Olympiad problems

2016 USAJMO, 2

Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation. [i]Proposed by Evan Chen[/i]

2016 USAJMO, 6

Tags: USAJMO , USA(J)MO , 2016 USAJMO , 2016 USAJMO Problem 6 , function , algebra , Hi

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$

2016 USAMO, 4

Tags: USAJMO , USA(J)MO , 2016 USAJMO , 2016 USAJMO Problem 6 , function , algebra , Hi

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$, $$(f(x)+xy)\cdot f(x-3y)+(f(y)+xy)\cdot f(3x-y)=(f(x+y))^2.$$

2016 USAJMO, 4

Tags: AMC , USA(J)MO , USAJMO , 2016 USAJMO , 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 $N$.