Olympiad problems

1981 Putnam, A5

Tags: polynomial , root

Let $P(x)$ be a polynomial with real coefficients and form the polynomial $$Q(x) = ( x^2 +1) P(x)P'(x) + x(P(x)^2 + P'(x)^2 ).$$ Given that the equation $P(x) = 0$ has $n$ distinct real roots exceeding $1$, prove or disprove that the equation $Q(x)=0$ has at least $2n - 1$ distinct real roots.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: algebra , vieta , polynomial

The equation $ x^3 \minus{} 9x^2 \plus{} 8x \plus{} 2 \equal{} 0$ has three real roots $ p$, $ q$, $ r$. Find $ \frac {1}{p^2} \plus{} \frac {1}{q^2} \plus{} \frac {1}{r^2}$.

2012-2013 SDML (High School), 13

Tags: polynomial , algebra

A polynomial $P$ is called [i]level[/i] if it has integer coefficients and satisfies $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$. What is the largest positive integer $d$ such that for any level polynomial $P$, $d$ is a divisor of $P\left(n\right)$ for all integers $n$? $\text{(A) }1\qquad\text{(B) }2\qquad\text{(C) }3\qquad\text{(D) }6\qquad\text{(E) }10$

1998 Greece JBMO TST, 4

Tags: polynomial , logarithm , algebra

(a) A polynomial $P(x)$ with integer coefficients takes the value $-2$ for at least seven distinct integers $x$. Prove that it cannot take the value $1996$. (b) Prove that there are irrational numbers $x,y$ such that $x^y$ is rational.

1997 AIME Problems, 15

Tags: rectangle , trigonometry , algebra , polynomial , geometry

The sides of rectangle $ABCD$ have lengths 10 and 11. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD.$ The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r,$ where $p, q,$ and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r.$

2022 239 Open Mathematical Olympiad, 1

Tags: algebra , polynomial , game

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial \[a_{99}x^{99} + \cdots + a_1x + a_0\]with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

2024 Indonesia TST, A

Tags: algebra , polynomial

Find all second degree polynomials $P(x)$ such that for all $a \in\mathbb{R} , a \geq 1$, then $P(a^2+a) \geq a.P(a+1)$

2012 Bosnia Herzegovina Team Selection Test, 4

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

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

2012 Dutch BxMO/EGMO TST, 1

Tags: algebra , polynomial , trinomial , quadratic , root

Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?

2018 PUMaC Live Round, 7.1

Tags: algebra , polynomial

Find the number of nonzero terms of the polynomial $P(x)$ if $$x^{2018}+x^{2017}+x^{2016}+x^{999}+1=(x^4+x^3+x^2+x+1)P(x).$$

2018 CHMMC (Fall), 4

Tags: polynomial , algebra

Find the sum of the real roots of $f(x) = x^4 + 9x^3 + 18x^2 + 18x + 4$.

1999 China National Olympiad, 2

Tags: polynomial , inequalities , algebra

Determine the maximum value of $\lambda$ such that if $f(x) = x^3 +ax^2 +bx+c$ is a cubic polynomial with all its roots nonnegative, then \[f(x)\geq\lambda(x -a)^3\] for all $x\geq0$. Find the equality condition.

2008 Hanoi Open Mathematics Competitions, 5

Tags: polynomial , algebra

Find all polynomials $P(x)$ of degree $1$ such that $\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$

2001 India IMO Training Camp, 1

Tags: algebra , polynomial , complex number , number theory

Complex numbers $\alpha$ , $\beta$ , $\gamma$ have the property that $\alpha^k +\beta^k +\gamma^k$ is an integer for every natural number $k$. Prove that the polynomial \[(x-\alpha)(x-\beta )(x-\gamma )\] has integer coefficients.

1951 Moscow Mathematical Olympiad, 202

Tags: polynomial , coefficient , algebra

Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.

1997 Irish Math Olympiad, 3

Tags: algebra , polynomial

Find all polynomials $ p(x)$ satisfying the equation: $ (x\minus{}16)p(2x)\equal{}16(x\minus{}1)p(x)$ for all $ x$.

2012 Belarus Team Selection Test, 2

Tags: algebra , circumcircle , polynomial , quadratic , complex number , geometry

Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that \[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\] [i]Proposed by Alexey Gladkich, Israel[/i]

2006 Stanford Mathematics Tournament, 4

Tags: algebra , polynomial

Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial?

PEN D Problems, 18

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

Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.

2005 ISI B.Stat Entrance Exam, 2

Tags: algebra , polynomial , integration , function , derivative , calculus

Let \[f(x)=\int_0^1 |t-x|t \, dt\] for all real $x$. Sketch the graph of $f(x)$. What is the minimum value of $f(x)$?

2011 AMC 12/AHSME, 23

Tags: inequalities , linear algebra , algebra , polynomial , function , matrix , induction

Let $f(z)=\frac{z+a}{z+b}$ and $g(z)=f(f(z))$, where $a$ and $b$ are complex numbers. Suppose that $|a|=1$ and $g(g(z))=z$ for all $z$ for which $g(g(z))$ is defined. What is the difference between the largest and smallest possible values of $|b|$? $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ \sqrt{2}-1 \qquad \textbf{(C)}\ \sqrt{3}-1 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

2020 Indonesia MO, 7

Tags: functional equation , polynomial , floor function , algebra

Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.

2007 Indonesia TST, 3

Tags: function , induction , number theory , polynomial , algebra

Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

2012 Today's Calculation Of Integral, 786

Tags: integration , limit , algebra , probability , calculus , induction , polynomial

For each positive integer $n$, define $H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}.$ (1) Find $H_1(x),\ H_2(x),\ H_3(x)$. (2) Express $\frac{d}{dx}H_n(x)$ interms of $H_n(x),\ H_{n+1}(x).$ Then prove that $H_n(x)$ is a polynpmial with degree $n$ by induction. (3) Let $a$ be real number. For $n\geq 3$, express $S_n(a)=\int_0^a xH_n(x)e^{-x^2}dx$ in terms of $H_{n-1}(a),\ H_{n-2}(a),\ H_{n-2}(0)$. (4) Find $\lim_{a\to\infty} S_6(a)$. If necessary, you may use $\lim_{x\to\infty}x^ke^{-x^2}=0$ for a positive integer $k$.

2021 Indonesia TST, A

Tags: algebra , polynomial

Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.