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Jharkhand RMO 2015

This time RMO of jharkhand was more difficult than last year .. what do you all reckon what will be cut off for this year?..

Note by Gyanendra Prakash
12 months ago

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last time cut off was 45. by the way how much are you getting this time? Aditya Kr · 11 months, 3 weeks ago

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find all triplets such that (a,b,c) of positive integers such that a<=b<=c: 3 does not divide b.c and (1+1/a)(2+1/b)(3+1/c)=13 Gyanendra Prakash · 12 months ago

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In a group of seven persons every person is asked to write the sum of ages of all the other six persons.if all the seven sums form a six element set {63,64,65,66,68,70}..then find the individual ages assuming ages are whole numbers.. Gyanendra Prakash · 12 months ago

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one question was: prove that the roots of the equation x^3-3x^2-1 = 0 are never rational. Gyanendra Prakash · 12 months ago

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@Gyanendra Prakash By rational root theorem, the only possible rational roots of the above polynomial could be \(\pm1\) which are not in this case. Therefore, the roots are Irrational of Imaginary.

Either it is not a RMO question or my approach is not complete. Vishal Yadav · 12 months ago

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@Vishal Yadav this is surely from RMO (jharkhand RMO 2015)...but i have solved this problem integrating 3 concepts : theory of equations, inequalities and then final proof by contradiction Gyanendra Prakash · 12 months ago

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@Gyanendra Prakash well i approached like this:- let the roots be a,b,c a+b+c=3 ab+bc+ac=0 abc=1 assuming roots to be rational..i took a=p1/q1,b as p2/q2 and c as p3/q3

so i got p1/q1+p2/q2+p3/q3=3---------eq.1 p1p2/q1q2 + p2p3/q2/q3 + p1p3/q1q3 = 0-----eq.2 and p1p2p3=q1q2q3 -----------eq-3

procceding with equation 1

i got after expanding and replacing q1q2q3 by p1p2p3...

recoiprocal of eq.2=3 (after three steps of monotonous algebraic expansion)

taking eq2 as x+y+z = 0 and then its reciprocal from above as 1/x+x/y+1/z = 3

by A.M-G.M we know that (x+y+z)(1/x+1/y+1/z)>=9

but here we are getting it as 3*0=0

therfore by contradiction roots can't be rational...

i suppose it's corrcet.. Gyanendra Prakash · 12 months ago

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@Gyanendra Prakash so does my solution deserve any credit? Gyanendra Prakash · 12 months ago

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@Gyanendra Prakash I don't know, but if it is a RMO Question it should be solved somewhat the way you did it. I can't actually understand your solution, should use LaTex and you left the most part of your solution that is the steps with which you solved. Vishal Yadav · 12 months ago

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@Vishal Yadav ok..i will improve it Gyanendra Prakash · 12 months ago

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If the roots of x^3+px^2+qx+r=0 are all real and +ve...prove pq-9r>=0 and q^3-27r^2>=0 Gyanendra Prakash · 12 months ago

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