97 Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle 60 Why does the discriminant in the Quadratic Formula reveal the number of real solutions? 52 How to simplify $\int{\sqrt{1-8{{x}^{2}}+8{{x}^{4}}-4x\sqrt{{{x}^{2}}-1}+8{{x}^{3}}\sqrt{{{x}^{2}}-1}}dx}$? 49 Can the product of three complex numbers ever be real? 41 Can sum of a rational number and its reciprocal be an integer?

### Reputation (170,603)

 +20 Why is $2-\sqrt{4-x}$ equal to $\frac{x}{2+\sqrt{4-x}}$? +25 For any real positive numbers $a, b, c$, prove that $3(a^2b+b^2c+c^2a)(ab^2+bc^2+ca^2) \geq abc(a+b+c)^3$ +20 Methods for showing that 4 points lie on the same circle? +10 Find $\sqrt a + \sqrt b + \sqrt c$ only in terms of $p$ .

### Questions (114)

 77 If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$ 39 Prove that $a\sqrt{a^2+bc}+b\sqrt{b^2+ac}+c\sqrt{c^2+ab}\geq\sqrt{2(a^2+b^2+c^2)(ab+ac+bc)}$ 32 Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$ 28 Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$ 26 Inequality with five variables

### Tags (432)

 5k inequality × 3176 1k contest-math × 679 3k algebra-precalculus × 1239 1k polynomials × 649 2k a.m.-g.m.-inequality × 771 1k cauchy-schwarz-inequality × 632 2k trigonometry × 801 1k real-analysis × 538 2k calculus × 779 1k geometry × 872

### Bookmarks (763)

 92 Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing 81 References for multivariable calculus 77 If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$ 73 Cardinality of set of real continuous functions 67 Examples of finite nonabelian groups.