MathDB

p1. Prove that in every group of

n

people (

n> 2

), there are always at least

2

people who they have the same number of friends within the group.
[url=https://artofproblemsolving.com/community/c4h1846780p12438053]p2. Given a parallelogram

ABCD

, let

E, F

the midpoints of sides

BC

and

CD

, respectively. Prove that

AE

and

AF

trisect

BD

.
p3. Calculate the value of:

(1 + 2) (1 + 2^2) (1 + 2^4) (1 + 2^8)... (1 + 2^{2^{1999}})

p4. Consider the following ordering of the first

n

positive integers:

1(\,\,\,) 2(\,\,\,) 3 (\,\,\,)\, ... \,(\,\,\,)n

Is it possible to place a sign

+

or a sign

-

in the place of each

(\,\,\,)

, so that when performing the operation, the resulting result is zero as the result?
p5. For the rectangles of the figure it is known that

\frac{a}{b}=\frac13

. Determine the value of the ratio

\frac{Area_{\vartriangle BEC}}{Area_{\vartriangle DCF}}

https://cdn.artofproblemsolving.com/attachments/a/2/4ba7e41affb58009a55393c59a00b330f7a9aa.png
p6. On an even-sided square board,

2\times 1

rectangular tokens are placed. Show that if we fill the board with the tokens without leaving and without overlapping tokens, then the tokens arranged horizontally they must be an even quantity.

Problem Statement