IMO Shortlist 2008 problem A3
Kvaliteta:
Avg: 3,0Težina:
Avg: 7,0 Let
be a set of real numbers. We say that a pair
of functions from
into
is a Spanish Couple on
, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e.
and
for all
,
with
;
(ii) The inequality
holds for all
.
Decide whether there exists a Spanish Couple on the set
of positive integers; on the set
Proposed by Hans Zantema, Netherlands
![S\subseteq\mathbb{R}](/media/m/b/e/4/be4829eb8c157b1d56b1ba9caa4c2c09.png)
![(f, g)](/media/m/3/3/a/33aa43a78efd6befde743ad1e77a8ae6.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
(i) Both functions are strictly increasing, i.e.
![f(x) < f(y)](/media/m/1/d/7/1d7a028b6dc4422f490c416e801e13ca.png)
![g(x) < g(y)](/media/m/0/5/2/0521cbf9ee82cd615ff4f9976c17b84f.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![y\in S](/media/m/2/4/8/248a78000e27674e524cef50fe46d185.png)
![x < y](/media/m/0/8/b/08b3f86331faf64ac3ba1d2d58aacb0a.png)
(ii) The inequality
![f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)](/media/m/5/6/9/5696fd5e1a62cd7c5d448bf77d78570b.png)
![x\in S](/media/m/b/6/5/b65adf7efa0225ddbbe87194ac8ca6ba.png)
Decide whether there exists a Spanish Couple on the set
![S = \mathbb{N}](/media/m/1/d/f/1df8134435e54df195a92c8b8b327db1.png)
![S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}](/media/m/6/6/1/6613dbc12c489194e7a2f32ee776516a.png)
Proposed by Hans Zantema, Netherlands
Izvor: Međunarodna matematička olimpijada, shortlist 2008