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

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Found problems: 3

1967 Putnam, B5
Tags: binomial , expansion
Show that the sum of the first $n$ terms in the binomial expansion of $(2-1)^{-n}$ is $\frac{1}{2},$ where $n$ is a positive integer.

1953 Putnam, B7
Tags: irrational number , expansion
Let $w\in (0,1)$ be an irrational number. Prove that $w$ has a unique convergent expansion of the form $$w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,$$ where $1\leq p_0 < p_1 < p_2 <\ldots $ are integers. If $w= \frac{1}{\sqrt{2}},$ find $p_0 , p_1 , p_2.$

1949 Putnam, B4
Tags: expansion
Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion $$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$ are positive integers.