with integer coefficients, it is possible to find another polynomial $q(x)=2x^2+rx+s$ with integer coefficients so that \[\{f(x)|x \in \mathbb{Z} \} \cap \{g(x)|x \in \mathbb{Z} \} = \emptyset .\]

1965 AMC 12/AHSME, 27

Tags: polynomial , algebra

When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \minus{} 1$ the quotient is $ f(y)$ and the remainder is $ R_1$. When $ y^2 \plus{} my \plus{} 2$ is divided by $ y \plus{} 1$ the quotient is $ g(y)$ and the remainder is $ R_2$. If $ R_1 \equal{} R_2$ then $ m$ is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \minus{} 1 \qquad \textbf{(E)}\ \text{an undetermined constant}$

2017 China Team Selection Test, 5

Tags: algebra , polynomial , number theory

Let $ \varphi(x)$ be a cubic polynomial with integer coefficients. Given that $ \varphi(x)$ has have 3 distinct real roots $u,v,w $ and $u,v,w $ are not rational number. there are integers $ a, b,c$ such that $u=av^2+bv+c$. Prove that $b^2 -2b -4ac - 7$ is a square number .

1970 Regional Competition For Advanced Students, 4

Tags: polynomial , algebra , greatest common divisor , number theory , algorithm

Find all real solutions of the following set of equations: \[72x^3+4xy^2=11y^3\] \[27x^5-45x^4y-10x^2y^3=\frac{-143}{32}y^5\]

2006 Pre-Preparation Course Examination, 1

Tags: polynomial , algebra

Find out wich of the following polynomials are irreducible. a) $t^4+1$ over $\mathbb{R}$; b) $t^4+1$ over $\mathbb{Q}$; c) $t^3-7t^2+3t+3$ over $\mathbb{Q}$; d) $t^4+7$ over $\mathbb{Z}_{17}$; e) $t^3-5$ over $\mathbb{Z}_{11}$; f) $t^6+7$ over $\mathbb{Q}(i)$.

2011 India IMO Training Camp, 2

Tags: algebra , polynomial , inequalities

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

2010 Harvard-MIT Mathematics Tournament, 6

Tags: polynomial , algebra

Suppose that a polynomial of the form $p(x)=x^{2010}\pm x^{2009}\pm \cdots \pm x \pm 1$ has no real roots. What is the maximum possible number of coefficients of $-1$ in $p$?

2017 Purple Comet Problems, 6

Tags: algebra , polynomial , quadratic

For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.

2018 Moldova Team Selection Test, 5

Tags: polynomial , algebra

Let $n, \in \mathbb {N^*} , n\ge 3$ a) Prove that the polynomial $f (x)=\frac {X^{2^n-1}-1}{X-1}-X^n $ has a divisor of form $X^p +1$ where $p\in\mathbb {N^*} $ b) Show that for $n=7$ the polynomial $f (X) $ has three divisors with integer coefficients .