Feeds:
Posts
Comments

Archive for April, 2008

09. PEN B6 Suppose that m does not have a primitive root. Show that

a{}^{{}^\frac{\varphi(m)}{2}} \equiv 1 \; (mod \; m)

for every a relatively prime to m.

Here is the official solution file: pen09.
You can also disscuss the problems here!

Read Full Post »

09. PEN B6 Suppose that m does not have a primitive root. Show that

a{}^{{}^\frac{\varphi(m)}{2}} \equiv 1 \; (mod \; m)

for every a relatively prime to m.

Read Full Post »

08. PEN O 35 ( Romania TST 1998 )

Let n \ge 3 be a prime number and a_{1} < a_{2} < \cdots < a_{n} be integers.
Prove that a_{1}, \cdots,a_{n} is an arithmetic progression if and only if there exists a partition of \{0, 1, 2, \cdots \} into sets A_{1},A_{2},\cdots,A_{n} such that

a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n}

where x + A denotes the set \{x + a \vert a \in A \}.

Here is the official solution file: PEN08S
You can also disscuss the problems here!

Read Full Post »

08. PEN O 35 ( Romania TST 1998 )

Let n \ge 3 be a prime number and a_{1} < a_{2} < \cdots < a_{n} be integers.
Prove that a_{1}, \cdots,a_{n} is an arithmetic progression if and only if there exists a partition of \{0, 1, 2, \cdots \} into sets A_{1},A_{2},\cdots,A_{n} such that

a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n}

where x + A denotes the set \{x + a \vert a \in A \}.

Read Full Post »

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}}.

Here is the official solution file: PEN07sol
You can also disscuss the problems here!

Read Full Post »