MathDB

2021 Alg/NT Div 1 P8

Source:

3/2/2021

There are integers

v,w,x,y,z

and real numbers

0\le \theta < \theta' \le \pi

such that

\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.

Given that

z\ne 0

and

v

is positive, find the sum of the

4

smallest possible values of

v

.
Proposed by Vijay Srinivasan

algebranumber theory

2021 Geo Div 1 P8

Source:

3/2/2021

Let

ABC

be a triangle with

AB < AC

and

\omega

be a circle through

A

tangent to both the

B

-excircle and the

C

-excircle. Let

\omega

intersect lines

AB, AC

at

X,Y

respectively and

X,Y

lie outside of segments

AB, AC

. Let

O

be the center of

\omega

and let

OI_C, OI_B

intersect line

BC

at

J,K

respectively. Suppose

KJ = 4

,

KO = 16

and

OJ = 13

. Find

\frac{[KI_BI_C]}{[JI_BI_C]}

.
Proposed by Grant Yu

geometry

2021 Combo Div 1 P8

Source:

3/2/2021

An augmentation on a graph

G

is defined as doing the following:
- Take some set

D

of vertices in

G

, and duplicate each vertex

v_i \in D

to create a new vertex

v_i'

.
- If there's an edge between a pair of vertices

v_i, v_j \in D

, create an edge between vertices

v_i'

and

v_j'

. If there's an edge between a pair of vertices

v_i \in D

,

v_j \notin D

, you can choose to create an edge between

v_i'

and

v_j

but do not have to.
A graph is called reachable from

G

if it can be created through some sequence of augmentations on

G

. Some graph

H

has

n

vertices and satisfies that both

H

and the complement of

H

are reachable from a complete graph of

2021

vertices. If the maximum and minimum values of

n

are

M

and

m

, find

M+m

.
Proposed by Oliver Hayman

combinatorics

1.8

Problems(3)