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I read a Putnam question recently. I am having trouble remembering which year it was, but it was probably from before the 1950's

Let $f\in[0,\pi]$ be a continuous function. If $$\int_0^\pi{f(\theta)\cos\theta \;{\rm d}\theta}=\int_0^\pi{f(\theta)\sin\theta\; {\rm d}\theta}$$ prove that $f$ has two zeroes in $(0,\pi)$

Consider the function $f(\theta)=e^{-\theta}$. It satisfies the condition given above, which can be seen through integration by parts, and does not have any zeroes. Where am I going wrong?

Florian
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  • You are quite right. I expect you misremembered the question. – TonyK Jan 27 '20 at 13:31
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    There is currently no $d$ in your integrals. Does the question read $f(\theta)d\cos(\theta)$ or $f(\theta)\cos(\theta)d\theta$? – quarague Jan 27 '20 at 13:33
  • With the condition $\int_0^\pi{f(\theta)\cos\theta}=\int_0^\pi{f(\theta)\sin\theta} \mathbf{ = 0}$ it is answered here: https://math.stackexchange.com/q/247385 . – Martin R Jan 27 '20 at 13:34
  • @MartinR- Thanks. I'm sure now that I misremembered the question. –  Jan 27 '20 at 13:37

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