7

Part of 2019 Durer Math Competition Finals

Problems(2)

area chasing inside a parallelogram

Source: 2019 Dürer Math Competition Finals Day2 E7 E+6 https://artofproblemsolving.com/community/c1621835_2019_

We choose a point on each side of a parallelogram

ABCD

, let these four points be

P, Q, R

S

. Then we divide the parallelogram into several regions using line segments as shown in the figure. The areas of the grey regions are given, except for one (see the figure). Find the area of the region marked with a question mark. https://cdn.artofproblemsolving.com/attachments/4/7/dbd009042dabdb2eafc8fc74960e9011038dae.png

geometryareasparallelogram

reciprocals of every non-zero digit of integers from 1 to 10^n is integer

Source: 2019 Dürer Math Competition Finals Day2 E+7 https://artofproblemsolving.com/community/c1621835_2019_

Find the smallest positive integer

n

with the following property: if we write down all positive integers from

1

10^n

and add together the reciprocals of every non-zero digit written down, we obtain an integer.

number theoryInteger