2016 USAJMO

Part of USAJMO

Subcontests

(6)

1

2016 Sets

Find, with proof, the least integer

N

such that if any

2016

elements are removed from the set

{1, 2,...,N}

, one can still find

2016

distinct numbers among the remaining elements with sum

N

1

Familiar Solution Set

Find all functions

f:\mathbb{R}\rightarrow \mathbb{R}

such that for all real numbers

x

y

(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.

1

Inversely Similiar Triangles

\triangle ABC

be an acute triangle, with

O

as its circumcenter. Point

H

is the foot of the perpendicular from

A

\overleftrightarrow{BC}

P

Q

are the feet of the perpendiculars from

H

\overleftrightarrow{AB}

\overleftrightarrow{AC}

, respectively.
Given that

AH^2=2\cdot AO^2,

prove that the points

O,P,

Q

1

Fixed point as P varies

The isosceles triangle

\triangle ABC

AB=AC

, is inscribed in the circle

\omega

P

be a variable point on the arc

\stackrel{\frown}{BC}

that does not contain

A

I_B

I_C

denote the incenters of triangles

\triangle ABP

\triangle ACP

, respectively.
Prove that as

P

varies, the circumcircle of triangle

\triangle PI_BI_C

passes through a fixed point.

1

Sequences of Subsets

X_1, X_2, \ldots, X_{100}

be a sequence of mutually distinct nonempty subsets of a set

S

X_i

X_{i+1}

are disjoint and their union is not the whole set

S

X_i\cap X_{i+1}=\emptyset

X_i\cup X_{i+1}\neq S

i\in\{1, \ldots, 99\}

. Find the smallest possible number of elements in

S

1

I Got Six Consecutive Zeros on JMO!

Prove that there exists a positive integer

n < 10^6

5^n

has six consecutive zeros in its decimal representation.
Proposed by Evan Chen