WMC 2018 Senior Problem 11

Algebra Level 3

Let f ( x ) f(x) be a function such that the system of equations below holds true for all x x .

{ f ( − x ) + f ( 1 − x ) = 15 f ( 1 + x ) = 5 + f ( − x ) \large {\begin{cases} f( -x) +f (1-x) =15 \\ f (1+x) =5+f(-x) \end{cases}}

Find f ( x ) + f ( − x ) f(x)+f(-x) .

0 20 5 15 10

Tiger Ang by Tiger Ang , 20, Malaysia

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2 solutions

Julian Yu
Oct 20, 2018

Plug − x -x into the second equation to get f ( 1 − x ) = 5 + f ( x ) f(1-x)=5+f(x) .

Now substitute this into the first equation: f ( − x ) + 5 + f ( x ) = 15 ⟹ f ( x ) + f ( − x ) = 10 f(-x)+5+f(x)=15\implies f(x)+f(-x)=10 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused
X X
Oct 14, 2018

f ( − x ) + f ( 1 − x ) = 15 f(-x)+f(1-x)=15 , so if − x = y -x=y , f ( y ) + f ( 1 + y ) = 15 f(y)+f(1+y)=15 .

It is same as f ( x ) + f ( 1 + x ) = 15 f(x)+f(1+x)=15

15 = f ( x ) + ( 5 + f ( − x ) ) , f ( x ) + f ( − x ) = 15 − 5 = 10 15=f(x)+(5+f(-x)),f(x)+f(-x)=15-5=10

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