Difficult but Interesting Inequalities Problems

Question

1.) Consider the identity

$$(px + (1-p)y)^2 = Ax^2 + Bxy + Cy^2.$$

Find the minimum of $\max(A,B,C)$ over $0 \leq p \leq 1$.

2.) Let $n$ be a positive integer. Show that the smallest integer greater than $(\sqrt{3} + 1)^{2n}$ is divisible by $2^{n+1}$.

I saw both of these problems as I was working on some algebra review problems. I was stuck on these for quite a while, with no clue how to do either. For the second problem, I tried proving $\lceil (\sqrt{3}+1)^{2n} \rceil = (\sqrt{3}+1)^{2n} + (\sqrt{3}-1)^{2n}$, in hopes of simplifying to get the answer, but I could not do so. Can I have some help as to how to do these problems? It would be greatly appreciated, thanks!

Answer 1

For the first one, the max should be minimized when two of $A,B,C$ are equal. Checking all three possibilities, you see that the minimum occurs when $B=C$ at $p=1/3$.

Answer 2

Thats two questions, the first you already have an answer for $$\min_p \max(A, B, C) = \min_p \max\left(p^2, 2p(1-p), (1-p)^2\right) = \frac49, \quad \text{when } p \in \{\tfrac13, \tfrac23 \} $$

For 2), note that $0< (\sqrt3-1)^{2n}< 1$, and $(\sqrt3+1)^{2n}+(\sqrt3-1)^{2n}$ is an integer (easily seen using the binomial theorem). Hence the smallest integer we seek is in fact $a_n = (\sqrt3+1)^{2n}+(\sqrt3-1)^{2n}$ and we need to show that $2^{n+1} \mid a_n$.

It follows from the theory of linear homogeneous recurrences that the characteristic equation for $a_n$ is $\left(x- (\sqrt3+1)^2\right)\left(x- (\sqrt3-1)^2\right)=0 \iff x^2-8x+4=0$, so $a_n$ satisfies $a_{n+2} = 4\cdot (2a_{n+1}-a_n)$. Now it is easy to show by induction that $2^{n+1} \mid a_n$.