4

Part of 2020 Durer Math Competition Finals

Problems(3)

n integers on a paper, sum of their 2^n subsets on the blackboard

Source: 2020 Dürer Math Competition Finals E1.4

n

(not necessarily distinct) integers on a paper. Then for each of the

2^n

subsets, Kelemen wrote their sum on the blackboard. a) For which values of

n

is it possible that two different

n

-tuples give the same numbers on the blackboard? b) Prove that if Endre only wrote positive integers on the paper and Ferenc only sees the numbers on the blackboard, then he can determine which integers are on the paper.

combinatoricsSum

sum of digits of 3n if sum of digits of n is 100 and of 4n is 800

Source: 2020 Dürer Math Competition Finals Day2 E4 https://artofproblemsolving.com/community/c1622639_2020_

We have a positive integer

n

, whose sum of digits is

100

. If the sum of digits of

44n

800

then what is the sum of digits of

3n

sum of digitsnumber theory

collinear wanted, perpend. to angle bisector passing through incenter

Source: 2020 Dürer Math Competition Finals E+ 1. 4

ABC

be a scalene triangle and its incentre

I

F_A

the intersection of the line

BC

and the perpendicular to the angle bisector at

A

I

. Let us define points

F_B

F_C

in a similar manner. Prove that points

F_A, F_B

F_C

geometrycollinearincenter