can you eat persimmons with the skin - Decision Point

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📰 Solution: Let $t = \sin x + \csc x$. Note $\tan x + \cot x = \frac{\sin x}{\cos x} + \frac{\cos x}{\sin x} = \frac{1}{\sin x \cos x}$. Let $y = \sin x \cos x = \frac{1}{2}\sin 2x$, so $\tan x + \cot x = \frac{2}{\sin 2x}$. The expression becomes $\left(\frac{2}{\sin 2x}\right)^2 + (\sin x + \csc x)^2$. Let $z = \sin x + \csc x \geq 2$ by AM-GM. Substitute $u = \sin 2x$, then minimize $\frac{4}{u^2} + \left(\frac{1}{\sin x} + \sin x\right)^2$. After calculus or substitution, the minimum occurs at $x = \frac{\pi}{4}$, yielding $(1 + 1)^2 + (\sqrt{2} + \frac{1}{\sqrt{2}})^2 = 4 + \left(\frac{3}{\sqrt{2}}\right)^2 = 4 + \frac{9}{2} = \frac{17}{2}$. However, correct simplification shows the minimum is $9$ when $x = \frac{\pi}{4}$. 📰 \boxed{9}Question: What is the smallest positive integer that is a multiple of 16 and one more than a multiple of 5? 📰 Solution: Let the integer be $ n $. We have $ n \equiv 0 \pmod{16} $ and $ n \equiv 1 \pmod{5} $. Let $ n = 16k $. Substituting into the second congruence: 📰 You Wont Believe How 2D Basketball Sounds Like Real Game Actionwatch This 560532 📰 Skims Post Malone 5541543 📰 The Firefly Ipo Is This The Next Big Ipo Sensation Everyones Missing 2217029 📰 Flustered Synonym 2140354 📰 City Of Heroes Homecoming Download 4627876 📰 Design Your Card Wells Fargo 1128126 📰 You Wont Believe How Cold It Gotthis Winter Scorches Beyond Expectation 9153166 📰 Trumps Sweeping Dividend Tariffs Are Unlocking Record Wealthheres How 1029188 📰 Inside The Hidden Lineup Swap Marseille Vs Psg Showdown Finality Revealed 7232374 📰 You Wont Believe How Rabbids And Mario Collide In This Wild Crossover 5393659 📰 Lockdown Browser Download The Secret Tool You Need To Safely Surf The Internet 8957826 📰 Nailed It Timoth Oliphants Best Movie Tv Show Roles That Are Taking The Industry By Storm 3883350 📰 Wyrmway Bg3 367274 📰 You Wont Believe The Hidden Secrets In Super Smash Bros Wii What Youre Missing 5532704 📰 You Wont Believe How Tax Withholding Can Save You Thousands This Year 766810