Olympiad problems

2010 AIME Problems, 6

Tags: quadratic , algebra , polynomial , function , ratio , analytic geometry , inequalities

Let $ P(x)$ be a quadratic polynomial with real coefficients satisfying \[x^2 \minus{} 2x \plus{} 2 \le P(x) \le 2x^2 \minus{} 4x \plus{} 3\] for all real numbers $ x$, and suppose $ P(11) \equal{} 181$. Find $ P(16)$.

2007 IMC, 5

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

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

2005 AIME Problems, 8

Tags: vieta , algebra , polynomial , number theory , logarithm

The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

2023 AMC 10, 12

Tags: polynomial

When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive? $\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

2013 AIME Problems, 10

Tags: algebra , polynomial

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let $p_{a,b}$ be the sum of the zeroes of $P(x)$. Find the sum of the $p_{a,b}$'s for all possible combinations of $a$ and $b$.

2012 Putnam, 1

Tags: function , algebra , polynomial , logarithm

Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies: (i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$ (ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$ (iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$ Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$

2023 Serbia Team Selection Test, P4

Tags: algebra , polynomial

Let $p$ be a prime and $P\in \mathbb{R}[x]$ be a polynomial of degree less than $p-1$ such that $\lvert P(1)\rvert=\lvert P(2)\rvert=\ldots=\lvert P(p)\rvert$. Prove that $P$ is constant.

2023 Taiwan TST Round 2, N

Tags: number theory , algebra , polynomial

Let $f_n$ be a polynomial with real coefficients for all $n \in \mathbb{Z}$. Suppose that \[f_n(k) = f_{n+k}(k) \quad n, k \in \mathbb{Z}.\] (a) Does $f_n = f_m$ necessarily hold for all $m,n \in \mathbb{Z}$? (b) If furthermore $f_n$ is a polynomial with integer coefficients for all $n \in\mathbb{Z}$, does $f_n = f_m$ necessarily hold for all $m, n \in\mathbb{Z}$? [i]Proposed by usjl[/i]

1970 AMC 12/AHSME, 11

Tags: algebra , polynomial , vieta

If two factors of $2x^3-hx+k$ are $x+2$ and $x-1$, the value of $|2h-3k|$ is $\textbf{(A) }4\qquad\textbf{(B) }3\qquad\textbf{(C) }2\qquad\textbf{(D) }1\qquad \textbf{(E) }0$

1995 Belarus Team Selection Test, 1

Tags: algebra , polynomial

Prove that the number of odd coefficients in the polynomial $(1+x)^n$ is a power of $2$ for every positive integer $N$

2014 Contests, 3

Tags: algebra , polynomial , inequalities , induction , trigonometry , calculus , derivative

Find all polynomials $P(x)$ with real coefficients that satisfy \[P(x\sqrt{2})=P(x+\sqrt{1-x^2})\]for all real $x$ with $|x|\le 1$.

MIPT student olimpiad autumn 2022, 2

Tags: geometry , polynomial

Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.

2019 Saudi Arabia Pre-TST + Training Tests, 1.2

Tags: arithmetic sequence , algebra , polynomial

Let $P(x)$ be a polynomial of degree $n \ge 2$ with rational coefficients such that $P(x)$ has $n$ pairwise different real roots forming an arithmetic progression. Prove that among the roots of $P(x)$ there are two that are also the roots of some polynomial of degree $2$ with rational coefficients.

2021 Thailand TSTST, 3

Tags: polynomial , algebra

Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$ cannot be expressed as a product of two non-constant polynomials with integral coeﬀicients.

2018 AIME Problems, 6

Tags: algebra , polynomial

A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2010 Harvard-MIT Mathematics Tournament, 9

Tags: algebra , polynomial

Let $f(x)=cx(x-1)$, where $c$ is a positive real number. We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$, all the roots of $f^n(x)$ are real. What is the smallest possible value of $c$?

2008 Silk Road, 4

Tags: polynomial , algebra

Find all polynomials $ P\in\mathbb{R}[x]$ such that for all $ r\in\mathbb{Q}$,there exist $ d\in\mathbb{Q}$ such that $ P(d)\equal{}r$

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 \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

2015 AMC 10, 16

Tags: vieta , algebra , polynomial , symmetry , rational root theorem , system of equations

If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$? $ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $

India EGMO 2025 TST, 2

Tags: algebra , polynomial

Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well. Proposed by Sutanay Bhattacharya

2012 NIMO Problems, 2

Tags: algebra , polynomial , vieta

If $r_1$, $r_2$, and $r_3$ are the solutions to the equation $x^3 - 5x^2 + 6x - 1 = 0$, then what is the value of $r_1^2 + r_2^2 + r_3^2$? [i]Proposed by Eugene Chen[/i]

2016 Thailand Mathematical Olympiad, 7

Tags: polynomial , absolute value , algebra

Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

1979 Austrian-Polish Competition, 2

Tags: algebra , polynomial

Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\dots,x_n$ satisfying $k \leq x_k \leq k+1$ for $k=1, \dots,n$.

1973 Spain Mathematical Olympiad, 5

Tags: polynomial , vector , linear algebra , algebra

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

2002 Taiwan National Olympiad, 5

Tags: polynomial , algebra

Suppose that the real numbers $a_{1},a_{2},...,a_{2002}$ satisfying $\frac{a_{1}}{2}+\frac{a_{2}}{3}+...+\frac{a_{2002}}{2003}=\frac{4}{3}$ $\frac{a_{1}}{3}+\frac{a_{2}}{4}+...+\frac{a_{2002}}{2004}=\frac{4}{5}$ $...$ $\frac{a_{1}}{2003}+\frac{a_{2}}{2004}+...+\frac{a_{2002}}{4004}=\frac{4}{4005}$ Evaluate the sum $\frac{a_{1}}{3}+\frac{a_{2}}{5}+...+\frac{a_{2002}}{4005}$.