IMO Shortlist 1993 problem A5
Dodao/la:
arhiva2. travnja 2012. ![a > 0](/media/m/7/9/1/791ecf93ca3ba277ca9acb1d38d57bd3.png)
and
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
,
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
are integers such that
![ac](/media/m/5/b/4/5b4687963f1b54d4381661aa2a11d123.png)
–
![b^2](/media/m/c/0/8/c086d4e67e4993c92c2005b15b9fb759.png)
is a square-free positive integer P. For example P could be 3*5, but not 3^2*5. Let
![f(n)](/media/m/d/3/e/d3e47283bffbbf24c97f0c6474d5a82d.png)
be the number of pairs of integers
![d, e](/media/m/5/f/b/5fb59d4c7ec8316289ff4d89609e480d.png)
such that
![ad^2 + 2bde + ce^2= n](/media/m/7/9/d/79d9654f2cdd26dffdd46d81e00bd5d5.png)
. Show that
![f(n)](/media/m/d/3/e/d3e47283bffbbf24c97f0c6474d5a82d.png)
is finite and that
![f(n) = f(P^{k}n)](/media/m/8/2/4/8240e9cec45ce89572edb9f197af9a64.png)
for every positive integer
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
.
Original Statement:
Let
![a,b,c](/media/m/3/6/4/36454fdb50fc50f021324b33a6b513e3.png)
be given integers
![ac-b^2 = P = P_1 \cdots P_n](/media/m/7/f/0/7f04a608ef13672732923edf6a01b523.png)
where
![P_1 \cdots P_n](/media/m/9/9/0/9907dc8c571661d2c2a0c81137d53654.png)
are (distinct) prime numbers. Let
![M(n)](/media/m/b/1/d/b1dbf4d979162e9e2429922979eba51b.png)
denote the number of pairs of integers
![(x,y)](/media/m/c/9/1/c91aec4078b932368ded863349deaec5.png)
for which
![ax^2 + 2bxy + cy^2 = n.](/media/m/c/8/7/c87375574b916f2881528fca9273cd2b.png)
Prove that
![M(n)](/media/m/b/1/d/b1dbf4d979162e9e2429922979eba51b.png)
is finite and
![M(n) = M(P_k \cdot n)](/media/m/8/6/8/8681660a67864b42ab945040a3322ec8.png)
for every integer
![k \geq 0.](/media/m/6/9/e/69e910d8f6b40453271ac9983ab78a42.png)
Note that the "
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
" in
![P_N](/media/m/3/8/e/38e4e79ed6aa6be58e08e8fc1fcb7096.png)
and the "
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
" in
![M(n)](/media/m/b/1/d/b1dbf4d979162e9e2429922979eba51b.png)
do not have to be the same.
%V0
$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. For example P could be 3*5, but not 3^2*5. Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$.
Original Statement:
Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which $$ax^2 + 2bxy + cy^2 = n.$$ Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.
Izvor: Međunarodna matematička olimpijada, shortlist 1993