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Archive for March, 2008

07. PEN D2 (Putnam 1991/B4) Suppose that p is an odd prime. Prove that

\sum_{j = 0}^{p}\binom{p}{j}\binom{p + j}{j}\equiv 2^{p} + 1 \pmod{p^{2}}.

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06. PEN A3 ( IMO 1988 ) Let a and b be positive integers such that ab+1 divides a^{2}+b^{2}. Show that

\frac{a^{2}+b^{2}}{ab+1}

is the square of an integer.

Here is the official solution file: PEN06S
You can also disscuss the problems here!

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06. PEN A3 ( IMO 1988 ) Let a and b be positive integers such that ab+1 divides a^{2}+b^{2}. Show that

\frac{a^{2}+b^{2}}{ab+1}

is the square of an integer.

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05a. [Saint-Petersburg 1998] Let d(n) denote the number of positive divisors of the number n. Prove that the sequence d(n^2+1) does not become strictly monotonic from some point onwards.

05b. PEN J11 Prove that d((n^2+1)^2) does not become monotonic from any given point onwards.

Here is the official solution file: PEN05S
You can also disscuss the problems here!

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