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Archive for the ‘PEN Solutions Archive’ Category

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

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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}.

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

Here is the official solution file: pen18.pdf

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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.

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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.

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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

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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.

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