MathDB

1

7-digit code by slide of finger - Portugal OPM 2016 p1

To unlock his cell phone, Joao slides his finger horizontally or vertically across a numerical box, similar to the one represented in the figure, describing a

7

-digit code, without ever passing through the same digit twice. For example, to indicate the code

1452369

, Joao follows the path indicated in the figure. https://cdn.artofproblemsolving.com/attachments/8/a/511018ba4e43c2c6f0be350d57161eb5ea7c2b.png João forgot his code, but he remembers that it is divisible by

9

. How many codes are there under these conditions?

6

1

coloring natural numbers in 2 colors

The natural numbers are colored green or blue so that:

\bullet

The sum of a green and a blue is blue;

\bullet

The product of a green and a blue is green. How many ways are there to color the natural numbers with these rules, so that

462

are blue and

2016

are green?

5

1

2^x +4^y +8^z +16^2 is a power of 2.

Determine all natural numbers

x, y

and

z

such that the number

2^x +4^y +8^z +16^2

is a power of

2

.

2

1

2016 = sum of consecutive natural numbers

In how many different ways can you write

2016

as the sum of a sequence of consecutive natural numbers?

4

1

angle wanted, parallelogram, tangents to circle, equal angles related

Let

[ABCD]

be a parallelogram with

AB <BC

and let

E, F

be points on the circle that passes through

A, B

and

C

such that

DE

and

DF

are tangents to this circle. Knowing that

\angle ADE = \angle CDF

, determine

\angle ABC

. https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png

3

1

min length of segment that is area bisector in an equilateral triangle

Let

[ABC]

be an equilateral triangle on the side

1

. Determine the length of the smallest segment

[DE]

, where

D

and

E

are on the sides of the triangle, which divides

[ABC]

into two figures with equal area.

2016 Portugal MO

Subcontests

7-digit code by slide of finger - Portugal OPM 2016 p1

coloring natural numbers in 2 colors

2^x +4^y +8^z +16^2 is a power of 2.

2016 = sum of consecutive natural numbers

angle wanted, parallelogram, tangents to circle, equal angles related

min length of segment that is area bisector in an equilateral triangle