Olympiad problems

1995 Vietnam Team Selection Test, 2

Tags: rational root theorem , search , polynomial , algebra , modular arithmetic

Find all integers $ k$ such that for infinitely many integers $ n \ge 3$ the polynomial \[ P(x) =x^{n+ 1}+ kx^n - 870x^2 + 1945x + 1995\] can be reduced into two polynomials with integer coefficients.

1980 AMC 12/AHSME, 24

Tags: quadratic , rational root theorem , algebra , polynomial

For some real number $r$, the polynomial $8x^3-4x^2-42x+45$ is divisible by $(x-r)^2$. Which of the following numbers is closest to $r$? $\text{(A)} \ 1.22 \qquad \text{(B)} \ 1.32 \qquad \text{(C)} \ 1.42 \qquad \text{(D)} \ 1.52 \qquad \text{(E)} \ 1.62$

2011 AIME Problems, 9

Tags: logarithm , rational root theorem , algebra , trigonometry , polynomial

Suppose $x$ is in the interval $[0,\pi/2]$ and $\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}$. Find $24\cot^2{x}$.

1998 Vietnam Team Selection Test, 1

Tags: polynomial , algebra , rational root theorem , calculus , integration , number theory

Find all integer polynomials $P(x)$, the highest coefficent is 1 such that: there exist infinitely irrational numbers $a$ such that $p(a)$ is a positive integer.

2009 Princeton University Math Competition, 3

Tags: polynomial , rational root theorem , algebra

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

2015 AMC 10, 16

Tags: symmetry , vieta , rational root theorem , algebra , polynomial , 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} $

2016 Hanoi Open Mathematics Competitions, 9

Tags: rational root theorem , number theory , relatively prime , algebra

Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in Z$. Prove that there exist two relative prime numbers $m, n$ such that $abc =\frac{m^2}{n^3}$ .

2009 Princeton University Math Competition, 1

Tags: rational root theorem , algebra , polynomial

Find the root that the following three polynomials have in common: \begin{align*} & x^3+41x^2-49x-2009 \\ & x^3 + 5x^2-49x-245 \\ & x^3 + 39x^2 - 117x - 1435\end{align*}

2011 India National Olympiad, 3

Tags: polynomial , integration , rational root theorem , calculus , algebra , number theory

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ and $Q(x)=b_nx^n+b_{n-1}x^{n-1}+\cdots+b_0$ be two polynomials with integral coefficients such that $a_n-b_n$ is a prime and $a_nb_0-a_0b_n\neq 0,$ and $a_{n-1}=b_{n-1}.$ Suppose that there exists a rational number $r$ such that $P(r)=Q(r)=0.$ Prove that $r\in\mathbb Z.$

2000 Putnam, 6

Tags: function , polynomial , rational root theorem , modular arithmetic , induction , algebra

Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0, a_1, \cdots $ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n \ge 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$.

2010 AIME Problems, 6

Tags: rational root theorem , algebra , polynomial

Find the smallest positive integer $ n$ with the property that the polynomial $ x^4 \minus{} nx \plus{} 63$ can be written as a product of two nonconstant polynomials with integer coefficients.

2018 China Team Selection Test, 1

Tags: polynomial , rational root theorem , sequence , algebra

Define the polymonial sequence $\left \{ f_n\left ( x \right ) \right \}_{n\ge 1}$ with $f_1\left ( x \right )=1$, $$f_{2n}\left ( x \right )=xf_n\left ( x \right ), \; f_{2n+1}\left ( x \right ) = f_n\left ( x \right )+ f_{n+1} \left ( x \right ), \; n\ge 1.$$ Look for all the rational number $a$ which is a root of certain $f_n\left ( x \right ).$

2004 Putnam, B1

Tags: algebra , rational root theorem , number theory , relatively prime , polynomial

Let $P(x)=c_nx^n+c_{n-1}x^{n-1}+\cdots+c_0$ be a polynomial with integer coefficients. Suppose that $r$ is a rational number such that $P(r)=0$. Show that the $n$ numbers $c_nr, c_nr^2+c_{n-1}r, c_nr^3+c_{n-1}r^2+c_{n-1}r, \dots, c_nr^n+c_{n-1}r^{n-1}+\cdots+c_1r$ are all integers.

2002 AMC 12/AHSME, 23

Tags: algebra , quadratic , complex number , quadratic formula , rational root theorem , polynomial

The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a$. $\textbf{(A) }\sqrt{118}\qquad\textbf{(B) }\sqrt{210}\qquad\textbf{(C) }2\sqrt{210}\qquad\textbf{(D) }\sqrt{2002}\qquad\textbf{(E) }100\sqrt2$

1991 India National Olympiad, 7

Tags: quadratic , complex number , rational root theorem , system of equations , polynomial , algebra

Solve the following system for real $x,y,z$ \[ \{ \begin{array}{ccc} x+ y -z & =& 4 \\ x^2 - y^2 + z^2 & = & -4 \\ xyz & =& 6. \end{array} \]

1967 AMC 12/AHSME, 16

Tags: rational root theorem , algebra , polynomial

Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is $\textbf{(A)}\ 43\qquad \textbf{(B)}\ 44\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 46\qquad \textbf{(E)}\ 47$

1995 Vietnam Team Selection Test, 2

1980 AMC 12/AHSME, 24

2011 AIME Problems, 9

1998 Vietnam Team Selection Test, 1

2009 Princeton University Math Competition, 3

2015 AMC 10, 16

2016 Hanoi Open Mathematics Competitions, 9

2009 Princeton University Math Competition, 1

2011 India National Olympiad, 3

2000 Putnam, 6

2010 AIME Problems, 6

2018 China Team Selection Test, 1

2004 Putnam, B1

2002 AMC 12/AHSME, 23

1991 India National Olympiad, 7

1967 AMC 12/AHSME, 16

1994 Polish MO Finals, 1

2006 Turkey MO (2nd round), 3

III Soros Olympiad 1996 - 97 (Russia), 9.1

1996 Canadian Open Math Challenge, 1

2006 Pan African, 2

1987 India National Olympiad, 6

PEN Q Problems, 7

2007 Princeton University Math Competition, 10

2011 Putnam, B2