# Kirthi Raman

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"You have enemies? Good. That means you've stood up for something, sometime in your life" - Winston Churchill

 47 Funny identities 28 If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square. 21 Which mathematicians have influenced you the most? 20 Numbers are too large to show $65^{64}+64^{65}$ is not a prime 17 Show by substitution that $\int_0^{\pi} \frac{x\sin x}{1+\cos^2 x} \,\mathrm dx = \frac{\pi}{2}\int_0^{\pi} \frac{\sin x}{1+\cos^2 x} \,\mathrm dx$

# 5,657 Reputation

 +15 Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$ +5 Find all solutions of $1/x+1/y+1/z=1$, where $x$, $y$ and $z$ are positive integers +10 If $x$ and $y$ are rational numbers and $x^5+y^5=2x^2y^2,$ then $1-xy$ is a perfect square. +5 Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$

# 25 Questions

 20 Evaluating $\int_0^1 \log \log \left(\frac{1}{x}\right) \frac{dx}{1+x^2}$ 11 Solve the integral $S_k = (-1)^k \int_0^1 (\log(\sin \pi x))^k dx$ 6 Let $a,b$ be positive real numbers. Prove $\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}} \geq \frac{2}{\sqrt{1+ab}}$ 6 Prove $1^a+2^a+\cdots+n^a < \frac{(n+1)^{(a+1)}-1}{a+1}$ for any $a >0$ and $n \in \mathbb{Z^+}$ 6 What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$

# 75 Tags

 96 elementary-number-theory × 29 40 sequences-and-series × 10 75 integration × 21 38 polynomials × 6 67 algebra-precalculus × 14 26 inequality × 9 61 calculus × 21 25 diophantine-equations × 6 49 number-theory × 11 23 trigonometry × 11

# 2 Accounts

 Mathematics 5,657 rep 1748 Stack Overflow 101 rep 2