5

Part of 2020 New Zealand MO

Problems(2)

NZMO Function Equation

Source: 2020 NZMO Round 1

Find all functions

f:\mathbb R \to \mathbb R

such that for all

x,y\in \mathbb R

f(x+f(y))=2x+2f(y+1)

functionfunctional equationalgebra

sum of the costs of all antipalindromic sequences of length 2020

Source: 2020 New Zealand MO Round 2 p5 NZMO

A

B

s is called antipalindromic if writing it backwards, then turning all the

A

B

s and vice versa, produces the original sequence. For example

ABBAAB

is antipalindromic. For any sequence of

A

B

s we define the cost of the sequence to be the product of the positions of the

A

s. For example, the string

ABBAAB

1\cdot 4 \cdot 5 = 20

. Find the sum of the costs of all antipalindromic sequences of length

2020