PEN Solutions Archive « Project PEN

June 25, 2008

PEN extended!

Filed under: PEN Solutions Archive — compactorange @ 3:32 pm

Hi, PEN FRIENDS!

Here is the pdf file of PEN extended (version 0.9)! Lots of interesting problems from recent mathematics olympiads are available, YAY! Good problems from this book will be taken, and included to the original PEN Problems Book. If you find any typos or errors, then please email us at pen@problem-solving.be

PEN Problems Book has 649 problems and PEN extended has 337 problems!

- PEN TEAM

One more thing. We have a Project PEN Group on facebook. Anyone who loves number theory is welcome!

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May 13, 2008

10S Partitions

Filed under: PEN Solutions Archive — compactorange @ 12:08 pm

10. PEN B6 Consider the set of all five-digit numbers whose decimal representation is a permutation of the digits $\text{[math]}$ . Prove that this set can be divided into two groups, in such a way that the sum of the squares of the numbers in each group is the same.

Here is the official solution file: PEN10S
You can also disscuss the problems here!

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April 30, 2008

09S Primitive Roots: Revisited

Filed under: PEN Solutions Archive — yiminge @ 11:50 am

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!

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April 17, 2008

08S An arithmetic partition

Filed under: PEN Solutions Archive — compactorange @ 5:40 pm

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!

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April 2, 2008

S07 A combinatorial congruence

Filed under: PEN Solutions Archive — compactorange @ 1:29 pm

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!

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March 19, 2008

S06 A historical divisibility.

Filed under: PEN Solutions Archive — compactorange @ 9:18 am

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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March 6, 2008

S05 On the monotonicity of the divisor function.

Filed under: PEN Solutions Archive — compactorange @ 2:19 am

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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February 21, 2008

S04 A hidden symmetry

Filed under: PEN Solutions Archive — compactorange @ 8:32 am

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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February 7, 2008

S03 A theorem on sum-free subsets

Filed under: PEN Solutions Archive — compactorange @ 3:43 am

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 $a+b=c$ .