Practice (127)
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Let $x$ and $y$ be two positive real numbers. Find the maximum value of $\frac{(3x+4y)^2}{x^2 + y^2}$.
Find all positive integers $n$ and $k_i$ $(1\le i \le n)$ such that $$k_1 + k_2 + \cdots + k_n = 5n-4$$ and $$\frac{1}{k_1} + \frac{1}{k_2} + \cdots + \frac{1}{k_n}=1$$
Let $a$ and $b$ be two positive real numbers. Show that if $\frac{1}{a}+\frac{1}{b}=1$. Prove that the following inequality holds for any positive integer $n$: $$(a+b)^n-a^n-b^n\ge 2^{2n}-2^{n+1}$$
As shown.
Let $a_1, a_2,\cdots, a_n > 0, n\ge 2,$ and $a_1+a_2+\cdots+a_n=1$. Prove $$\frac{a_1}{2-a_1} + \frac{a_2}{2-a_2}+\cdots+\frac{a_n}{2-a_n}\ge\frac{n}{2n-1}$$
Let $a, b, c$ be the lengths of the sides of triangle $ABC$. Show that
$$\sqrt{a}(c+a-b) + \sqrt{b}(a+b-c)+\sqrt{c}(b+c-a)\le\sqrt{(a^2 + b^2 + c^2)(a+b+c)}$$