Practice (125)
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A farmer wants to fence in 60,000 square feet of land in a rectangular plot along a straight highway. The fence he plans to use along the highway costs \$2 per foot, while the fence for the other three sides costs \$1 per foot. How much of each type of fence will he have to buy in order to keep expenses to a minimum? What is the minimum expense?
Prove the Maximum Area of a Triangle with Fixed Perimeter is Equilateral
Use the Arithmetic Mean-Geometric Mean Inequality to find the maximum volume of a box made from a 25 by 25 square sheet of cardboard by removing a small square from each corner and folding up the sides to form a lidless box.
As shown.
Let $n$ be a positive integer. Show that $$\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{3^2}\Big)\cdots\Big(1+\frac{1}{3^n}\Big) < 2$$
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 $\{a_n\}$ be an increasing geometric sequence satisfying $a_1+a_2=6$ and $a_3+a_4=24$. Let $\{b_n\}$ be another sequence satisfying $b_n=\frac{a_n}{(a_n-1)^2}$. If $T_n$ is the sum of first $n$ terms in $\{b_n\}$, show that for any positive integer $n$, it always holds that $T_n < 3$.
Show that $$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
Show that for any integer $n\ge 2$: $n! < \Big(\frac{n+1}{2}\Big)^n$
Let $a_1, a_2, \cdots, a_n$ be positive real numbers such that $a_1\cdot a_2\cdots a_n=1$. Show that $$(1+a_1)(1+a_2)\cdots(1+a_n)\ge 2^n$$
Let ${{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}$ be positive real numbers that satisfy ${{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1$ . Prove that $$(a_2+1)^2\cdot (a_3+1)^3\cdots (a_n+1)^n\ge n^n$$
Let $a, b$ be positive real numbers. Prove $$(a+b)\sqrt{\frac{a+b}{2}} \ge a\sqrt{b} + b \sqrt{a}$$.
Let $a, b$ be positive numbers, show that $$\frac{1}{2}(a+b)+\frac{1}{4}\ge \sqrt{\frac{a+b}{2}}$$
If $a>1$, then $$\frac{1}{a-1}+\frac{1}{a} + \frac{1}{a+1} > \frac{3}{a}$$ holds.
Let $a, b$ be positive numbers such that $a+b=1$. Show that $$\Big(a+\frac{1}{a}\Big)^2 +\Big(b+\frac{1}{b}\Big)^2\ge \frac{25}{2}$$
Let $a, b, c$ be positive real numbers. Show that $$6a+4b+5c\ge 5\sqrt{ab} + 3\sqrt{bc} + 7\sqrt{ca}$$
Let $a, b, c$ be the lengths of the sides of triangle $ABC$. Show that
$$\sqrt{a}(c+a-b) + \sqrt{b}(a+b-c)+\sqrt{c}(b+c-a)\le\sqrt{(a^2 + b^2 + c^2)(a+b+c)}$$
Let $a, b, c$ be positive numbers such that $a+b+c=1$. Prove $$\Big(1+\frac{1}{a}\Big)\Big(1+\frac{1}{b}\Big)\Big(1+\frac{1}{c}\Big)\ge 64$$
(Nesbitt's Inequality) Let $a, b, c$ be positive numbers. Show that $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}$$
Let $a, b, c$ be positive numbers lying in the interval $(0, 1]$. Show that $$\frac{a}{1+b+ca}+\frac{b}{1+c+ab}+\frac{c}{1+a+bc}\le 1$$
Let $x, y, z$ be strictly positive real numbers. Prove that $$\Big(\frac{x}{y}+\frac{z}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{y}{z}+\frac{x}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{z}{x}+\frac{y}{\sqrt[3]{xyz}}\Big)^2 \ge 12$$
Let $x, y, z$ be three distinct positive real numbers such that $x+\sqrt{y+\sqrt{z}}=z+\sqrt{y+\sqrt{x}}$. Show that $40xz<1$
(Rearrangement Theorem) Let $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ be sequences of positive real numbers, and let $c_1, c_2, \cdots, c_n$ be a permutation of $b_1, b_2, \cdots, b_n$. The sum $S=a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximal if the two sequences $a_1, a_2, \cdots, a_n$ and $b_1, b_2, \cdots, b_n$ are sorted in the same way and minimal if the two sequences are sorted oppositely, one increasing and the other decreasing.
Let real numbers $a_1$, $a_2$, $\cdots$, $a_{2016}$ satisfy $9a_i\ge 11a_{i+1}^2$ for $i=1, 2,\cdots, 2015$. Define $a_{2017}=a_1$, find the maximum value of $$P=\displaystyle\prod_{i=1}^{2016}(a_i-a_{i+1}^2)$$
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]
Proposed by Angelo Di Pasquale, Australia