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## 19S. Binomial Sum Divisible by Primes

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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## 19. Binomial Sum Divisible by Primes

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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## 18S.Fractions Mod p and Wolstenholme’s Theorem

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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## 18. Fractions Mod p and Wolstenholme’s Theorem

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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