MathDB

Two-part geometry

Source: Romania JBMO TST 2024 Day 1 P4

7/31/2024

Let

ABC

be a triangle. An arbitrary circle which passes through the points

B,C

intersects the sides

AC,AB

for the second time in

D,E

respectively. The line

BD

intersects the circumcircle of the triangle

AEC

at

P{}

and

Q{}

and the line

CE

intersects the circumcircle of the triangle

ABD

at

R{}

and

S{}

such that

P{}

is situated on the segment

BD{}

and

R{}

lies on the segment

CE.

Prove that:
[*]The points

P,Q,R

and

S{}

are concyclic. [*]The triangle

APQ

is isosceles.
Petru Braica

geometry

Welsh darts board

Source: Romania JBMO TST 2024 Day 2 P4

7/31/2024

Let

n\geqslant 2

be an integer. A Welsh darts board is a disc divided into

2n

equal sectors, half of them being red and the other half being white. Two Welsh darts boards are matched if they have the same radius and they are superimposed so that each sector of the first board comes exactly over a sector of the second board.
Suppose that two given Welsh darts boards can be matched so that more than half of the paurs of superimposed sectors have different colours. Prove that these Welsh darts boards can be matched so that at least

2\lfloor n/2\rfloor +2

pairs of superimposed sectors have the same colour.

combinatorics

Point-set duality

Source: Romania JBMO TST 2024 Day 3 P4

7/31/2024

Let

n\geqslant 3

be a positive integer and

N=\{1,2,\ldots,n\}

and let

k>0

be a real number. Let's associate each non-empty of

N{}

with a point in the plane, such that any two distinct subsets correspond to different points. If the absolute value of the difference between the arithmetic means of the elements of two distinct non-empty subsets of

N{}

is at most

k{}

we connect the points associated with these subsets with a segment. Determine the minimum value of

k{}

such that the points associated with any two distinct non-empty subsets of

N{}

are connected by a segment or a broken line.
Cristi Săvescu

combinatoricsset theory

Combinatorial geometry with circle interiors

Source: Romania JBMO TST 2024 Day 4 P4

7/31/2024

Let

n\geqslant 2

be an integer and

A{}

a set of

n

points in the plane. Find all integers

1\leqslant k\leqslant n-1

with the following property: any two circles

C_1

and

C_2

in the plane such that

A\cap\text{Int}(C_1)\neq A\cap\text{Int}(C_2)

and

|A\cap\text{Int}(C_1)|=|A\cap\text{Int}(C_2)|=k

have at least one common point.
Cristi Săvescu

combinatoricscombinatorial geometry

P4

Problems(4)