Olympiad problems

1998 Harvard-MIT Mathematics Tournament, 6

How many pairs of positive integers $(a,b)$ with $a\leq b$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}$?

symmetry sfft number theory

2012 Pan African, 1

The numbers $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$ are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say $x$ and $y$, erases them and she writes instead the number $x + y + xy$. She continues to do this until only one number is left on the board. What are the possible values of the final number?

special factorizations induction geometry 3d geometry combinatorics sfft invariant factorization invariants mathematical induction solid geometry telescoping

2001 AMC 10, 10

If $ x$, $ y$, and $ z$ are positive with $ xy \equal{} 24$, $ xz \equal{} 48$, and $ yz \equal{} 72$, then $ x \plus{} y \plus{} z$ is $ \textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\textbf{(E) }24$

sfft algebra

2010 AIME Problems, 5

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}$.

special factorizations logarithm sfft factorization logarithms