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## 20S. Sequences of Consecutive Integers

A37 If $n$ is a natural number, prove that the number

$(n + 1)(n + 2)\cdots(n + 10)$

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

Here is the official solution file: PEN A9 A37 O51

## 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.

## 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

## 17S. A Hidden Divisibility

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.

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.

## 15. Exponential Congruence Sequence

D5. Prove that for , $\underbrace{2^{2^{\cdots^{2}}}}_{n\text{ terms}}\equiv \underbrace{2^{2^{\cdots^{2}}}}_{n-1\text{ terms}}\; \pmod{n}.$
D6. Show that, for any fixed integer  the sequence $2, \; 2^{2}, \; 2^{2^{2}}, \; 2^{2^{2^{2}}}, \cdots \pmod{n}$ is eventually constant.

Sorry all, for the delay of the problem 15.  Here goes the solution: pen-15.pdf

## 14S. Different Approaches to an Intuitive Problem

The fourth problem of the second season of PEN is as follows:

N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers.

Here is the official solution file: pen14.pdf

You can discuss the problem in MathLinks here.