Olympiad problems

1955 AMC 12/AHSME, 18

The discriminant of the equation $ x^2\plus{}2x\sqrt{3}\plus{}3\equal{}0$ is zero. Hence, its roots are: $ \textbf{(A)}\ \text{real and equal} \qquad \textbf{(B)}\ \text{rational and equal} \qquad \textbf{(C)}\ \text{rational and unequal} \\ \textbf{(D)}\ \text{irrational and unequal} \qquad \textbf{(E)}\ \text{imaginary}$

1994 AMC 12/AHSME, 20

Tags: ratio , quadratics , geometric sequence , arithmetic sequence , AMC

Suppose $x,y,z$ is a geometric sequence with common ratio $r$ and $x \neq y$. If $x, 2y, 3z$ is an arithmetic sequence, then $r$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4$

1988 IMO Longlists, 45

Tags: quadratics , number theory unsolved , number theory

Let $g(n)$ be defined as follows: \[ g(1) = 0, g(2) = 1 \] and \[ g(n+2) = g(n) + g(n+1) + 1, n \geq 1. \] Prove that if $n > 5$ is a prime, then $n$ divides $g(n) \cdot (g(n) + 1).$

2014 AIME Problems, 9

Tags: algebra , polynomial , Vieta , AMC , AIME , quadratics , trigonometry

Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

2011 All-Russian Olympiad, 1

Tags: algebra , polynomial , quadratics , function , algebra proposed

Given are two distinct monic cubics $F(x)$ and $G(x)$. All roots of the equations $F(x)=0$, $G(x)=0$ and $F(x)=G(x)$ are written down. There are eight numbers written. Prove that the greatest of them and the least of them cannot be both roots of the polynomial $F(x)$.

2013 APMO, 2

Tags: modular arithmetic , inequalities , quadratics , function , algebra , floor function , Hi

Determine all positive integers $n$ for which $\dfrac{n^2+1}{[\sqrt{n}]^2+2}$ is an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

2006 Czech and Slovak Olympiad III A, 2

Tags: quadratics , algebra unsolved , algebra

Let $m,n$ be positive integers such that the equation (in respect of $x$) \[(x+m)(x+n)=x+m+n\] has at least one integer root. Prove that $\frac{1}{2}n<m<2n$.

1973 Putnam, B5

Tags: Putnam , quadratics , algebra , rational function , function

(a) Let $z$ be a solution of the quadratic equation $$az^2 +bz+c=0$$ and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$ (b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$

MathLinks Contest 7th, 7.1

Tags: modular arithmetic , quadratics , Vieta , number theory , relatively prime , algebra

Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

2002 Turkey MO (2nd round), 1

Tags: modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved

Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition \[y^2 \equiv x^3 - x \pmod p\] is exactly $p.$

2002 Greece National Olympiad, 1

Tags: inequalities , quadratics , analytic geometry , inequalities proposed

The real numbers $a,b,c$ with $bc\neq0$ satisfy $\frac{1-c^2}{bc}\geq0.$ Prove that $10(a^2+b^2+c^2-bc^3)\geq2ab+5ac.$

1951 AMC 12/AHSME, 47

Tags: quadratics

If $ r$ and $ s$ are the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$, the value of $ \frac {1}{r^2} \plus{} \frac {1}{s^2}$ is: $ \textbf{(A)}\ b^2 \minus{} 4ac \qquad\textbf{(B)}\ \frac {b^2 \minus{} 4ac}{2a} \qquad\textbf{(C)}\ \frac {b^2 \minus{} 4ac}{c^2} \qquad\textbf{(D)}\ \frac {b^2 \minus{} 2ac}{c^2}$ $ \textbf{(E)}\ \text{none of these}$

1988 IMO Longlists, 9

Tags: limit , quadratics , algebra unsolved , algebra

If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by: \[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \] Show that there exist a real number $a > 0$ such that: [b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$ [b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$

2012 ELMO Shortlist, 5

Tags: quadratics , modular arithmetic , combinatorics proposed , combinatorics

Form the infinite graph $A$ by taking the set of primes $p$ congruent to $1\pmod{4}$, and connecting $p$ and $q$ if they are quadratic residues modulo each other. Do the same for a graph $B$ with the primes $1\pmod{8}$. Show $A$ and $B$ are isomorphic to each other. [i]Linus Hamilton.[/i]

2001 All-Russian Olympiad, 1

Tags: quadratics , algebra , polynomial , algebra unsolved

Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals. Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.

2007 Serbia National Math Olympiad, 3

Tags: modular arithmetic , inequalities , quadratics , number theory , greatest common divisor , number theory proposed

Determine all pairs of natural numbers $(x; n)$ that satisfy the equation \[x^{3}+2x+1 = 2^{n}.\]

2003 China Team Selection Test, 2

Tags: quadratics , modular arithmetic , combinatorics unsolved , combinatorics

Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality.

2010 South East Mathematical Olympiad, 1

Tags: quadratics , algebra , number theory unsolved , number theory

Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.

2005 Harvard-MIT Mathematics Tournament, 10

Tags: quadratics , algebra , quadratic formula

Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.

2012 USA TSTST, 6

Tags: inequalities , algebra , polynomial , Vieta , quadratics , USA TSTST , Hi

Positive real numbers $x, y, z$ satisfy $xyz+xy+yz+zx = x+y+z+1$. Prove that \[ \frac{1}{3} \left( \sqrt{\frac{1+x^2}{1+x}} + \sqrt{\frac{1+y^2}{1+y}} + \sqrt{\frac{1+z^2}{1+z}} \right) \le \left( \frac{x+y+z}{3} \right)^{5/8} . \]

PEN G Problems, 29

Tags: algebra , polynomial , quadratics , Irrational numbers

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?

2012 Today's Calculation Of Integral, 798

Tags: calculus , integration , function , geometry , derivative , quadratics , analytic geometry

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

1986 China Team Selection Test, 3

Tags: inequalities , quadratics , inequalities unsolved

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

Oliforum Contest IV 2013, 2

Tags: geometry , circumcircle , quadratics , algebra , quadratic formula , power of a point , geometry proposed

Given an acute angled triangle $ABC$ with $M$ being the mid-point of $AB$ and $P$ and $Q$ are the feet of heights from $A$ to $BC$ and $B$ to $AC$ respectively. Show that if the line $AC$ is tangent to the circumcircle of $BMP$ then the line $BC$ is tangent to the circumcircle of $AMQ$.

1988 Romania Team Selection Test, 13

Tags: quadratics , algebra proposed , algebra

Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$, $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$. Prove that $(y,x)$ is a solution of the equation \[ |y^2 - axy - x^2 | = 1 \] if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$. [i]Serban Buzeteanu[/i]