## USAMO 1972 #4 July 26, 2009

Posted by lumixedia in Problem-solving.
Tags: , , , ,

USAMO 1972 #4. Let ${R}$ denote a non-negative rational number. Determine a fixed set of integers ${a}$, ${b}$, ${c}$, ${d}$, ${e}$, ${f}$ such that, for every choice of ${R}$,

$\displaystyle |\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}|<|R-\sqrt[3]{2}|.$

Solution. From the desired inequality, we can conclude that

$\displaystyle 0\le\lim_{R\rightarrow\sqrt[3]{2}}|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}|$

$\displaystyle \le\lim_{R\rightarrow\sqrt[3]{2}}||R-\sqrt[3]{2}|=0$

So

$\displaystyle \lim_{R\rightarrow\sqrt[3]{2}}|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}|=0$

Since ${\sqrt[3]{2}}$ is not the root of any quadratic polynomial with integer coefficients, ${\frac{aR^2+bR+c}{dR^2+eR+f}}$ is continuous at ${\sqrt[3]{2}}$ and therefore

$\displaystyle \frac{a(\sqrt[3]{2})^2+b\sqrt[3]{2}+c}{d(\sqrt[3]{2})^2+e\sqrt[3]{2}+f}-\sqrt[3]{2}=0$

$\displaystyle a2^{2/3}+b2^{1/3}+c=e2^{2/3}+f2^{1/3}+2d$

For this equation to be satisfied, we must have ${a=e}$, ${b=f}$, ${c=2d}$. So the LHS of the inequality becomes the absolute value of

$\displaystyle \frac{aR^2+bR+2d}{dR^2+aR+b}-\sqrt[3]{2}$

$\displaystyle =\frac{aR^2+bR+2d-dR^2\sqrt[3]{2}-aR\sqrt[3]{2}-b\sqrt[3]{2}}{dR^2+aR+b}$

$\displaystyle =\frac{aR(R-\sqrt[3]{2})+b(R-\sqrt[3]{2})-d\sqrt[3]{2}(R+\sqrt[3]{2})(R-\sqrt[3]{2})}{dR^2+aR+b}$

$\displaystyle =(R-\sqrt[3]{2})\frac{aR+b-d\sqrt[3]{2}(R+\sqrt[3]{2})}{dR^2+aR+b}$

Dividing both sides of the inequality by ${|R-\sqrt[3]{2}|}$ and multiplying by ${|dR^2+aR+b|}$, we find that it is equivalent to

$\displaystyle |aR+b-d\sqrt[3]{2}(R+\sqrt[3]{2})|<|dR^2+aR+b|$

This is certainly satisfied if we choose ${(a,b,d)}$ so both ${aR+b-d\sqrt[3]{2}(R+\sqrt[3]{2})}$ and ${dR^2+aR+b}$ are always positive for nonnegative ${R}$. Any ${(a,b,d)}$ with ${a}$,${b}$,${d>0}$, ${a>d\sqrt[3]{2}}$, and ${b>d2^{2/3}}$ will work. One example is ${(a,b,c,d,e,f)=(2,2,2,1,2,2)}$.

It might possibly be interesting to find conditions on ${(a,b,d)}$ which are necessary and sufficient for the inequality to hold. It might also not be very interesting at all.

I suggested in the AoPS thread on the problem here that an interesting follow-up would be to figure out which choices take convergents of the continued fraction of $\sqrt[3]{2}$ to convergents. It’s probable no choice with this property exists since the continued fraction of non-square roots are aperiodic.