Olympiad problems

2011 All-Russian Olympiad, 2

Tags: arithmetic sequence , quadratic , vieta , invariant , polynomial , induction , algebra

In the notebooks of Peter and Nick, two numbers are written. Initially, these two numbers are 1 and 2 for Peter and 3 and 4 for Nick. Once a minute, Peter writes a quadratic trinomial $f(x)$, the roots of which are the two numbers in his notebook, while Nick writes a quadratic trinomial $g(x)$ the roots of which are the numbers in [i]his[/i] notebook. If the equation $f(x)=g(x)$ has two distinct roots, one of the two boys replaces the numbers in his notebook by those two roots. Otherwise, nothing happens. If Peter once made one of his numbers 5, what did the other one of his numbers become?

2012 France Team Selection Test, 2

Tags: polynomial , algebra

Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).

2018 CCA Math Bonanza, I13

Tags: polynomial , algebra

$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\right)$? [i]2018 CCA Math Bonanza Individual Round #13[/i]

2013 IFYM, Sozopol, 2

Tags: quadratic , vieta , polynomial , number theory , modular arithmetic , algebra

Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$? [i]Proposed by Mahan Malihi[/i]

2011 AIME Problems, 7

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

Find the number of positive integers $m$ for which there exist nonnegative integers $x_0,x_1,\ldots,x_{2011}$ such that \[ m^{x_0}=\sum_{k=1}^{2011}m^{x_k}. \]

2004 239 Open Mathematical Olympiad, 1

Tags: polynomial , algebra , function

Given non-constant linear functions $p_1(x), p_2(x), \dots p_n(x)$. Prove that at least $n-2$ of polynomials $p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1$ have a real root.

2014 USAMO, 3

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

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 $a+b+c=2014$.

2021 Taiwan Mathematics Olympiad, 5.

Tags: integer polynomial , polynomial , algebra

Let $n$ be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial $P(x)$ with degree no more than $n$. Bob doesn’t know $P(x)$, and his goal is to determine whether there exists an integer $k$ such that no integer roots of $P(x) = k$ exist. In each round, Bob can choose a constant $c$. Alice will tell Bob an integer $k$, representing the number of integer $t$ such that $P(t) = c$. Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal. [i]Proposed by ltf0501[/i]

2000 Romania National Olympiad, 4

Tags: polynomial , algebra

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

2001 Poland - Second Round, 3

Tags: polynomial , algebra

Let $n\ge 3$ be a positive integer. Prove that a polynomial of the form \[x^n+a_{n-3}x^{n-3}+a_{n-4}x^{n-4}+\ldots +a_1x+a_0,\] where at least one of the real coefficients $a_0,a_1,\ldots ,a_{n-3}$ is nonzero, cannot have all real roots.

2006 IMO Shortlist, 4

Tags: root , polynomial , algebra

Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.

2001 AMC 12/AHSME, 23

Tags: algebra , quadratic , complex number , polynomial

A polynomial of degree four with leading coefficient 1 and integer coefficients has two zeros, both of which are integers. Which of the following can also be a zero of the polynomial? $ \textbf{(A)} \ \frac {1 \plus{} i \sqrt {11}}{2} \qquad \textbf{(B)} \ \frac {1 \plus{} i}{2} \qquad \textbf{(C)} \ \frac {1}{2} \plus{} i \qquad \textbf{(D)} \ 1 \plus{} \frac {i}{2} \qquad \textbf{(E)} \ \frac {1 \plus{} i \sqrt {13}}{2}$

1986 Poland - Second Round, 5

Tags: polynomial , algebra

Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .

1996 Baltic Way, 11

Tags: algebra , polynomial

Real numbers $x_1,x_2,\ldots ,x_{1996}$ have the following property: For any polynomial $W$ of degree $2$ at least three of the numbers $W(x_1),W(x_2),\ldots ,W(x_{1996})$ are equal. Prove that at least three of the numbers $x_1,x_2,\ldots ,x_{1996}$ are equal.

2009 Math Prize For Girls Problems, 2

Tags: algebra , polynomial , function

If $ a$, $ b$, $ c$, $ d$, and $ e$ are constants such that every $ x > 0$ satisfies \[ \frac{5x^4 \minus{} 8x^3 \plus{} 2x^2 \plus{} 4x \plus{} 7}{(x \plus{} 2)^4} \equal{} a \plus{} \frac{b}{x \plus{} 2} \plus{} \frac{c}{(x \plus{} 2)^2} \plus{} \frac{d}{(x \plus{} 2)^3} \plus{} \frac{e}{(x \plus{} 2)^4} \, ,\] then what is the value of $ a \plus{} b \plus{} c \plus{} d \plus{} e$?

2003 China Team Selection Test, 2

Tags: polynomial , algebra

Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$, \[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.

2000 IMC, 3

Tags: complex number , complex analysis , polynomial , algebra

Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.

2013 Brazil National Olympiad, 3

Tags: function , algebra , limit , polynomial , functional equation , induction

Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.

2020 HMIC, 4

Tags: catalan , polynomial , number theory , algebra

Let $C_k=\frac{1}{k+1}\binom{2k}{k}$ denote the $k^{\text{th}}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set \[\left\{\sum_{k=1}^{p-1}C_kn^k\,\middle\vert\, n\in\{1,2,\ldots,p-1\}\right\}\] are divisible by $p$. [i]Tristan Shin[/i]

2014 All-Russian Olympiad, 3

Tags: algebra , polynomial , derivative , calculus , function

If the polynomials $f(x)$ and $g(x)$ are written on a blackboard then we can also write down the polynomials $f(x)\pm g(x)$, $f(x)g(x)$, $f(g(x))$ and $cf(x)$, where $c$ is an arbitrary real constant. The polynomials $x^3-3x^2+5$ and $x^2-4x$ are written on the blackboard. Can we write a nonzero polynomial of form $x^n-1$ after a finite number of steps?

2024 China Team Selection Test, 15

Tags: polynomial , algebra

$n>1$ is an integer. Let real number $x>1$ satisfy $$x^{101}-nx^{100}+nx-1=0.$$ Prove that for any real $0<a<b<1$, there exists a positive integer $m$ so that $a<\{x^m\}<b.$ [i]Proposed by Chenjie Yu[/i]

2003 AIME Problems, 15

Tags: algebra , polynomial , function

Let \[P(x)=24x^{24}+\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). \] Let $z_{1},z_{2},\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\ldots,r,$ where $i=\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let \[\sum_{k=1}^{r}|b_{k}|=m+n\sqrt{p}, \] where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$

2003 Moldova Team Selection Test, 1

Tags: algebra , vieta , polynomial

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

VI Soros Olympiad 1999 - 2000 (Russia), 11.6

Tags: integer polynomial , polynomial , algebra

Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.

2014 ELMO Shortlist, 3

Tags: binomial theorem , polynomial , induction , algebra , number theory

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]