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Archive for February, 2008

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.

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04. PEN I11 (Korea 2000) Let p be a prime number of the form 4k + 1. Show that

\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right\rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) =\frac{p-1}{2}.

Here is the official solution file: PEN04S
You can also disscuss the problems here!

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O4. PEN I11 (Korea 2000) Let p be a prime number of the form 4k + 1. Show that

\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right\rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) =\frac{p-1}{2}.

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03. PEN O53 (Schur Theorem) Suppose the set M=\{1,2,\ldots,n\} is partitioned into t disjoint subsets M_1,\ldots,M_t. Show that if n\ge\lfloor t!\cdot e\rfloor then at least one class M_z contains three elements a,b,c with the property that a+b=c.

Here is the official solution file: PEN03S
You can also disscuss the problems here!

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