Olympiad problems

1984 Putnam, A3

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by $m_{ij}=x$ if $i=j$, $m_{ij}=a$ if $i \not= j$ and $i+j$ is even, $m_{ij}=b$ if $i \not= j$ and $i+j$ is odd. For example $ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}$. Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ . P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)

1953 AMC 12/AHSME, 44

Tags: quadratic , algebra , polynomial

In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $ 8$ and $ 2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $ \minus{}9$ and $ \minus{}1$ for the roots. The correct equation was: $ \textbf{(A)}\ x^2\minus{}10x\plus{}9\equal{}0 \qquad\textbf{(B)}\ x^2\plus{}10x\plus{}9\equal{}0 \qquad\textbf{(C)}\ x^2\minus{}10x\plus{}16\equal{}0\\ \textbf{(D)}\ x^2\minus{}8x\minus{}9\equal{}0 \qquad\textbf{(E)}\ \text{none of these}$

1948 Putnam, B1

Tags: polynomial , ratio , algebra

Let $f(x)$ be a cubic polynomial with roots $x_1 , x_2$ and $x_3.$ Assume that $f(2x)$ is divisible by $f'(x)$ and compute the ratio $x_1 : x_2: x_3 .$

1955 Moscow Mathematical Olympiad, 314

Tags: root , algebra , polynomial

Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.

2021 Philippine MO, 4

Tags: polynomial , number theory , algebra

Determine the set of all polynomials $P(x)$ with real coefficients such that the set $\{P(n) | n \in \mathbb{Z}\}$ contains all integers, except possibly finitely many of them.

1982 Tournament Of Towns, (022) 4

Tags: algebra , polynomial

A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.

2009 USA Team Selection Test, 5

Tags: algebra , polynomial , quadratic , vieta , inequalities , number theory

Find all pairs of positive integers $ (m,n)$ such that $ mn \minus{} 1$ divides $ (n^2 \minus{} n \plus{} 1)^2$. [i]Aaron Pixton.[/i]

2017 China Team Selection Test, 6

Tags: polynomial , algebra , number theory , prime number

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

1947 Moscow Mathematical Olympiad, 130

Tags: coefficient , algebra , polynomial

Which of the polynomials, $(1+x^2 -x^3)^{1000}$ or $(1-x^2 +x^3)^{1000}$, has the greater coefficient of $x^{20}$ after expansion and collecting the terms?

1999 China National Olympiad, 2

Tags: function , polynomial , induction , algebra

Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$. a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$. b) Find an explicit expression for $f_n(x)$.

2009 Indonesia TST, 1

Tags: combinatorics , algebra , search , polynomial

Let $ n \ge 1$ and $ k \ge 3$ be integers. A circle is divided into $ n$ sectors $ a_1,a_2,\dots,a_n$. We will color the $ n$ sectors with $ k$ different colors such that $ a_i$ and $ a_{i \plus{} 1}$ have different color for each $ i \equal{} 1,2,\dots,n$ where $ a_{n \plus{} 1}\equal{}a_1$. Find the number of ways to do such coloring.

2005 Purple Comet Problems, 25

Tags: algebra , polynomial

Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

2022 Federal Competition For Advanced Students, P2, 4

Tags: integer polynomial , number theory , polynomial

Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$. [i](Walther Janous)[/i]

1987 Spain Mathematical Olympiad, 6

Tags: polynomial , algebra , limit , polynomial equation , real root

For all natural numbers $n$, consider the polynomial $P_n(x) = x^{n+2}-2x+1$. (a) Show that the equation $P_n(x)=0$ has exactly one root $c_n$ in the open interval $(0,1)$. (b) Find $lim_{n \to \infty}c_n$.

2007 CentroAmerican, 3

Tags: quadratic , polynomial , algebra

Let $S$ be a finite set of integers. Suppose that for every two different elements of $S$, $p$ and $q$, there exist not necessarily distinct integers $a \neq 0$, $b$, $c$ belonging to $S$, such that $p$ and $q$ are the roots of the polynomial $ax^{2}+bx+c$. Determine the maximum number of elements that $S$ can have.

2023 ELMO Shortlist, A1

Tags: polynomial , functional equation , algebra

Find all polynomials $P(x)$ with real coefficients such that for all nonzero real numbers $x$, \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

2010 Today's Calculation Of Integral, 549

Tags: limit , calculus , polynomial , calculus computations , function , integration , algebra

Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.

2024 Bangladesh Mathematical Olympiad, P6

Tags: algebra , substitution , polynomial

Find all polynomials $P(x)$ for which there exists a sequence $a_1, a_2, a_3, \ldots$ of real numbers such that \[a_m + a_n = P(mn)\] for any positive integer $m$ and $n$.

2012 Romania National Olympiad, 4

Tags: superior algebra , algebra , polynomial

[color=darkred] Let $m$ and $n$ be two nonzero natural numbers. Determine the minimum number of distinct complex roots of the polynomial $\prod_{k=1}^m\, (f+k)$ , when $f$ covers the set of $n^{\text{th}}$ - degree polynomials with complex coefficients. [/color]

2017 239 Open Mathematical Olympiad, 4

Tags: polynomial , algebra

A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)

1976 IMO Shortlist, 8

Tags: algebra , coefficient , polynomial

Let $P$ be a polynomial with real coefficients such that $P(x) > 0$ if $x > 0$. Prove that there exist polynomials $Q$ and $R$ with nonnegative coefficients such that $P(x) = \frac{Q(x)}{R(x)}$ if $x > 0.$

2024 Vietnam Team Selection Test, 1

Tags: function , functional equation , polynomial , algebra

Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

2004 Bulgaria Team Selection Test, 1

Tags: algebra , number theory , induction , polynomial

Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.

2013 Peru IMO TST, 2

Tags: absolute value , algebra , polynomial

Let $a \geq 3$ be a real number, and $P$ a polynomial of degree $n$ and having real coefficients. Prove that at least one of the following numbers is greater than or equal to $1:$ $$|a^0- P(0)|, \ |a^1-P(1)| , \ |a^2-P(2)|, \cdots, |a^{n + 1}-P(n + 1)|.$$

1988 Swedish Mathematical Competition, 4

Tags: analysis , algebra , polynomial

A polynomial $P(x)$ of degree $3$ has three distinct real roots. Find the number of real roots of the equation $P'(x)^2 -2P(x)P''(x) = 0$.