This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 1148

1959 AMC 12/AHSME, 16
Tags: quadratic , factoring quadratics , algebra
The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is: $ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$

2007 Stanford Mathematics Tournament, 9
Tags: quadratic
Find $a^2+b^2$ given that $a, b$ are real and satisfy \[a=b+\frac{1}{a+\frac{1}{b+\frac{1}{a+\cdots}}}; b=a-\frac{1}{b+\frac{1}{a-\frac{1}{b+\cdots}}}\]

1956 AMC 12/AHSME, 7
Tags: quadratic
The roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ will be reciprocal if: $ \textbf{(A)}\ a \equal{} b \qquad\textbf{(B)}\ a \equal{} bc \qquad\textbf{(C)}\ c \equal{} a \qquad\textbf{(D)}\ c \equal{} b \qquad\textbf{(E)}\ c \equal{} ab$

1950 AMC 12/AHSME, 13
Tags: quadratic
The roots of $ (x^2\minus{}3x\plus{}2)(x)(x\minus{}4)\equal{}0$ are: $\textbf{(A)}\ 4\qquad \textbf{(B)}\ 0\text{ and }4 \qquad \textbf{(C)}\ 1\text{ and }2 \qquad \textbf{(D)}\ 0,1,2\text{ and }4\qquad \textbf{(E)}\ 1,2\text{ and }4$

1960 AMC 12/AHSME, 39
Tags: quadratic , algebra , quadratic formula
To satisfy the equation $\frac{a+b}{a}=\frac{b}{a+b}$, $a$ and $b$ must be: $ \textbf{(A)}\ \text{both rational} \qquad\textbf{(B)}\ \text{both real but not rational} \qquad\textbf{(C)}\ \text{both not real}\qquad$ $\textbf{(D)}\ \text{one real, one not real}\qquad\textbf{(E)}\ \text{one real, one not real or both not real} $

2017 Purple Comet Problems, 6
Tags: algebra , polynomial , quadratic
For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.

2007 All-Russian Olympiad Regional Round, 9.1
Tags: quadratic , polynomial , geometry , geometric transformation , algebra
Pete chooses $ 1004$ monic quadratic polynomial $ f_{1},\cdots,f_{1004}$, such that each integer from $ 0$ to $ 2007$ is a root of at least one of them. Vasya considers all equations of the form $ f_{i}\equal{}f_{j}(i\not \equal{}j)$ and computes their roots; for each such root , Pete has to pay to Vasya $ 1$ ruble . Find the least possible value of Vasya's income.

2012 AMC 10, 17
Tags: quadratic , number theory , relatively prime , algebra , quadratic formula
Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\tfrac{a^3-b^3}{(a-b)^3}=\tfrac{73}{3}$. What is $a-b$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5 $

1961 AMC 12/AHSME, 22
Tags: algebra , polynomial , quadratic
If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $

2010 India IMO Training Camp, 2
Tags: polynomial , quadratic , algebra
Two polynomials $P(x)=x^4+ax^3+bx^2+cx+d$ and $Q(x)=x^2+px+q$ have real coefficients, and $I$ is an interval on the real line of length greater than $2$. Suppose $P(x)$ and $Q(x)$ take negative values on $I$, and they take non-negative values outside $I$. Prove that there exists a real number $x_0$ such that $P(x_0)<Q(x_0)$.

2014 Contests, 2
Tags: quadratic residue , quadratic , induction , modular arithmetic , number theory
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

1988 IberoAmerican, 5
Tags: quadratic , system of equations , algebra
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that \[(x+yt+z^2)(u+vt+wt^2)=1\]

2007 China Team Selection Test, 1
Tags: trigonometry , quadratic , function , inequalities
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$ prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$

1969 AMC 12/AHSME, 14
Tags: quadratic , inequalities
The complete set of $x$-values satisfying the inequality $\dfrac{x^2-4}{x^2-1}>0$ is the set of all $x$ such that: $\textbf{(A) }x>2\text{ or }x<-2\text{ or }-1<x<1\qquad\, \textbf{(B) }x>2\text{ or }x<-2$ $\textbf{(C) }x>1\text{ or }x<-2\qquad\qquad\qquad\qquad\,\,\,\,\,\,\, \textbf{(D) }x>1\text{ or }x<-2\qquad$ $\textbf{(E) }x\text{ is any real number except }1\text{ or }-1$

1986 AIME Problems, 1
Tags: quadratic , algebra , polynomial , sum of roots
What is the sum of the solutions to the equation $\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}$?

2010 Polish MO Finals, 2
Tags: modular arithmetic , quadratic , trigonometry , number theory
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.

2004 Harvard-MIT Mathematics Tournament, 5
Tags: trigonometry , quadratic
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.

2010 CHKMO, 4
Tags: quadratic , number theory
Find all non-negative integers $ m$ and $ n$ that satisfy the equation: \[ 107^{56}(m^2\minus{}1)\minus{}2m\plus{}5\equal{}3\binom{113^{114}}{n}\] (If $ n$ and $ r$ are non-negative integers satisfying $ r\le n$, then $ \binom{n}{r}\equal{}\frac{n}{r!(n\minus{}r)!}$ and $ \binom{n}{r}\equal{}0$ if $ r>n$.)

2007 Mediterranean Mathematics Olympiad, 3
Tags: inradius , trigonometry , quadratic , geometry
In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

2013 Indonesia MO, 5
Tags: polynomial , quadratic , algebra
Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that: - $P(x) = P_1(x) + P_2(x) + P_3(x)$ - $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root) - The roots of $P_1, P_2, P_3$ are different

2011 India Regional Mathematical Olympiad, 3
Tags: quadratic
A natural number $n$ is chosen strictly between two consecutive perfect squares. The smaller of these two squares is obtained by subtracting $k$ from $n$ and the larger by adding $l$ to $n.$ Prove that $n-kl$ is a perfect square.

1970 AMC 12/AHSME, 14
Tags: quadratic
Consider $x^2+px+q=0$ where $p$ and $q$ are positive numbers. If the roots of this equation differ by $1$, then $p$ equals $\textbf{(A) }\sqrt{4q+1}\qquad\textbf{(B) }q-1\qquad\textbf{(C) }-\sqrt{4q+1}\qquad\textbf{(D) }q+1\qquad \textbf{(E) }\sqrt{4q-1}$

2003 China Team Selection Test, 2
Tags: modular arithmetic , quadratic , number theory
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2001 All-Russian Olympiad, 1
Tags: algebra , polynomial , quadratic , quadratic formula
The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.

2012 Bulgaria National Olympiad, 2
Tags: quadratic , function , algebra
Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.