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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.


A13 Show that for all prime numbers p,

Q(p) = \prod^{p - 1}_{k = 1}k^{2k - p - 1}

is an integer.

Here is the official solution file: pen17.pdf.

A13 Show that for all prime numbers p,

Q(p) = \prod^{p - 1}_{k = 1}k^{2k - p - 1}

is an integer.

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C2 The positive integers a and b are such that the numbers 15a+16b and 16a-15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Here is the official solution file: pen16.pdf

You can discuss the problem in MathLinks here.

C2 The positive integers a and b are such that the numbers 15a+16b and 16a-15b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?