# Xeing

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# 68 Questions

 23 Find integer in the form: $\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b}$ 15 Find $x,y,z \in \mathbb Q$ such that $x + \frac 1y, y + \frac 1z, z+ \frac 1x \in \mathbb Z$ 13 Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots 12 Find $x\in \mathbb{Z}$ such that $54x^3+1$ is a cube 10 Find minimum of $\frac x {x^2+1} + \frac y{y^2+1} + \frac z{z^2+1}$

# 1,348 Reputation

 +5 Prove inequality: $\sum \frac{a^4}{a^3+b^3} \ge \frac{a+b+c}{2}$ +15 How to prove $\frac{a}{a+bc}+\frac{b}{b+ac}+\frac{c}{c+ab}\geq \frac{3}{2}$ +10 Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$? +5 How many binary string are there such that there are no k consecutive characters are the same?

 5 The number $n^4 + 4$ is never prime for $n>1$ 5 Find maxi,minimum $f(x,y)=x^3+y^3 (\text{where} ~~~ x,y\in \mathbb{R}, x^2+y^2=1)$ 2 Prove $x^2+y^2+z^2 \ge 14$ with constraints 1 Assume $n$ is even. Prove that $323$ divides $20^n+16^n-3^n-1$. 1 Let $a,b,c>0$ and $a+b+c= 1$, how to prove the inequality $\frac{\sqrt{a}}{1-a}+\frac{\sqrt{b}}{1-b}+\frac{\sqrt{c}}{1-c}\geq \frac{3\sqrt{3}}{2}$?

# 35 Tags

 6 elementary-number-theory × 13 0 number-theory × 22 5 divisibility × 2 0 diophantine-equations × 11 5 calculus 0 algorithms × 11 3 inequality × 32 0 geometry × 4 2 optimization × 9 0 combinatorics × 4

# 5 Accounts

 Mathematics 1,348 rep 322 Stack Overflow 194 rep 8 Programmers 101 rep 3 Information Security 101 rep 2 Computer Science 97 rep 2