MathDB

p1. In the rectangle

ABCD

there exists a point

P

on the side

AB

such that

\angle PDA = \angle BDP = \angle CDB

and

DA = 2

. Find the perimeter of the triangle

PBD

.
p2. Write in each of the empty boxes of the following pyramid a number natural greater than

1

, so that the number written in each box is equal to the product of the numbers written in the two boxes on which it is supported. https://cdn.artofproblemsolving.com/attachments/3/f/395e1c09fd955ed7ba6f9fea23ef68a00a4880.png
p3. In how many ways can seven numbers be chosen from

1

9

such that their sum is a multiple of

3

?
p4. Simplify the product

p = (10 + 1) (10^2 + 1) ((10^2)^2 + 1) ((10^2)^4 + 1) ((10^2)^8 + 1)...((10^2)^{1024} +1)

p5. A lattice point is a point on the Cartesian plane with integer coordinates. We randomly select five lattice points. Show that there are two of them that form a line segment whose midpoint is also a lattice point.

Problem Statement