## Problems from the GRE

To be accepted in a US University for a Ph.D. in Mathematics program, a student must pass two Graduate Record Examinations (GREs), a general one that checks literacy and quickness of wits, and a professional one in mathematics. The Mathematics GRE examination is usually a small masterpiece of multiple choice question setting, and tests both for solid knowledge and for the ability to spot a shortcut. A number of years ago I prepared one of our graduates for the Mathematics GRE, and I recently discovered the material we used. Some of the questions seem suitable for the blog.

1. Let $f(x)$ be a function whose graph passes through the origin. If $f(2n)=n^2+f((2(n-1))$ for every integer $n$, what is the value of $f(8)$?

Possible answers are: (A) 24; (B) 30; (C) 32; (D) 36; (E) Cannot be determined from this data.

Comment: this is a nice question, but not quite suitable for a multiple choice one as one immediately tries to compute the answer.

2. (This one is for the MM*03 afficionados.) Consider the set $X$ of all real-valued functions defined on the interval $[a,b]$, $a. Which of the following conditions, if true, would ensure that the function $f \in X$ is constant on $[a,b]$?

I. The inverse of $f$ is a constant function;

II. For each $g \in X$, $g \circ f$ is a constant function;

III. $|f(x)-f(y)| \leq (x-y)^2$ for every $x$ and $y$ in $[a,b]$.

Possible answers: (A) II only; (B) I and II only; (C) I and III only; (D) II and III only; (E) I, II, and III.

Comment: Almost immediately here one is left with only two possible answers, and then it is non-trivial to work out which of these is true! Just don’t give up.

By the way, the exam comprises 65 questions and you have 170 minutes to do them…

(MG)