MathDB

p1. Juan was born before the year

2000

. On August

25

2001

he is as old as the sum of the digits of the year of his birth. Determine your date of birth and justify that it is the only possible solution.
p2. Triangle

ABC

is right isosceles. The figure below shows two basic ways to inscribe a square in it. The square

ADEF

is known to have area

2250

. Determine the area of the square

GHIJ

. https://cdn.artofproblemsolving.com/attachments/9/3/84d0cda6e109aaf01fb2f3de4d1933f0f16f82.jpg
p3. Given

9

people, show that there exists a value of

n

such that with people you can form

n

groups of

3

, so that each pair of people is in exactly one of these groups. Also, show one of the possible formations of the groups.
p4. Show that there are no positive integers

a, b, c

such that

a^2 + b^2 = 8c + 6

.
p5. In a circular roulette the numbers from

1

36

are randomly placed. Show that necessarily there must be

3

consecutive numbers whose sum is at least

55

.
p6. Let

ABCD

be a quadrilateral inscribed in a circle of radius

r

, such that its diagonals are perpendicular at point

E

. Prove that

AE^2 + BE^2 + CE^2 + DE^2 = 4r^2.

p7. In a rectangular board of

m

rows and

n

columns, place

1

0

in each of the

mn

cells, so that the numbers of each row have the same sum

f

and those of each column have the same sum

c

. For a board with

m = 15

n = 10

, and for

f = 4

, determine the value of

c

that makes this assignment possible and show a way to do this assignment.
PS. Juniors p2, p6, were posted also as [url=https://artofproblemsolving.com/community/c4h2691996p23368695]Seniors variation of p2, part of p6.

Problem Statement