MathDB

p1. How many

10

-digit strings are there, such that all its digits are only zeros or ones and the sum of its even-place digits equals the sum of the odd-place digits.
p2. Find all pairs

(x, y)

of nonnegative integers, such that

x! + 24 = y^2

.
p3. Consider a function

f: Z \to Q

such that

f(1) = 2015 \,\,\, \,\, f (1) + f (2) + ...+ f (n) = n^2 f (n).

Determine the value of

f(2015)

.
p4. Let

ABC

be an isosceles triangle with

AB = AC

, and let

D

be the midpoint of

BC

E

the foot of the perpendicular on

AB

from

D

, and F the midpoint of

DE

. Show that

AF

is perpendicular to

CE

.
p5. On an island there are only two tribes: the Mienteretes who always lie and the Veritas who always tell the truth. On a certain day there is an assembly attended by

2015

inhabitants of the island. They sit at random in a circle and each one declares: "the two people next to me are Mienteretes." The next day the assembly continues but one of them became ill, for which

2014

inhabitants attend, again they sit at random in a circle and each one declares: “The two people next to me are from the same tribe, which is not mine ”. Deduct the number of inhabitants of each tribe and the type of inhabitant to which the patient belongs.

Problem Statement