Olympiad problems

2007 AMC 12/AHSME, 23

Tags: inradius , geometry , perimeter , sfft , special factorizations

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters? $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$

2023 Chile Classification NMO Seniors, 4

Tags: algebra , sfft

When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product. Find the six-digit number.

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 $m+n.$

2010 Stanford Mathematics Tournament, 9

Tags: sfft

Suppose $xy-5x+2y=30$, where $x$ and $y$ are positive integers. Find the sum of all possible values of $x$