1

Part of 2023 Durer Math Competition Finals

Problems(4)

2 lines intersect on a circle tangent to 2 sides of a triangle

Source: (2022 -) 2023 XVI Dürer Math Competition Finals Day 1 E1

ABC

is an isosceles triangle. The base

BC

1

cm long, and legs

AB

AC

2

cm long. Let the midpoint of

AB

F

, and the midpoint of

AC

G

. Additionally,

k

is a circle, that is tangent to

AB

C

, and it’s points of tangency are

F

G

accordingly. Prove, that the intersection of

CF

BG

falls on the circle

k

geometryconcurrentConcyclic

a whole basket of cherries

Source: (2022-) 2023 XVI 16th Dürer Math Competition Finals Day 2 E1

Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?

Covering perimeter of rectangle with disks

Source: Dürer Competition Finals 2023/E+ 1

Prove that for any real

r>0

, one can cover the circumference of a

1\times r

rectangle with non-intersecting disks of unit radius.

geometryperimeterrectangle

ages are 3 distinct prime numbers

Source: (2022-) 2023 XVI 16th Dürer Math Competition Finals Day 2 E+1

Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?

number theoryalgebraprime numbers