IMO Shortlist 1969 problem 59
Dodao/la:
arhiva2. travnja 2012. ![(SWE 2)](/media/m/e/6/5/e65e844d0c53c388ba8ed948ffe399e0.png)
For each
![\lambda (0 < \lambda < 1](/media/m/e/f/a/efa0ef022cd106418654fc228eb62906.png)
and
![\lambda = \frac{1}{n}](/media/m/a/b/3/ab38c7748ccab7ccca87cd65d3533cb7.png)
for all
![n = 1, 2, 3, \cdots)](/media/m/2/8/2/2823a8ee02502835ff40e99cb8460598.png)
, construct a continuous function
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
such that there do not exist
![x, y](/media/m/2/7/9/279a699b10f7b70e7160f4aaaa89e453.png)
with
![0 < \lambda < y = x + \lambda \le 1](/media/m/4/7/b/47bbfe3610ab8f75d09311f1b07cf15c.png)
for which
%V0
$(SWE 2)$ For each $\lambda (0 < \lambda < 1$ and $\lambda = \frac{1}{n}$ for all $n = 1, 2, 3, \cdots)$, construct a continuous function $f$ such that there do not exist $x, y$ with $0 < \lambda < y = x + \lambda \le 1$ for which $f(x) = f(y).$
Izvor: Međunarodna matematička olimpijada, shortlist 1969