MathDB

p1. The sequence of numbers

123456789101112131415...

is obtained by writing the positive integers in order, one after the other. What position is

2

in the first time does

2019

appear in succession?
p2. Consider a square in the plane with vertices

(a_1, b_1), (a_2, b_2), (a_3, b_3), (a_4, b_4)

where

a_i

b_i

are integer numbers for each

i = 1, ..., 4

. Suppose the area of the square is a power of

3

. Prove that its sides are parallel to the axes.
p3. Prove that for every integer

n> 2

, it is true that

\frac{1}{n + 1}+ \frac{1}{n + 2}+ ...+ \frac{1}{2n}< \frac56

p4.

ABC

is a triangle of area

4

with circumcenter

O

and

M

is the midpoint of

AO

. We choose the points

P, Q

on the sides

AB

and

AC

respectively such that

M

is at

PQ

and segments

BC

and

PQ

are parallel. Suppose the area of the triangle

APQ

1

. Calculate the angle

BAC

Problem Statement