IMO Shortlist 1997 problem 1
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 In the plane the points with integer coordinates are the vertices of unit squares. The squares are coloured alternately black and white (as on a chessboard). For any pair of positive integers
and
, consider a right-angled triangle whose vertices have integer coordinates and whose legs, of lengths
and
, lie along edges of the squares. Let
be the total area of the black part of the triangle and
be the total area of the white part. Let
.
a) Calculate
for all positive integers
and
which are either both even or both odd.
b) Prove that
for all
and
.
c) Show that there is no constant
such that
for all
and
.
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![S_2](/media/m/c/1/1/c11855875777bfedb764b27ccc108413.png)
![f(m,n) = | S_1 - S_2 |](/media/m/0/7/7/077694ded48614856e81de31b18da944.png)
a) Calculate
![f(m,n)](/media/m/c/d/f/cdf87ccace47df2a9b8c2fd7780237dc.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
b) Prove that
![f(m,n) \leq \frac 12 \max \{m,n \}](/media/m/d/e/5/de59945f6bfab232b528d69cc7ecf5be.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
c) Show that there is no constant
![C\in\mathbb{R}](/media/m/d/a/1/da174be93f3b797ac88af0fb745e31d5.png)
![f(m,n) < C](/media/m/b/c/c/bcc3a448738f2eae606f23a57ed15446.png)
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1997