5

Part of 2015 Harvard-MIT Mathematics Tournament

Problems(2)

2015 Algebra #5: Maximum of Minimum of Expression

Source:

a,b,c

be positive real numbers such that

a+b+c=10

ab+bc+ca=25

m=\min\{ab,bc,ca\}

. Find the largest possible value of

m

algebrainequalitiesmaximization

2015 Geometry #5

Source:

I

be the set of points

(x,y)

in the Cartesian plane such that

x>\left(\frac{y^4}{9}+2015\right)^{1/4}

f(r)

denote the area of the intersection of

I

x^2+y^2\le r^2

r>0

centered at the origin

(0,0)

. Determine the minimum possible real number

L

f(r)<Lr^2

r>0