Olympiad problems

2001 Romania National Olympiad, 1

Tags: polynomial , algebra

a) Consider the polynomial $P(X)=X^5\in \mathbb{R}[X]$. Show that for every $\alpha\in\mathbb{R}^*$, the polynomial $P(X+\alpha )-P(X)$ has no real roots. b) Let $P(X)\in\mathbb{R}[X]$ be a polynomial of degree $n\ge 2$, with real and distinct roots. Show that there exists $\alpha\in\mathbb{Q}^*$ such that the polynomial $P(X+\alpha )-P(X)$ has only real roots.

1997 China Team Selection Test, 1

Tags: polynomial , vieta , inequalities , induction , algebra

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

2013 Israel National Olympiad, 3

Tags: arctangent , algebra , polynomial

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

2019 Final Mathematical Cup, 2

Tags: sum , polynomial , algebra

Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .

2023 IRN-SGP-TWN Friendly Math Competition, 6

Tags: polynomial with integer coeffi , algebra , number theory , functional equation , polynomial

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

1959 AMC 12/AHSME, 27

Tags: quadratic , vieta , algebra , quadratic equation , quadratic formula , polynomial

Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$? $ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$ $\textbf{(B)}\ \text{The discriminant is 9}\qquad$ $\textbf{(C)}\ \text{The roots are imaginary}\qquad$ $\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$ $\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $

2011 USA Team Selection Test, 5

Tags: binomial theorem , induction , limit , function , modular arithmetic , algebra , polynomial

Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that \[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]

2023 Turkey EGMO TST, 4

Tags: inequalities , polynomial , algebra

Let $n$ be a positive integer and $P,Q$ be polynomials with real coefficients with $P(x)=x^nQ(\frac{1}{x})$ and $P(x) \geq Q(x)$ for all real numbers $x$. Prove that $P(x)=Q(x)$ for all real number $x$.

2018 Harvard-MIT Mathematics Tournament, 10

Tags: polynomial , probability , algebra

Let $S$ be a randomly chosen $6$-element subset of the set $\{0,1,2,\ldots,n\}.$ Consider the polynomial $P(x)=\sum_{i\in S}x^i.$ Let $X_n$ be the probability that $P(x)$ is divisible by some nonconstant polynomial $Q(x)$ of degree at most $3$ with integer coefficients satisfying $Q(0) \neq 0.$ Find the limit of $X_n$ as $n$ goes to infinity.

2019 Canadian Mathematical Olympiad Qualification, 3

Tags: root , polynomial , algebra

Let $f(x) = x^3 + 3x^2 - 1$ have roots $a,b,c$. (a) Find the value of $a^3 + b^3 + c^3$ (b) Find all possible values of $a^2b + b^2c + c^2a$

1998 Romania National Olympiad, 2

Tags: algebra , polynomial

Let $P(x) = a_{1998}X^{1998} + a_{1997}X^{1997} +...+a_1X + a_0$ be a polynomial with real coefficients such that $P(0) \ne P (-1)$, and let $a, b$ be real numbers. Let $Q(x) = b_{1998}X^{1998} + b_{1997}X^{1997} +...+b_1X + b_0$ be the polynomial with real coefficients obtained by taking $b_k = aa_k + b$ ,$\forall k = 0, 1,2,..., 1998$. Show that if $Q(0) = Q (-1) \ne 0$ , then the polynomial $Q$ has no real roots.

1974 Polish MO Finals, 3

Tags: quadratic polynomial , algebra , polynomial

Let $r$ be a natural number. Prove that the quadratic trinomial $x^2 - rx- 1$ does not divide any nonzero polynomial whose coefficients are integers with absolute values less than $r$.

