2018 Japan TST

Part of Japan TST

Subcontests

(3)

1

Similar to some G7

A convex quadrilateral

ABCD

has an inscribed circle with center

I

I_a, I_c

be the incenters of the triangles

DAB, BCD

respectively. Suppose that the common external tangents of the circles

BI_aI_c

DI_aI_c

X

X,I,I_a,I_c

1

Product of any two is perfect power

Prove that there are finitely many positive integers

n

such that there exists a subset

S

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

satisfying the following conditions:
[*]

S

\lfloor \sqrt{n}\rfloor +1

elements [*] For any

x,y\in S

xy

is a perfect power, i.e.

xy

a^b

for positive integers

a

b\ge 2

1

Min sum of products of number of points

n\ge 3

be a positive integer.

S

n

points on the plane, with no three collinear.

L

is the set of all lines passing through any two points in

S

\ell \in L

, the separating number of

\ell

is the product of the number of points in

S

in the two sides of

\ell

, excluding the two points on

\ell

itself. Determine the minimum possible total sum of all separating numbers.