Library / Irish Mathematical Olympiad /25 of 236

Irish Mathematical Olympiad, 2014

Algebra Difficulty 4.6 AIME Prove it Ireland

Suppose $t$ is a real number and $\sin(2t) > 0$ . Prove that
$1 + 6 \sin t \cos t \geq (\sin^3 t + \cos^3 t)(\sin t + \cos t)^3 + 16 \sin^3 t \cos^3 t,$
with equality iff $\cos(2t) = 0$ .

Solution

Let $s$ stand for $\sin t$ , $c$ for $\cos t$ , and let $x = 2sc = \sin(2t)$ . Then
$\begin{aligned} & (\sin^3 t + \cos^3 t)(\sin t + \cos t)^3 + 16 \sin^3 t \cos^3 t \\ &= (s^3 + c^3)(s + c)^3 + 2x^3 \\ &= (s^2 - sc + c^2)(s + c)^4 + 2x^3 \\ &= (1 - sc)(s^2 + 2sc + c^2)^2 + 2x^3 \\ &= \left(1 - \frac{1}{2}x\right) (1 + x)^2 + 2x^3 = 1 + \frac{3}{2}x + \frac{3}{2}x^3. \end{aligned}$
Hence, the inequality reduces to proving that
$1 + 3x \ge 1 + \frac{3}{2}x + \frac{3}{2}x^3 \quad \text{or equivalently} \quad x(1 - x^2) \ge 0,$
which is true since $0 < x \le 1$ . There is equality iff $\sin^2(2t) = x^2 = 1$ , equivalently, iff $\cos^2(2t) = 0$ as stated.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.