University of South Carolina High School Math Contest/1993 Exam/Problem 5

Problem

Suppose that $f$ is a function with the property that for all $x$ and $y$ , $f(x + y) = f(x) + f(y) + 1$ and $f(1) = 2.$ What is the value of $f(3)$ ?

$\mathrm{(A) \ }4 \qquad \mathrm{(B) \ }5 \qquad \mathrm{(C) \ }6 \qquad \mathrm{(D) \ }7 \qquad \mathrm{(E) \ }8$

Solution

Notice that $f(3)=f(2+1)=f(2)+f(1)+1=f(2)+3$ . Also, $f(2)=f(1+1)=f(1)+f(1)+1=5$ . Thus, $f(3)=3+5=8$ .

In general, $f(x + 1) = f(x) + f(1) + 1 = f(x) + 3$ , so we have a simple recursive definition for the function $f$ . From here we can see that $f(n) = 3n - 1$ for all positive integers $n$ .

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University of South Carolina High School Math Contest/1993 Exam/Problem 5

Problem

Solution