2010 Romania Team Selection Test, 1

Tags: arithmetic sequence , polynomial , induction , algebra , modular arithmetic , calculus , integration

A nonconstant polynomial $f$ with integral coefficients has the property that, for each prime $p$, there exist a prime $q$ and a positive integer $m$ such that $f(p) = q^m$. Prove that $f = X^n$ for some positive integer $n$. [i]AMM Magazine[/i]

2016 India Regional Mathematical Olympiad, 8

Tags: integer , polynomial , algebra

At some integer points a polynomial with integer coefficients take values $1, 2$ and $3$. Prove that there exist not more than one integer at which the polynomial is equal to $5$.

1998 Romania Team Selection Test, 3

Tags: irreducible , polynomial , algebra

Show that for any positive integer $n$ the polynomial $f(x)=(x^2+x)^{2^n}+1$ cannot be decomposed into the product of two integer non-constant polynomials. [i]Marius Cavachi[/i]

2016 NZMOC Camp Selection Problems, 5

Tags: algebra , polynomial

Find all polynomials $P(x)$ with real coefficients such that the polynomial $$Q(x) = (x + 1)P(x-1) -(x-1)P(x)$$ is constant.

2000 National Olympiad First Round, 20

Tags: polynomial , algebra

For every real $x$, the polynomial $p(x)$ whose roots are all real satisfies $p(x^2-1)=p(x)p(-x)$. What can the degree of $p(x)$ be at most? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ \text{There is no upper bound for the degree of } p(x) \qquad\textbf{(E)}\ \text{None} $

1987 USAMO, 3

Tags: induction , polynomial , algebra

Construct a set $S$ of polynomials inductively by the rules: (i) $x\in S$; (ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$. Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.

2000 China Team Selection Test, 1

Tags: function , polynomial , algebra

Let $F$ be the set of all polynomials $\Gamma$ such that all the coefficients of $\Gamma (x)$ are integers and $\Gamma (x) = 1$ has integer roots. Given a positive intger $k$, find the smallest integer $m(k) > 1$ such that there exist $\Gamma \in F$ for which $\Gamma (x) = m(k)$ has exactly $k$ distinct integer roots.

2006 Irish Math Olympiad, 5

Tags: algebra , function , polynomial

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

2002 China Team Selection Test, 1

Tags: inequalities , polynomial , algebra

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]

1989 IMO Longlists, 78

Tags: polynomial , algebra

Let $ P(x)$ be a polynomial with integer coefficients such that \[ P(m_1) \equal{} P(m_2) \equal{} P(m_3) \equal{} P(m_4) \equal{} 7\] for given distinct integers $ m_1,m_2,m_3,$ and $ m_4.$ Show that there is no integer m such that $ P(m) \equal{} 14.$

1977 Poland - Second Round, 5

Tags: polynomial , algebra

Let the polynomials $ w_n $ be given by the formulas: $$ w_1(x) = x^2 - 1, \quad w_{n+1}(x) = w_n(x)^2 - 1, \quad (n = 1, 2, \ldots)$$ and let $a$ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have?

PEN M Problems, 8

Tags: recursive sequences , algebra , polynomial

The Bernoulli sequence $\{B_{n}\}_{n \ge 0}$ is defined by \[B_{0}=1, \; B_{n}=-\frac{1}{n+1}\sum^{n}_{k=0}{{n+1}\choose k}B_{k}\;\; (n \ge 1)\] Show that for all $n \in \mathbb{N}$, \[(-1)^{n}B_{n}-\sum \frac{1}{p},\] is an integer where the summation is done over all primes $p$ such that $p| 2k-1$.

The Golden Digits 2024, P3

Tags: number theory , algebra , polynomial

Prove that there exist infinitely many positive integers $d$ such that we can find a polynomial $P\in\mathbb{Z}[x]$ of degree $d$ and $N\in\mathbb{N}$ such that for all integers $x>N$ and any prime $p$, we have $$\nu_p(P(x)^3+3P(x)^2-3)<\frac{d\cdot\log(x)}{2024^{2024}}.$$ [i]Proposed by Marius Cerlat[/i]