Olympiad problems

2012 Harvard-MIT Mathematics Tournament, 10

Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?

2020 Brazil Team Selection Test, 4

Tags: polynomial , algebra

Let $\mathbb{Z}$ denote the set of all integers. Find all polynomials $P(x)$ with integer coefficients that satisfy the following property: For any infinite sequence $a_1$, $a_2$, $\dotsc$ of integers in which each integer in $\mathbb{Z}$ appears exactly once, there exist indices $i < j$ and an integer $k$ such that $a_i +a_{i+1} +\dotsb +a_j = P(k)$.

1987 AMC 12/AHSME, 28

Tags: polynomial , algebra , complex number

Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ -a \qquad\textbf{(E)}\ -b $

2014 Contests, 3

Tags: algebra , polynomial , quadratic

Find all real numbers $p$ for which the equation $x^3+3px^2+(4p-1)x+p=0$ has two real roots with difference $1$.

2009 India IMO Training Camp, 9

Tags: algebra , polynomial

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

1974 Swedish Mathematical Competition, 4

Tags: algebra , polynomial

Find all polynomials $p(x)$ such that $p(x^2) = p(x)^2$ for all $x$. Hence find all polynomials $q(x)$ such that \[ q\left(x^2 - 2x\right) = q\left(x-2\right)^2 \]

2014 BMT Spring, P2

Tags: polynomial

Define $\eta(f)$ to be the number of roots that are repeated of the complex-valued polynomial $f$ (e.g. $\eta((x-1)^3\cdot(x+1)^4)=5$). Prove that for nonconstant, relatively prime $f,g\in\mathbb C[x]$, $$\eta(f)+\eta(g)+\eta(f+g)<\deg f+\deg g$$

2010 Contests, 2

Tags: number theory , algebra , polynomial , floor function

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

2015 Moldova Team Selection Test, 1

Tags: algebra , polynomial

Find all polynomials $P(x)$ with real coefficients which satisfies \\ $P(2015)=2025$ and $P(x)-10=\sqrt{P(x^{2}+3)-13}$ for every $x\ge 0$ .

2014 ELMO Shortlist, 7

Tags: polynomial , algebra

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

2024 USEMO, 4

Tags: polynomial , algebra

Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.) [i]Kornpholkrit Weraarchakul[/i]

1994 Vietnam Team Selection Test, 3

Tags: polynomial , algebra

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation \[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\] has also 4 positive roots.

2009 IMC, 4

Tags: algebra , polynomial , modular arithmetic

Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following : [list] (a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$ (b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list] How many polynomials are in $\mathbf{W}?$

2014 Contests, 3

Tags: linear algebra , matrix , analytic geometry , polynomial , graphing lines

Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.

2012 Iran MO (3rd Round), 1

Tags: algebra , induction , polynomial , modular arithmetic , function

Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

2021 AMC 10 Fall, 25

Tags: algebra , polynomial

A quadratic polynomial $p(x)$ with real coefficients and leading coefficient $1$ is called disrespectful if the equation $p(p(x)) = 0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(1)?$ $\textbf{(A) }\dfrac5{16} \qquad \textbf{(B) }\dfrac12 \qquad \textbf{(C) }\dfrac58 \qquad \textbf{(D) }1 \qquad \textbf{(E) }\dfrac98$

PEN Q Problems, 2

Tags: algebra , integration , calculus , pigeonhole principle , polynomial

Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.

2013 ELMO Problems, 5

Tags: polynomial , number theory , algebra , 3d geometry , geometry , modular arithmetic

For what polynomials $P(n)$ with integer coefficients can a positive integer be assigned to every lattice point in $\mathbb{R}^3$ so that for every integer $n \ge 1$, the sum of the $n^3$ integers assigned to any $n \times n \times n$ grid of lattice points is divisible by $P(n)$? [i]Proposed by Andre Arslan[/i]

1978 IMO Shortlist, 15

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

Let $p$ be a prime and $A = \{a_1, \ldots , a_{p-1} \}$ an arbitrary subset of the set of natural numbers such that none of its elements is divisible by $p$. Let us define a mapping $f$ from $\mathcal P(A)$ (the set of all subsets of $A$) to the set $P = \{0, 1, \ldots, p - 1\}$ in the following way: $(i)$ if $B = \{a_{i_{1}}, \ldots , a_{i_{k}} \} \subset A$ and $\sum_{j=1}^k a_{i_{j}} \equiv n \pmod p$, then $f(B) = n,$ $(ii)$ $f(\emptyset) = 0$, $\emptyset$ being the empty set. Prove that for each $n \in P$ there exists $B \subset A$ such that $f(B) = n.$

1985 Putnam, A6

Tags: polynomial , algebra

If $p(x)=a_{0}+a_{1} x+\cdots+a_{m} x^{m}$ is a polynomial with real coefficients $a_{i},$ then set $$ \Gamma(p(x))=a_{0}^{2}+a_{1}^{2}+\cdots+a_{m}^{2}. $$ Let $F(x)=3 x^{2}+7 x+2 .$ Find, with proof, a polynomial $g(x)$ with real coefficients such that (i) $g(0)=1,$ and (ii) $\Gamma\left(f(x)^{n}\right)=\Gamma\left(g(x)^{n}\right)$ for every integer $n \geq 1.$

Gheorghe Țițeica 2025, P3

Tags: function , real analysis , polynomial

Let $\mathcal{P}_n$ be the set of all real monic polynomial functions of degree $n$. Prove that for any $a<b$, $$\inf_{P\in\mathcal{P}_n}\int_a^b |P(x)|\, dx >0.$$ [i]Cristi Săvescu[/i]

1971 Canada National Olympiad, 4

Tags: linear algebra , matrix , polynomial , algebra , quadratic , quadratic formula

Determine all real numbers $a$ such that the two polynomials $x^2+ax+1$ and $x^2+x+a$ have at least one root in common.

2017 Hong Kong TST, 3

Tags: polynomial , algebra , combinatorics

Let a sequence of real numbers $a_0, a_1,a_2, \cdots$ satisfies the condition: $$\sum_{n=0}^ma_n\cdot(-1)^n\cdot{m\choose n}=0$$ for all sufficiently large values of $m$. Show that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\geq 0$

VMEO III 2006 Shortlist, A2

Tags: polynomial , algebra

Given a polynomial $P(x)=x^4+3x^2-9x+1$. Calculate $P(\alpha^2+\alpha+1)$ where\[ \alpha=\sqrt[3]{\frac{1+\sqrt{5}}{2}}+\sqrt[3]{\frac{1-\sqrt{5}}{2}} \]

2022 USEMO, 5

Tags: algebra , number theory , polynomial

Let $\tau(n)$ denote the number of positive integer divisors of a positive integer $n$ (for example, $\tau(2022) = 8$). Given a polynomial $P(X)$ with integer coefficients, we define a sequence $a_1, a_2,\ldots$ of nonnegative integers by setting \[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\] for each positive integer $n$. We then say the sequence [i]has limit infinity[/i] if every integer occurs in this sequence only finitely many times (possibly not at all). Does there exist a choice of $P(X)$ for which the sequence $a_1$, $a_2$, . . . has limit infinity? [i]Jovan Vuković[/i]