Prove that the equation is satisfied at least once

[giokrutoi]

Homework Statement

f(n) is function that takes input n and outputs the smallest integer grater that n^pi
prove that there exists natural numbers abcdwxyz that are all not smaller than 2015 such that equation is satisfied
f(x) + f(y) + f(z) + f(w) = f(a) + f(b) + f(c) + f(d)
and they abcd and wxyz are not trivial meaning that a,b,c,d is not equal to w,x,y,z

Homework Equations

well ordering principal
or maybe induction

The Attempt at a Solution

f(x) + f(y) + f(z) + f(w) = f(a) + f(b) + f(c) + f(d)
f(x) = [x^pi + 1]
f(y) = [y^pi + 1]
and so on
[x] means greatest number smaller than x for example [32.23] = 32 [12.99] = 12
let x₀ be the smallest x that satisfies the equation
and i am stuck i don't know what to do afterwards

 

[mfb]

The pigeonhole principle can help here.