An Arcy problem by Mr. Harsh Shrivastava

Geometry Level 5

$\arctan (3+a) + \arctan (3+b) = \dfrac{\pi}{2} + \text{arccot} \left(\dfrac{1}{3}\right)$

Let $a,b > 0$ be real numbers satisfying the relation above.

Find the minimum value of $a+b$ .

The answer is 6.32.

1 solution

Rishabh Jain
Jan 31, 2016

Taking tan of both the sides and using $\small {\color{forestgreen}{ \tan (A +B)=\dfrac{ \tan A + \tan B}{1- \tan A \tan B} \\\color{forestgreen}{ \tan (arctan \theta)=\theta~~\text{and}~\tan(\frac{\pi}{2}+\theta)=-\cot \theta }}}$ $\dfrac{(3+a)+(3+b)}{1-(3+a)(3+b)}=-\cot (arccot \frac{1}{3})=-\frac{1}{3}$ $\Rightarrow 18+\not{3a} + \not{3b}=ab+\not{3a}+\not{3b}+8$ $\Rightarrow \Large ab=10$ $\Large a+b\geq 2\sqrt{ab}~~~(\small{\color{darkviolet}{AM-GM}})\\ \Large~~~~~~~~~~~\color{#624F41}{=2\sqrt{10}=6.32}$ $\Large \color{#0C6AC7}{\text{For equality a=b=}\sqrt{10}}$

highly overated problem.

Shivam Jadhav - 5 years, 4 months ago

Yeah, I initially set the level to 4.

Harsh Shrivastava - 5 years, 4 months ago

See only 13% people got this write which has something to do with the problem being a level fiver

Rishabh Jain - 5 years, 4 months ago

Maybe,

105 views;14 solvers

Harsh Shrivastava - 5 years, 4 months ago

there is a typo first line should be tan(a+b) and not tan a+tan b

Kaustubh Miglani - 5 years, 4 months ago

Yup.... corrected

Rishabh Jain - 5 years, 4 months ago

An Arcy problem by Mr. Harsh Shrivastava

The answer is 6.32.

1 solution

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