MathDB

p1. Given the square

ABCD

, find all points

P

interior squared such that the area of quadrilateral

ABCP

is equal to three times the area of quadrilateral

APCD

.
p2. Find all possible pairs of positive integers

(x, y)

such that neither

x

nor

y

has a prime factor greater than

5

, and are also a solution of the equation

x^2-y^2= 2^{100}

.
p3. On the right pan of a scale is a bag that weighs

11,111

grams. A person place weights on one or another plate of the balance; the first weight is one gram, and each weight placed has twice the weight of the previous one, that is, the weights are successively

1, 2, 4, 8, ...

grams. At some point the plates on the scale are in balance. Determine and argue on which the

16

gram plate weight is, the right or the left one.
p4. How many integer numbers between

1

and

1000

inclusive can be written as the sum of a positive multiple of

7

plus a positive multiple of

4

?
p5. We will say that two numbers are concatenated when one is written below from the other, for example by concatenating 200 and

6

you get

2006

and by concatenating

6

and

200

gives

6200

. Find two

6

-digit numbers such that when you concatenate them the resulting number is divisible by its product.

Problem Statement