Find the greatest integer less than $$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{1000000}}$$
Solve $$\Big|\frac{1}{\log_{\frac{1}{2}}x+2}\Big|> \frac{3}{2}$$
Let real numbers $a$, $b$, and $c$ satisfy $a+b+c=2$ and $abc=4$. Find
the minimal value of the largest among $a$, $b$, and $c$.
the minimal value of $\mid a\mid +\mid b \mid +\mid c \mid$.
Let real numbers $a, b, c$ satisfy $a > 0$, $b>0$, $2c>a+b$, and $c^2>ab$. Prove $$c-\sqrt{c^2-ab} < a < c +\sqrt{c^2-ab}$$
Let real numbers $x$, $y$, and $z$ satisfy $0 < x, y, z < 1$. Prove $$x(1-y)+y(1-z)+z(1-x)< 1$$
Find the minimal value of $y=\sqrt{x^2+2x+5}+\sqrt{x^2-4x+5}$.
Find the minimum possible value of \[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]given that $a$, $b$, $c$, $d$ are nonnegative real numbers such that $a+b+c+d=4$.
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$
(Cauchy–Schwarz inequality) Show that if $u$ and $v$ a two vectors, then $|\langle u, v\rangle|^2\le \langle u, u\rangle\cdot\langle v, v\rangle$. This inequality can also be written as $$|u_1v_1+u_2v_2+\cdots +u_nv_n|^2 \le (|u_1|^2+|u_2|^2+\cdots|u_n|^2)(|v_1|^2+|v_2|^2+\cdots|v_n|^2)$$
Let $x$, $y$, $z$ be three positive real numbers satisfying $xyz+x+z=y$. Find the maximum value of $$P=\frac{2}{x^2 + 1}-\frac{2}{y^2+1}+\frac{3}{z^2+1}$$
Let real numbers $x$ and $y$ satisfy the relation $4x^2-5xy+4y^2=5$. Find the maximum and minimal value of $x^2+y^2$.
Given non-negative real numbers $x$, $y$ and $z$, prove
$$\sqrt{x^2+y^2-xy}+\sqrt{y^2 + z^2 - yz}\ge\sqrt{x^2+z^2+xz}$$
$$|\sin x + \cos x + \tan x + \cot x + \sec x + \csc x|$$
where $x$ is a real number.
Let $a$ and $b$ be two positive real numbers not exceeding $1$. Prove $$\frac{1}{\sqrt{a^2 + 1}}+\frac{1}{\sqrt{b^2 +1}}\le\frac{2}{\sqrt{1+ab}}$$
Solve this inequality $$\frac{x}{\sqrt{x^2 +1}}+\frac{1-x^2}{1+x^2} > 0$$
Let real numbers $a$ and $b$ satisfy $0 < a < a +\frac{1}{2} \le b$ and $a^{40}+b^{40}=1$. Show that all the twelve digits after the decimal point are $9$ if $b$ is expressed in decimal.
Does the expression $x+\sqrt{2x^2-2x+1}$ has either maximum or minimal value?
Show
$$\frac{1}{2} \cdot \frac{3}{4} \cdots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n}}$$
Find the minimal value of $\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8}$.
Let $m$ and $n$ be two positive integers, find the minimal value of $\mid 12^m - 5^n\mid$.
Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB + CD = 12$, and $BC + AD = 13$.
Find the greatest possible area for $ABCD$.
Let $f(x)=a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a $n$-degree polynomial and all its coefficients $a_i$ $(0\le i\le n)$ be either $1$ or $-1$. If $f(x)$ has only real roots, what is the maximum value of $n$?
Let $m$ and $n$ be positive integers satisfying $1 < m < n$. Show that $(1+m)^n > (1+n)^m$.
Show that the following relation holds for any positive integers $1 < k \le m < n$: $$\binom{n}{k}m^k > \binom{m}{k}n^k$$
Let $m$ and $n$ be positive integers. Show that $$\frac{(m+n)!}{(m+n)^{m+n}}<\frac{m!}{m^m}\frac{n!}{n^n}$$