Olympiad problems

2013 AMC 12/AHSME, 6

Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$? $ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $

1988 China Team Selection Test, 1

Tags: quadratic , inequalities

Suppose real numbers $A,B,C$ such that for all real numbers $x,y,z$ the following inequality holds: \[A(x-y)(x-z) + B(y-z)(y-x) + C(z-x)(z-y) \geq 0.\] Find the necessary and sufficient condition $A,B,C$ must satisfy (expressed by means of an equality or an inequality).

2004 USA Team Selection Test, 1

Tags: inequalities , quadratic , algebra , polynomial , trigonometry , geometry , parallelogram

Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.

1993 AMC 12/AHSME, 20

Tags: quadratic , complex number , algebra , quadratic formula

Consider the equation $10z^2-3iz-k=0$, where $z$ is a complex variable and $i^2=-1$. Which of the following statements is true? $ \textbf{(A)}\ \text{For all positive real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(B)}\ \text{For all negative real numbers}\ k,\ \text{both roots are pure imaginary.} \\ \qquad\textbf{(C)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and rational.} \\ \qquad\textbf{(D)}\ \text{For all pure imaginary numbers}\ k,\ \text{both roots are real and irrational.} \\ \qquad\textbf{(E)}\ \text{For all complex numbers}\ k,\ \text{neither root is real.} $

2014 Singapore Senior Math Olympiad, 21

Tags: trigonometry , quadratic , vieta , algebra

Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$

2003 Purple Comet Problems, 13

Tags: algebra , polynomial , function , quadratic

Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.

2005 Korea - Final Round, 2

Tags: induction , quadratic , inequalities

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , \[\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}. \]

2006 USAMO, 4

Tags: quadratic , algebra

Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$

1959 IMO, 3

Tags: trigonometric equation , quadratic , trigonometry , algebra

Let $a,b,c$ be real numbers. Consider the quadratic equation in $\cos{x}$ \[ a \cos^2{x}+b \cos{x}+c=0. \] Using the numbers $a,b,c$ form a quadratic equation in $\cos{2x}$ whose roots are the same as those of the original equation. Compare the equation in $\cos{x}$ and $\cos{2x}$ for $a=4$, $b=2$, $c=-1$.

2015 India National Olympiad, 6

Tags: quadratic , modular arithmetic , pigeonhole principle , number theory

Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.

2010 Stanford Mathematics Tournament, 19

Tags: algebra , polynomial , quadratic , quadratic formula

Find the roots of $6x^4+17x^3+7x^2-8x-4$

2006 China Second Round Olympiad, 7

Tags: trigonometry , quadratic

Let $f(x)=\sin^4x-\sin x\cos x+cos^4 x$. Find the range of $f(x)$.

2008 Harvard-MIT Mathematics Tournament, 1

Tags: quadratic

Determine all pairs $ (a,b)$ of real numbers such that $ 10, a, b, ab$ is an arithmetic progression.

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}$

2007 German National Olympiad, 6

Tags: inequalities , calculus , derivative , quadratic , algebra

For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$

2011 Math Prize For Girls Problems, 2

Tags: quadratic

Express $\sqrt{2 + \sqrt{3}}$ in the form $\frac{a + \sqrt{b}}{\sqrt{c}}$, where $a$ is a positive integer and $b$ and $c$ are square-free positive integers.

2011 USA TSTST, 6

Tags: function , quadratic , inequalities

Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that \[ 1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 + \frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}. \]

PEN A Problems, 8

Tags: quadratic , search , modular arithmetic , algebra , divisibility theory

The integers $a$ and $b$ have the property that for every nonnegative integer $n$ the number of $2^n{a}+b$ is the square of an integer. Show that $a=0$.

2005 IberoAmerican, 1

Tags: quadratic , algebra

Determine all triples of real numbers $(a,b,c)$ such that \begin{eqnarray*} xyz &=& 8 \\ x^2y + y^2z + z^2x &=& 73 \\ x(y-z)^2 + y(z-x)^2 + z(x-y)^2 &=& 98 . \end{eqnarray*}

1962 AMC 12/AHSME, 21

Tags: quadratic

It is given that one root of $ 2x^2 \plus{} rx \plus{} s \equal{} 0$, with $ r$ and $ s$ real numbers, is $ 3\plus{}2i (i \equal{} \sqrt{\minus{}1})$. The value of $ s$ is: $ \textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \minus{}13 \qquad \textbf{(E)}\ 26$

2012 AIME Problems, 8

Tags: quadratic , complex number , algebra , quadratic formula

The complex numbers $z$ and $w$ satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of $|zw|^2$.

2010 All-Russian Olympiad, 1

Tags: algebra , induction , topology , geometry , 3d geometry , sphere , quadratic

Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.

2024 All-Russian Olympiad Regional Round, 9.2

Tags: quadratic , function graphing , analytic geometry , parabola , trapezoid , conic , geometry

On a cartesian plane a parabola $y = x^2$ is drawn. For a given $k > 0$ we consider all trapezoids inscribed into this parabola with bases parallel to the x-axis, and the product of the lengths of their bases is exactly $k$. Prove that the diagonals of all such trapezoids share a common point.

1963 AMC 12/AHSME, 24

Tags: quadratic

Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected from the set of integers $\{1,2,3, 4, 5,6\}$? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$

1975 AMC 12/AHSME, 22

Tags: calculus , integration , quadratic

If $ p$ and $ q$ are primes and $ x^2 \minus{} px \plus{} q \equal{} 0$ has distinct positive integral roots, then which of the following statements are true? $ \text{I. The difference of the roots is odd.}$ $ \text{II. At least one root is prime.}$ $ \text{III. } p^2 \minus{} q \text{ is prime.}$ $ \text{IV. } p \plus{} q \text{ is prime.}$ $ \textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad$ $ \textbf{(D)}\ \text{I, II and IV only} \qquad \textbf{(E)}\ \text{All are true.}$