MathDB

p1. Some natural numbers can be written as the sum of consecutive natural numbers. For example:

5 = 2 + 3

6 = 1 + 2 + 3

10 = 1 + 2 + 3 + 4

25 = 3 + 4 + 5 + 6 + 7

. However, this is not possible with all natural numbers. Determine with what numbers it is not possible and prove such impossibility.
p2. Show that number

2^2+2^5+2^j

is a perfect square only for the case

j = 6

.
p3. You have the numbers from

1

100

and each one of them is painted in one of the four colors red, blue, yellow or green. Show that there are two numbers of the same color, whose difference is also the same color.
p4.

ABCD

is a trapezoid,

ABIE

is a parallelogram where

E

is the midpoint of

AD

;

F

is the midpoint of

BC

; the points

G

and

H

are the intersection of the segment

EI

with the diagonals of the trapezoid. Show that if

AB/FI = 2001

, then

DC/GH = 1999

.
p5. Determine all integer solutions of the equation

\frac{1}{x}+\frac{1}{y}=\frac{1}{2}

.
PS. Wording of P4 has been corrected thanks to vanstraelen.

Problem Statement