MathDB

1

Combinatorial geometry featuring walls

n^2

segments in the plane (read walls), no two of which are parallel or intersecting.
Prove that there are at least

n

points in the plane such that no two of them see each other (meaning there is a wall separating them).

P5

1

Weird NT with even weirder scores

For positive integers

a

and

b

, define

a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i

Let

p

be a prime and

n>3

a positive integer. Show that there exist at least 2 different positive integers

t

such that

1<t<p^n

and

t!_p\equiv 1\pmod {p^n}

.

P4

1

(Well known?) polynomial

Let

p

be a prime and

P\in \mathbb{R}[x]

be a polynomial of degree less than

p-1

such that

\lvert P(1)\rvert=\lvert P(2)\rvert=\ldots=\lvert P(p)\rvert

. Prove that

P

is constant.

P2

1

Depressed geometry

A circle centered at

A

intersects sides

AC

and

AB

of

\triangle ABC

at

E

and

F

, and the circumcircle of

\triangle ABC

at

X

and

Y

. Let

D

be the point on

BC

such that

AD

,

BE

,

CF

concur. Let

P=XE\cap YF

and

Q=XF\cap YE

. Prove that the foot of the perpendicular from

D

to

EF

lies on

PQ

.

P1

1

Ad hoc extremal graph theory

In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

P3

1

Positive integers partitioned into 2 sequences

The positive integers are partitioned into 2 sequences

a_1<a_2<\dots

and

b_1<b_2<\dots

such that

b_n=a_n+n

for every positive integer

n

.
Show that

a_n+b_n=a_{b_n}

.

2023 Serbia Team Selection Test

Subcontests

Combinatorial geometry featuring walls

Weird NT with even weirder scores

(Well known?) polynomial

Depressed geometry

Ad hoc extremal graph theory

Positive integers partitioned into 2 sequences