## IMC 2016 – Day 1 – Problem 1

**Problem 1.** Let be continuous on and differentiable on . Suppose that has infinitely many zeros, but there is no with .

- (a) Prove that .
- (b) Give an example of such a function.

*Solution sketch:* Start from the fact that has infinitely many zeros in and pick an accumulation point of a sequence of zeros .

Now we have two cases. First we may have . But then and .

Otherwise we know that is inside . Then

since . This contradicts the hypothesis.

As an example take for and .

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Categories: Analysis, Olympiad
Analysis, convergence, derivative, function

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