MathDB

5 points concyclic wanted

Source: 2021 Dürer Math Competition Finals Day 1 E3 E+1

1/2/2022

Let

A

and

B

different points of a circle

k

centered at

O

in such a way such that

AB

is not a diagonal of

k

. Furthermore, let

X

be an arbitrary inner point of the segment

AB

. Let

k_1

be the circle that passes through the points

A

and

X

, and

A

is the only common point of

k

and

k_1

. Similarly, let

k_2

be the circle that passes through the points

B

and

X

, and

B

is the only common point of

k

and

k_2

. Let

M

be the second intersection point of

k_1

and

k_2

. Let

Q

denote the center of circumscribed circle of the triangle

AOB

. Let

O_1

and

O_2

be the centers of

k_1

and

k_2

. Show that the points

M,O,O_1,O_2,Q

are on a circle.

geometryConcyclic

a line intersecting a square lattice

Source: 2021 Dürer Math Competition Finals Day2 E3 https://artofproblemsolving.com/community/c2749870_

1/9/2022

The figure shows a line intersecting a square lattice. The area of some arising quadrilaterals are also indicated. What is the area of the region with the question mark? https://cdn.artofproblemsolving.com/attachments/0/d/4d5741a63d052e3f6971f87e60ca7df7302fb0.png

areasgeometry

n kids with k bars of chocolate

Source: 2021 Dürer Math Competition Finals Day 1 E+3

1/2/2022

On the evening of Halloween a group of

n

kids collected

k

bars of chocolate of the same type. At the end of the evening they wanted to divide the bars so that everybody gets the same amount of chocolate, and none of the bars is broken into more than two pieces. For which

n

and

k

is this possible?

combinatorics

3

Problems(3)