IMO Shortlist 1975 problem 9
Dodao/la:
arhiva2. travnja 2012. Let
![f(x)](/media/m/3/f/4/3f40d68090aa4fb60a440be4675c7aca.png)
be a continuous function defined on the closed interval
![0 \leq x \leq 1](/media/m/7/7/b/77bd022353f5e779e66731bb2b68bb27.png)
. Let
![G(f)](/media/m/0/5/4/05456b2d0c5bc1cc58778740e4188e14.png)
denote the graph of
![x \leq 1, y = f(x) \}](/media/m/b/9/e/b9eb405da10e884384bb81c6f8a4c394.png)
. Let
![G_a(f)](/media/m/9/0/f/90ff75050fc6e3814c46cc11d1f83cd8.png)
denote the graph of the translated function
![f(x - a)](/media/m/9/8/b/98bce54ab551d17f518f4416fe3e1b33.png)
(translated over a distance
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
), defined by
![G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}](/media/m/9/f/1/9f146e9c3518b3cd9e33844801277b39.png)
. Is it possible to find for every
![a, \ 0 < a < 1](/media/m/4/c/9/4c9c9391e14cb3d595306aaa3e46d9a1.png)
, a continuous function
![f(x)](/media/m/3/f/4/3f40d68090aa4fb60a440be4675c7aca.png)
, defined on
![0 \leq x \leq 1](/media/m/7/7/b/77bd022353f5e779e66731bb2b68bb27.png)
, such that
![f(0) = f(1) = 0](/media/m/9/8/9/98962154b9343e5baa31adb620baa187.png)
and
![G(f)](/media/m/0/5/4/05456b2d0c5bc1cc58778740e4188e14.png)
and
![G_a(f)](/media/m/9/0/f/90ff75050fc6e3814c46cc11d1f83cd8.png)
are disjoint point sets ?
%V0
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$ $x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
Izvor: Međunarodna matematička olimpijada, shortlist 1975