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Archive for April, 2009

E16 Prove that for any prime p in the interval \left]n, \frac {4n}{3}\right], p divides

\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.

Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.

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E16 Prove that for any prime p in the interval \left]n, \frac {4n}{3}\right], p divides

\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.

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A23 Prove that if

1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}

is expressed as a fraction, where p>3 is a prime, then p^2 divides the numerator.

A24 Let p>3 be a prime number and k=\lfloor\frac{2p}{3}\rfloor. Prove that

{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}

is divisible by p^2.

Here is the official solution file: pen18.pdf

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A23 Prove that if

1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}

is expressed as a fraction, where p>3 is a prime, then p^2 divides the numerator.

A24 Let p>3 be a prime number and k=\lfloor\frac{2p}{3}\rfloor. Prove that

{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}

is divisible by p^2.


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