My discussions $1$

Some awesome problems that I loved to solve, Try solving them they are fun...

( $1$ )

If $a_1,a_2,a_3...............a_{24}$ are such that

$\dfrac{a_2}{a_1}= \dfrac{a_3}{a_2}=............ =\dfrac{a_{24}}{a_{23}}$

and $\displaystyle\sum_{i=1}^{24}$ $a_i = 2$

$\displaystyle\sum_{i=1}^{24}$ $\dfrac{1}{a_i} = 1$

Find $\displaystyle\prod_{i=1}^{24}$ $a_i$

( $2$ )

For an acute-angled $\triangle{ABC}$ ] $0,\pi/2$ ] , proove that (without using AM-GM)

$\sin(A). \sin(B). \sin(C) \leq \dfrac{3\sqrt3}{8}$

( $3$ )

Find the largest constant $k$ such that

$\dfrac{kabc}{a+b+c} \leq (a+b) ^2+(a+b+4c)^2$

( $4$ )

Find the real roots of the equation

$x^2+2ax+\dfrac{1}{16} = - a+\sqrt{a^2+x-\dfrac{1}{16}}$

( $5$ )

Proove that

$1 <\dfrac{1}{1001} + \dfrac{1}{1002} +............ +\dfrac{1}{3001} < \dfrac{4}{3}$

Hope you will enjoy them!!

#Algebra #Geometry #NumberTheory #Inequality #Summation

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Solution for( 2)

F(x)= $\log(\sin(x))$

F'(x)= $\cot(x)$

F"(x)= - $cosec(x)$

Apply jensens inequality

$F(\dfrac{A+B+C}{3}) \geq \dfrac{F(A)+F(B) +F(C)}{3}$

log(sin( $\pi/3$ ) $\geq \dfrac{log(sin(A)) +log(sin(B))+log(sin(C))}{3}$

3 log( $\dfrac{\sqrt{3}}{2}$ ) $\geq \log(\sin A+\sin B+\sin C)$

Thus

$\boxed{(\sin A. \sin B. \sin C) \leq \dfrac{3\sqrt3}{8}}$

Parth Lohomi - 6 years, 4 months ago

Rather directly apply on $\sin x$ .

Sudeep Salgia - 6 years, 4 months ago

(2)

$A,B,C<\pi \Rightarrow\sin A, \sin B, \sin C >0$

Now consider $\sin A , \sin B , \sin C$ .

We know that $GM≤AM$ , and hence the maximum value will occur only when $GM=AM$ , i.e. the terms are equal.

$\therefore \sin A = \sin B = \sin C \Rightarrow A=B=C$

As $A+B+C=\pi$ , we have $A=B=C=\frac{\pi}{3}$

Therefore, the maximum value of $\sin A \sin B \sin C$ occurs at $A=B=C=\frac{\pi}{3}$ .

$\therefore \sqrt[3]{\sin A \sin B \sin C} \le \frac{\sin A + \sin B + \sin C}{3}$

$\sqrt[3]{\sin A \sin B \sin C} \le \sin\frac{\pi}{3}$

$\sqrt[3]{\sin A \sin B \sin C} \le \frac{\sqrt{3}}{2}$

$\boxed{\sin A \sin B \sin C \le \frac{3\sqrt{3}}{8}}$

Omkar Kulkarni - 6 years, 4 months ago

Sorry I forgot to mention without AM-GM

Parth Lohomi - 6 years, 4 months ago

Oh :P

Omkar Kulkarni - 6 years, 4 months ago

5) Let $1001 \leq k \leq 3001$

A.M - H.M inequality -

$\displaystyle \left(\sum_{k=1001}^{3001}k\right)\left( \sum_{k=1001}^{3001} \dfrac{1}{k}\right) > (2001)^2$

$\displaystyle \sum_{k=1001}^{3001} k = (2001)^2$

Thus, $\displaystyle \sum_{k=1001}^{3001} \dfrac{1}{k} > 1$

$M = \displaystyle \sum_{k=1001}^{3001} \dfrac{1}{k} < \dfrac{500}{1000} + \dfrac{500}{1500} + \dfrac{500}{2000} + \dfrac{1}{3001}$ ( grouped 500 terms simultaneously)

$< \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{3000} = \dfrac{3851}{3000} < \dfrac{4}{3}$

$\boxed{1 < \dfrac{1}{1001} + .... + \dfrac{1}{3001} < \dfrac{4}{3}}$

U Z - 6 years, 4 months ago

Good solution @megh choksi

Parth Lohomi - 6 years, 4 months ago

I think the last question should be -

$1 < \dfrac{1}{1001} + ...... + \dfrac{1}{3001} < \dfrac{4}{3}$

U Z - 6 years, 4 months ago

Oops corrected! !

Parth Lohomi - 6 years, 4 months ago

4) Good one , we can see RHS is inverse of LHS , thus

$x^2 + 2ax + \dfrac{1}{16} = x$

$x^2 + x(2a - 1) + \dfrac{(2a - 1)^2}{4} = \dfrac{(2a - 1)^2}{4} + \dfrac{1}{16}$

$(x + (\dfrac{2a - 1}{2})^2 = \dfrac{(2a - 1)^2}{4} + \dfrac{1}{16}$

$\implies x = \dfrac{2a - 1}{2} \pm \sqrt{\dfrac{(2a - 1)^2}{4} + \dfrac{1}{16}}$

Real roots depends on a.

From our equation - $x^2 + x(2a - 1) - \dfrac{1}{16}$ , we can say for real roots discriminant should be greater than or equal to zero thus

$(2a - 1)^2 - \dfrac{1}{16} \geq 0$

$\implies a \leq \dfrac{1}{4} ~or~ a \geq \dfrac{3}{4}$

U Z - 6 years, 4 months ago

@Krishna Ar @Satvik Golechha @shubhendra singh @Calvin Lin @Sanjeet Raria @Sandeep Bhardwaj @Pranjal Jain @Jake Lai @Ronak Agarwal @Joel Tan @Aritra Jana and everyone else................

Parth Lohomi - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

My discussions \(1\)

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