1983 IMO Problems/Problem 1

Problem

Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy the conditions: (i) $f(xf(y))=yf(x)$ for all $x,y$ ; (ii) $f(x)\to0$ as $x\to \infty$ .

Solution 1

Let $x=y=1$ and we have $f(f(1))=f(1)$ . Now, let $x=1,y=f(1)$ and we have $f(f(f(1)))=f(1)f(1)\Rightarrow f(1)=[f(1)]^2$ since $f(1)>0$ we have $f(1)=1$ .

Plug in $y=x$ and we have $f(xf(x))=xf(x)$ . If $a=1$ is the only solution to $f(a)=a$ then we have $xf(x)=1\Rightarrow f(x)=\frac{1}{x}$ . We prove that this is the only function by showing that there does not exist any other $a$ :

Suppose there did exist such an $a\ne1$ . Then, letting $y=a$ in the functional equation yields $f(xa)=af(x)$ . Then, letting $x=\frac{1}{a}$ yields $f(\frac{1}{a})=\frac{1}{a}$ . Notice that since $a\ne1$ , one of $a,\frac{1}{a}$ is greater than $1$ . Let $b$ equal the one that is greater than $1$ . Then, we find similarly (since $f(b)=b$ ) that $f(xb)=bf(x)$ . Putting $x=b$ into the equation, yields $f(b^2)=b^2$ . Repeating this process we find that $f(b^{2^k})=b^{2^k}$ for all natural $k$ . But, since $b>1$ , as $k\to \infty$ , we have that $b^{2^k}\to\infty$ which contradicts the fact that $f(x)\to0$ as $x\to \infty$ .

Solution 2

Let $x=1$ so $f(f(y))=yf(1).$ If $f(a)=f(b),$ then $af(1)=f(f(a))=f(f(b))=bf(1)\implies a=b$ because $f(1)$ goes to the real positive integers, not $0.$ Hence, $f$ is injective. Let $x=y$ so $f(xf(x))=xf(x)$ so $xf(x)$ is a fixed point of $f.$ Then, let $y=1$ so $f(xf(1))=f(x)\implies f(1)=1$ as $x$ can't be $0$ so $1$ is a fixed point of $f.$ We claim $1$ is the only fixed point of $f.$ Suppose for the sake of contradiction that $a,b$ be fixed points of $f$ so $f(a)=a$ and $f(b)=b.$ Then, setting $x=a,y=b$ in (i) gives $f(ab)=f(af(b))=bf(a)=ab$ so $ab$ is also a fixed point of $f.$ Also, let $x=\frac{1}{a},y=a$ so $1=f(1)=f(\frac{1}{a}\cdot a)=f(\frac{1}{a}\cdot f(a))=af(\frac{1}{a})\implies f(\frac{1}{a})=\frac{1}{a}$ so $\frac{1}{a}$ is a fixed point of $f.$ If $f(a)=a$ with $a>1,$ then $f(a^n)$ is a fixed point of $f$ , contradicting (ii). If $f(a)=a$ with $0<a<1,$ then $f(\frac{1}{a^n})=\frac{1}{a^n}$ so $\frac{1}{a^n}$ is a fixed point, contradicting (ii). Hence, the only fixed point is $1$ so $xf(x)=1$ so $f(x)=\boxed{\frac{1}{x}}$ and we can easily check that this solution works.

Video Solution

https://youtu.be/Fu0jBKKa4Lc [Video Solution by little fermat]

1983 IMO (Problems) • Resources
Preceded by First question	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 2
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1983 IMO Problems/Problem 1

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Problem

Solution 1

Solution 2

Video Solution