MathDB

p1. A vote is made between three candidates:

A

B

and

C

, with an electorate of

20

people. Each elector votes in order for the three candidates, each possible combination receiving at least one vote. Of the voters,

11

preferred

A

over

B

and

12

preferred

C

over

A

. Given this, the electoral commission asks

B

to withdraw to make an election only between

A

and

C

. Faced with this request, the candidate

B

protests and argues that there are

14

voters who prefer him over

C

. Given the confusion, the commission Electoral decides to make an election between the first two largest ace. Determine which of the candidates are the ones who got the first highest ace, and how many votes did they get.
p2. For each positive integer

n

, prove that

n^2 + n + 1

is not a perfect square.
p3. Find all integer solutions of the equation

x^3 + 2y^3 = 4z^3

p4.

\vartriangle ABC

has a circumcircle

K

. The interior bisectors of the triangle intersect again

K

D

E

F

\vartriangle DEF

turns out to be equilateral. Prove that

\vartriangle ABC

is equilateral.
p5. Let

n\ge 2

be a natural.

0\le x_1\le x_2\le ... \le x_n

are real, which have sum

1

. If

x_n \le \frac23

, prove that there exists

k

, with

1\le k\le n

, such that

\frac13 \le \sum_{j = 1}^{k} x_j \le \frac23

p6. Let

p, q, r

integers, where

r

is not a perfect square. Suppose that

x = \sqrt[3]{p +\sqrt{r}} + \sqrt[3]{q -\sqrt{r}}

is a rational. Prove that

p = q

.
p7. Consider a circle

C

, with center

O

and radius

r> 0

. Let

Q

be a point in the interior of

C

. A point

P \in C

is drawn as well as the

\angle OPQ

. Where should point

P

lie such that the

\angle OPQ

has maximum measure?

Problem Statement