Matematické Fórum

stuart clark · 24. 10. 2012 17:20

If $\bf{a\;,b\;,c \in \mathbb{N}}$ and $\bf{f(x) = ax^2-bx+c = 0}$ has $\bf{2}$ distinct real roots in $\bf{(0,1)}$

Then minimum value of $\bf{a\;,b\;,c}$ and $\bf{abc}$ are

Brano · 26. 02. 2013 15:18 — Editoval Brano (26. 02. 2013 15:22)

Let $0<p<q<1$ be roots of $f(x)$ . Then $c=apq<a$ . And also holds
1) $b^2-4ac>0$
2) $2aq=b+\sqrt{b^2-4ac}<2a$
From 2) we get $b<a+c$ and since these are integers we have $a+c-1\ge b>2\sqrt{ac}$ so $(a+c-1)^2\ge 4ac+1$ and from this we have $(a-c)^2\ge 2(a+c)$ . Now put $d=a-c(>0)$ and subtitute for $a$ to obtain
$d^2\ge 2d+4c\ge 2d+4$ so $d\ge 4$ and since $c\ge 1$ we have $a\ge 5$ and since $b>2\sqrt{ac}$ we have $b\ge 5$ and therefore $abc\ge 25$ . Now it is enough to check that $5x^2-5x+1$ has desired roots.