IMO Shortlist 1969 problem 22
Dodao/la:
arhiva2. travnja 2012. ![(FRA 5)](/media/m/f/f/8/ff834472ce41af5b56573ee72ad36101.png)
Let
![\alpha(n)](/media/m/2/b/a/2ba91a633148e5989cf20d1a5d6e6289.png)
be the number of pairs
![(x, y)](/media/m/1/5/2/1520b43353795b60686f7df83802e90a.png)
of integers such that
![x+y = n, 0 \le y \le x](/media/m/c/e/b/cebb6f1f3dd3bbad27e588d35dfc31df.png)
, and let
![\beta(n)](/media/m/2/1/7/217f71028aea4a4049542e6d0e3f0b59.png)
be the number of triples
![(x, y, z)](/media/m/f/2/d/f2d4c9b9b3e7f29445f7a1063c15263f.png)
such that
![x + y + z = n](/media/m/b/e/3/be3de3c88fb706de5c84a43397fddcc3.png)
and
![0 \le z \le y \le x.](/media/m/2/8/e/28e95bd2e7e65a17d107c4c1de699f11.png)
Find a simple relation between
![\alpha(n)](/media/m/2/b/a/2ba91a633148e5989cf20d1a5d6e6289.png)
and the integer part of the number
![\frac{n+2}{2}](/media/m/d/f/b/dfbb701c5a48b13648be04693129eae0.png)
and the relation among
![\beta(n), \beta(n -3)](/media/m/d/f/d/dfdc8985f75c17ca7fe760643607653c.png)
and
![\alpha(n).](/media/m/b/3/e/b3e9bbd9fb070229acaf90067064715e.png)
Then evaluate
![\beta(n)](/media/m/2/1/7/217f71028aea4a4049542e6d0e3f0b59.png)
as a function of the residue of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
modulo
![6](/media/m/e/e/e/eeec330d59a70f8ed1d6882474cb02a3.png)
. What can be said about
![\beta(n)](/media/m/2/1/7/217f71028aea4a4049542e6d0e3f0b59.png)
and
![1+\frac{n(n+6)}{12}](/media/m/5/4/a/54aad3bed4549a85891f280e9456e0c3.png)
? And what about
![\frac{(n+3)^2}{6}](/media/m/7/e/2/7e20e5ec80f80b9805418c3934943c65.png)
?
Find the number of triples
![(x, y, z)](/media/m/f/2/d/f2d4c9b9b3e7f29445f7a1063c15263f.png)
with the property
![x+ y+ z \le n, 0 \le z \le y \le x](/media/m/b/0/e/b0e095eac368fd15e892b8f1020c9b25.png)
as a function of the residue of
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
modulo
![6.](/media/m/5/b/8/5b8d6cf04efde2c9bbe2f755fd45be6d.png)
What can be said about the relation between this number and the number
![\frac{(n+6)(2n^2+9n+12)}{72}](/media/m/2/1/2/212ee1a5bf58fc267b9de942d11f606f.png)
?
%V0
$(FRA 5)$ Let $\alpha(n)$ be the number of pairs $(x, y)$ of integers such that $x+y = n, 0 \le y \le x$, and let $\beta(n)$ be the number of triples $(x, y, z)$ such that$x + y + z = n$ and $0 \le z \le y \le x.$ Find a simple relation between $\alpha(n)$ and the integer part of the number $\frac{n+2}{2}$ and the relation among $\beta(n), \beta(n -3)$ and $\alpha(n).$ Then evaluate $\beta(n)$ as a function of the residue of $n$ modulo $6$. What can be said about $\beta(n)$ and $1+\frac{n(n+6)}{12}$? And what about $\frac{(n+3)^2}{6}$?
Find the number of triples $(x, y, z)$ with the property $x+ y+ z \le n, 0 \le z \le y \le x$ as a function of the residue of $n$ modulo $6.$What can be said about the relation between this number and the number $\frac{(n+6)(2n^2+9n+12)}{72}$?
Izvor: Međunarodna matematička olimpijada, shortlist 1969