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 $T_n < 3$.
Given a sequence $\{a_n\}$, if $a_n\ne 0$, $a_1=1$, and $3a_na_{n-1}+a_n+a_{n-1}=0$ for any $n\ge 2$, find the general term of $a_n$.
If a sequence $\{a_n\}$ satisfies $a_1=1$ and $a_{n+1}=\frac{1}{16}\big(1+4a_n+\sqrt{1+24a_n}\big)$, find the general term of $a_n$.
Let sequence $\{a_n\}$ satisfy $a_0=1$ and $a_n=\frac{\sqrt{1+a_{n-1}^2}-1}{a_{n-1}}$. Prove $a_n > \frac{\pi}{2^{n+2}}$.
Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?
Show that $$\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}=\binom{2n}{n-1}$$
Prove that $7\mid 8^n-1$ for $n\ge 1$.
Show that $5\mid 4^{2n}-1$ for $n\ge 1$.
Prove that $15\mid 4^{2n}-1$ for $n\ge 1$.
Show that $|\sin(nx)|\le n|\sin(x)|$ for any positive integer $n$.
Solve the equation $x^4 -97x^3+2012x^2-97x+1=0$.
Show that if $a, b, c$ are the lengths of the sides of a triangle, then the equation $$b^2x^2 +(b^2+c^2-a^2)x+c^2=0$$ does not have real roots.
Solve the equation in real numbers $$\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6$$
If the product of two roots of polynomial $x^4 - 18x^3 + kx^2 + 200x - 1984 = 0$ is $- 32$. Find the value of $k$.
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 $$\Big(\frac{x}{y}+\frac{z}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{y}{z}+\frac{x}{\sqrt[3]{xyz}}\Big)^2+\Big(\frac{z}{x}+\frac{y}{\sqrt[3]{xyz}}\Big)^2 \ge 12$$