Inequalities (Standard)

Inequalities: core idea

• 1. Max $xy$ given $x+y=10$, $x,y>0$?

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• 2. Solve $x^2-5x+6\le 0$: number of integer solutions?

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• 3. Find the minimum value of $x+\dfrac1x$ for $x>0$.

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• 4. Solve $\dfrac{x-1}{x+1}>0$: number of integers from $-5$ to $5$ inclusive satisfying?

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• 5. Cauchy on $(1,1,1)$ and $(a,b,c)$: $(\sum a)^2\le3\sum a^2$. For $a=b=c=1$: left?

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• 6. Max $-x^2+8x-7$?

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• 7. Rearrangement: sorted same order max sum products — for $(1,2)$ and $(3,4)$ max dot?

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• 8. Solve $|x-3|<2$: greatest integer?

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• 9. Min $a+b$ for $ab=16$, $a,b>0$?

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