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: 73

1981 AMC 12/AHSME, 14
Tags: ratio , geometric sequence , geometric series , algebra , difference of squares , special factorizations
In a geometric sequence of real numbers, the sum of the first two terms is 7, and the sum of the first 6 terms is 91. The sum of the first 4 terms is $\text{(A)}\ 28 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 49 \qquad \text{(E)}\ 84$

2002 AMC 10, 14
Tags: quadratic , vieta , algebra , number theory , prime number , special factorizations
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

2004 Harvard-MIT Mathematics Tournament, 10
Tags: algebra , polynomial , difference of squares , special factorizations
There exists a polynomial $P$ of degree $5$ with the following property: if $z$ is a complex number such that $z^5+2004z=1$, then $P(z^2)=0$. Calculate the quotient $\tfrac{P(1)}{P(-1)}$.

1985 AMC 12/AHSME, 19
Tags: function , quadratic , special factorizations
Consider the graphs $ y \equal{} Ax^2$ and and $ y^2 \plus{} 3 \equal{} x^2 \plus{} 4y$, where $ A$ is a positive constant and $ x$ and $ y$ are real variables. In how many points do the two graphs intersect? $ \textbf{(A)}\ \text{exactly } 4 \qquad \textbf{(B)}\ \text{exactly } 2$ $ \textbf{(C)}\ \text{at least } 1, \text{ but the number varies for different positive values of } A$ $ \textbf{(D)}\ 0 \text{ for at least one positive value of } A \qquad \textbf{(E)}\ \text{none of these}$

2009 Princeton University Math Competition, 1
Tags: special factorizations
Find the number of pairs of integers $x$ and $y$ such that $x^2 + xy + y^2 = 28$.

2010 ELMO Shortlist, 2
Tags: modular arithmetic , algebra , difference of squares , special factorizations , number theory
Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

2004 AMC 12/AHSME, 13
Tags: function , algebra , difference of squares , special factorizations
If $ f(x) \equal{} ax \plus{} b$ and $ f^{ \minus{} 1}(x) \equal{} bx \plus{} a$ with $ a$ and $ b$ real, what is the value of $ a \plus{} b$? $ \textbf{(A)} \minus{} \!2 \qquad \textbf{(B)} \minus{} \!1 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 2$

2007 Harvard-MIT Mathematics Tournament, 3
Tags: trigonometry , special factorizations
The equation $x^2+2x=i$ has two complex solutions. Determine the product of their real parts.

1995 All-Russian Olympiad, 5
Tags: algebra , difference of squares , special factorizations , number theory
Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$. [i]A. Golovanov[/i]

1987 AIME Problems, 5
Tags: sfft , special factorizations
Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.

2010 AIME Problems, 5
Tags: sfft , special factorizations , logarithm
Positive numbers $ x$, $ y$, and $ z$ satisfy $ xyz \equal{} 10^{81}$ and $ (\log_{10}x)(\log_{10} yz) \plus{} (\log_{10}y) (\log_{10}z) \equal{} 468$. Find $ \sqrt {(\log_{10}x)^2 \plus{} (\log_{10}y)^2 \plus{} (\log_{10}z)^2}$.

2002 AIME Problems, 4
Tags: sfft , special factorizations
Consider the sequence defined by $a_k=\frac 1{k^2+k}$ for $k\ge 1.$ Given that $a_m+a_{m+1}+\cdots+a_{n-1}=1/29,$ for positive integers $m$ and $n$ with $m<n$, find $m+n.$

2004 Iran Team Selection Test, 1
Tags: modular arithmetic , quadratic , special factorizations , number theory
Suppose that $ p$ is a prime number. Prove that for each $ k$, there exists an $ n$ such that: \[ \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)\]

2011 AMC 12/AHSME, 21
Tags: quadratic , algebra , difference of squares , special factorizations
The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is $|x-y|$? $ \textbf{(A)}\ 24 \qquad \textbf{(B)}\ 48 \qquad \textbf{(C)}\ 54 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 70 $

2013 AMC 10, 11
Tags: quadratic , special factorizations , algebra , quadratic formula
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 $

1981 AMC 12/AHSME, 21
Tags: trigonometry , algebra , difference of squares , special factorizations , geometry
In a triangle with sides of lengths $a,b,$ and $c,$ $(a+b+c)(a+b-c) = 3ab.$ The measure of the angle opposite the side length $c$ is $\displaystyle \text{(A)} \ 15^\circ \qquad \text{(B)} \ 30^\circ \qquad \text{(C)} \ 45^\circ \qquad \text{(D)} \ 60^\circ \qquad \text{(E)} \ 150^\circ$

1999 AIME Problems, 3
Tags: algebra , polynomial , vieta , quadratic , special factorizations
Find the sum of all positive integers $n$ for which $n^2-19n+99$ is a perfect square.

2010 Purple Comet Problems, 24
Tags: sfft , special factorizations
Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.

2002 AMC 12/AHSME, 12
Tags: quadratic , vieta , algebra , number theory , prime number , special factorizations
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

1999 Finnish National High School Mathematics Competition, 3
Tags: modular arithmetic , geometric series , algebra , difference of squares , special factorizations , number theory
Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]

1971 AMC 12/AHSME, 14
Tags: algebra , difference of squares , special factorizations
The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are $\textbf{(A) }61,63\qquad\textbf{(B) }61,65\qquad\textbf{(C) }63,65\qquad\textbf{(D) }63,67\qquad \textbf{(E) }67,69$

2008 Puerto Rico Team Selection Test, 4
Tags: special factorizations
If the sides of a triangle have lengths $ a, b, c$, such that $ a \plus{} b \minus{} c \equal{} 2$, and $ 2ab \minus{} c^{2} \equal{} 4$, prove that the triangle is equilateral.

2011 USAMTS Problems, 2
Tags: inequalities , special factorizations
Find all integers $a$, $b$, $c$, $d$, and $e$ such that \begin{align*}a^2&=a+b-2c+2d+e-8,\\b^2&=-a-2b-c+2d+2e-6,\\c^2&=3a+2b+c+2d+2e-31,\\d^2&=2a+b+c+2d+2e-2,\\e^2&=a+2b+3c+2d+e-8.\end{align*}