MathDB

p1. Determine which of the following numbers is greater

10^{10^{10^{10}}}, (10^{10})^{(10^{10})}.

[url=https://artofproblemsolving.com/community/c1068820h2935461p26269201]p2. The integers

a, b

satisfy the following identity

2a^2 + a = 3b^2 + b.

Prove that

a- b

2a + 2b + 1

, and

3a + 3b + 1

are perfect squares.
[url=https://artofproblemsolving.com/community/c4h1845719p12426408]p3. Let

ABCD

be a square and

M

be its center. Consider the point

E

on line

AC

such that

| MC | = | CE |

. Let

S

be the circle circumscribed to triangle

\triangle EDB

. Show that

S

passes through the midpoint of

AM

.
p4. Find the sum of all positive divisors of the number

2.010.000.000

.
p5. It is known that the sides and the diagonal of a rectangle are integers. Prove that the area of the rectangle is divisible by

12

.
[url=https://artofproblemsolving.com/community/c4h1845722p12426437]p6. Consider a line

L

in the plane and let

B_1, B_2, B_3

be points different in

L

. Let

A

be a point that does not lie in

L

. Show that there are

P, Q

\{B_1, B_2, B_3\}

with

P \ne Q

such that the distance from

A

L

to be greater than the distance from

P

to the line that passes through

A

and

Q

.
PS. Problem 2 was also proposed as Seniors P1.

Problem Statement