MathDB

An elegant geometry problem

Source: The Golden Digits Contest, April 2024, P3

4/21/2024

Let

ABC

be an acute scalene triangle with orthocentre

H{}

and circumcentre

O.{}

Let

P{}

be an arbitrary point on the segment

OH

and

O_a

be the circumcentre of

PBC.{}

The line

PO_a

intersects the line

HA

at

X_a.{}

Define

X_b

and

X_c

similarly. Let

Q{}

be the isogonal conjugate of

P{}

and

X{}

be the circumcentre of

X_aX_bX_c.{}

Prove that

PQ

and

HX

are parallel.
Proposed by David Anghel

geometry

A beautiful combination of NT and polynomials

Source: KoMaL & The Golden Digits Contest, May 2024, P3

5/19/2024

Let

p

be a prime number and

\mathcal{A}

be a finite set of integers, with at least

p^k

elements. Denote by

N_{\text{even}}

the number of subsets of

\mathcal{A}

with even cardinality and sum of elements divisible by

p^k

. Define

N_{\text{odd}}

similarly. Prove that

N_{\text{even}}\equiv N_{\text{odd}}\bmod{p}.

number theorypolynomialkomal

Combinatorial geometry bomb

Source: The Golden Digits Contest, March 2024, P3

4/8/2024

On the surface of a sphere, a non-intersecting closed curve comprised of finitely many circle arcs is drawn. It divides the surface of the sphere in two regions, which are coloured red and blue. Prove that there exist two antipodes of different colours. Note: the curve is considered to be colourless.
Proposed by Vlad Spătaru

combinatoricsgeometry

The chocolate division theorem

Source: The Golden Digits Contest, February 2024, P3

4/8/2024

There are

m

identical rectangular chocolate bars and

n

people. Each chocolate bar may be cut into two (possibly unequal) pieces at most once. For which

m

and

n

is it possible to split the chocolate evenly among all the people?
Selected from the Kvant Magazine (D. Bugaenko and N. Konstantinov)

combinatoricsChocolate

P3

Problems(4)