**Problem**: (1983 IMO - #1) Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy:$ f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.

**Solution**: First, it would help a lot if we showed the function was one-to-one (that is, for each $y$ there is at most one $x$ such that $f(x) = y$).

So suppose $ f(a) = f(b) $.

Then $ f(af(b) = f(af(a)) = bf(a) $. But we also have $ f(af(a)) = af(a) $ so $ f(af(a)) = bf(a) = af(a)$. And since $f$ takes positive real values, we know can divide by $f(a)$ to get $a = b$. Hence $f$ is one-to-one and we take take its inverse.

Now let's experiment with some values.

Let $x = y = 1$. We have $ f(f(1)) = f(1) \Rightarrow f(1) = 1 $ (because we can take its inverse).

Now take $x = y$. We have $ f(xf(x)) = xf(x) $. Suppose $k = xf(x)$.

Assume $k > 1$. We have $ f(k) = k \Rightarrow f(kf(k)) = f(k^2) = k^2 $, and, repeating this process, we have $ f(k^{2n}) = k^{2n} $. Then if we take $ n \to \infty$, we have $ f(x) \to \infty $ as $x \to \infty $, which contradcits the given condition.

Now assume $k < 1$. We have $ f(k) = k \Rightarrow f\left(\frac{1}{k}f(k)\right) = f(1) = 1 = kf\left(\frac{1}{k}\right) \Rightarrow f\left(\frac{1}{k}\right) = \frac{1}{k} $. But again $ f\left(\frac{1}{k}\right) = \frac{1}{k} \Rightarrow f\left(\frac{1}{k}f\left(\frac{1}{k}\right)\right) = f\left(\frac{1}{k^2}\right) = \frac{1}{k^2} $ which, repeating the process, becomes $ f\left(\frac{1}{k^{2n}\right) = \frac{1}{k^{2n} $. Then if we take $ n \to \infty$, we have $ f(x) \to \infty $ as $ x \to \infty $, again giving a contradiction.

Therefore $k = 1 = xf(x) \Rightarrow f(x) = \frac{1}{x} $ is the only such function. QED.

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Practice Problem: (1986 IMO - #5) Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x) \neq 0$ for $0\le x<2$, and$ f(xf(y))f(y)=f(x+y)$ for all $x,y$.

x = 0 gives us f(0) f(y) = f(y), and because f(x) \neq 0 for 0 \le x

ReplyDeleteDamn lack of HTML filtering. Anyway, f(0) = 1 and f(0) f(2) = f(x+2) = 0 so f(x) = 0 for x \ge 2. Then if x + y = 2 for y \neq 2 we get f(x f(y)) f(y) = 0, so f(x f(y)) = 0 and x f(y) \ge 2 so f(y) \ge 2 / (2 - y) . And, err, I have to eat and do my homework now. XD

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