Monthly Archives: April 2021

2020 Mathcounts State Sprint Round #30

Posted on April 22, 2021 by kevin

Hank builds an increasing sequence of positive integers as follows: The first term is 1 and the second term is 2. Each subsequent term is the smallest positive integer that does NOT form a three-term arithmetic sequence with any previous … Continue reading →

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Cyclic System of Equations

Posted on April 11, 2021 by kevin

Find $abc$. $$ \begin{array}{ll} a+\dfrac{1}{bc} = \dfrac{7}{6}\\ b+\dfrac{1}{ca} = \dfrac{7}{3}\\ c+\dfrac{1}{ab} = \dfrac{7}{2} \end{array} $$ Solution Multiply the first equation by $bc$, the second by $ca$, and the third by $ab$: $$ \begin{array}{ll} abc+1 = \dfrac{7}{6}bc\\ abc+1 = \dfrac{7}{3}ac\\ abc+1 … Continue reading →

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