MathDB

Delightful sets of vertices

Source: Russian TST 2020, Day 4 P2

3/21/2023

There are 10,000 vertices in a graph, with at least one edge coming out of each vertex. Call a set

S{}

of vertices delightful if no two of its vertices are connected by an edge, but any vertex not from

S{}

is connected to at least one vertex from

S{}

. For which smallest

m

is there necessarily a delightful set of at most

m

vertices?

graph theorycombinatorics

Inequality with gcd

Source: Russian TST 2020, Day 6 P2

3/21/2023

Given a natural number

n{}

find the smallest

\lambda

such that

\gcd(x(x + 1)\cdots(x + n - 1), y(y + 1)\cdots(y + n - 1)) \leqslant (x-y)^\lambda,

for any positive integers

y{}

and

x \geqslant y + n

.

number theoryinequalitiesgreatest common divisor

Inscribed octagon, geometry

Source:

8/3/2021

Octagon

A_1A_2A_3A_4A_5A_6A_7A_8

is inscribed in a circle

\Omega

with center

O

. It is known that

A_1A_2\|A_5A_6

,

A_3A_4\|A_7A_8

and

A_2A_3\|A_5A_8

. The circle

\omega_{12}

passes through

A_1

,

A_2

and touches

A_1A_6

; circle

\omega_{34}

passes through

A_3

,

A_4

and touches

A_3A_8

; the circle

\omega_{56}

passes through

A_5

,

A_6

and touches

A_5A_2

; the circle

\omega_{78}

passes through

A_7

,

A_8

and touches

A_7A_4

. The common external tangent to

\omega_{12}

and

\omega_{34}

cross the line passing through

{A_1A_6}\cap{A_3A_8}

and

{A_5A_2}\cap{A_7A_4}

at the point

X

. Prove that one of the common tangents to

\omega_{56}

and

\omega_{78}

passes through

X

.

octagoncirclesradical axisconcurrencycommon tangentsgeometry

P2

Problems(3)