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6 problems match “AM-GM inequality”. Clear

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1. Minimum area with given hypotenuse altitude ●●○○○ 20 ptsIn right triangle $ABC$, the altitude to the hypotenuse has length 4. Find the minimum value of the area of triangle $ABC$.
2. Minimum sum of squares after error ●●○○○ 20 ptsAshok multiplied three real numbers. He used 73 instead of 37 for one of the numbers, and as a result the product increased by 720. Find the minimum possible value of the sum of the squares of the other two numbers.
3. Minimum of a trigonometric sum ●●●○○ 50 ptsFor real numbers $x$ for which all six functions are defined, find the minimum value of $\left|\sin x+\cos x+\tan x+\cot x+\sec x+\csc x\right|$. Round your answer to two decimal places.
4. Maximum of tan beta under a constraint ●●●○○ 50 ptsGiven $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$ with $\alpha+\beta \neq \frac{\pi}{2}$ and $\sin \beta=2 \cos (\alpha+\beta) \sin \alpha$, find the maximum value of $\tan \beta$. Round to two decimal places.
5. Maximum traffic flow over average speed ●●●○○ 50 ptsOn a certain road section, the traffic flow $y$ (in thousands of vehicles per hour) and the average speed $v$ (in km/h) of vehicles satisfy $y=\frac{670(v-5)}{v^2-8v+915}$ for $v>5$. Find the average speed $v$ at which the traffic flow is largest, and the maximum traffic flow. Round the maximum traffic flow to one decimal place.
6. Maximum of |bc| for a bounded quadratic ●●●●● 200 ptsLet complex numbers $a$, $b$, $c$ satisfy: for any complex number $z$ with $|z|\leq 1$, we have $|az^2+bz+c|\leq 1$. Find the maximum value of $|bc|$. Round to two decimal places.