1. Suppose \( \frac{8^x + 27^x}{12^x + 18 ^x} = 7/6 \). Find x.
2. There is a parallelogram ABCD. O is any point in the interior such that angle AOB + angle DOC = 180. Prove that angle ODC= angle OBC
3. Find the number of natural number n for which \(n^6+n^4+1 \) is a perfect square
4. \(\phi_k (m) = \phi_1 (\phi_{k-1} (m)) \) and \(\phi_1 (m) \) is number of divisors of m . Find k for which \( \phi_k (2019^2019) =2 \)
5. If there exists a calculator with 12 buttons, 10 being the buttons for the digits and A and B being two buttons being processes where if n is displayed on the calculator if A is pressed it increases the displayed number by 1 and if B is pressed it multiplies n by 2 hence 2n. Hence find the minimum number of moves to get 260 from 1
6. Find the sum 1+111+11111+1111111+.....1....111(2k+1) ones
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Founder - Faculty at Cheenta.Pursuing Ph.D. at University of Wisconsin, Milwaukee. USAResearch Interest: Geometric Group Theory
[/et_pb_blurb][/et_pb_column][et_pb_column type="1_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_blurb title="Srijit Mukherjee" image="https://cheenta.com/wp-content/uploads/2019/03/48081B40-8E1C-4330-8F97-51CA1690AC73.jpeg" _builder_version="3.22.4" body_font="Raleway||||||||"]Faculty - Admin at CheentaPursuing B.Stat at Indian Statistical Institute, India
[/et_pb_blurb][/et_pb_column][et_pb_column type="1_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_blurb title="Writabrata Bhattacharya" image="https://cheenta.com/wp-content/uploads/2016/11/Writabrata.jpg" _builder_version="3.22.4" body_font="Raleway||||||||"]Faculty at CheentaPursuing B.Sc. Math at Chennai Mathematical Institute, IndiaResearch Interest - Algebraic Geometry
[/et_pb_blurb][/et_pb_column][et_pb_column type="1_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_blurb title="Ishan Sengupta" image="data:image/svg+xml;base64,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" _builder_version="3.22.4" body_font="Raleway||||||||"]Associate Faculty at Cheenta
Pursuing B.Stat from Indian Statistical Institute, India
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Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
Evaluate the integral
Integral from 0 to e^x of log(t)cos^4(t)dt
Part B of the paper :
Integral-> 0 to infinity
dx/(1+x^2)^(m+1)
Thank you
Probability of two rooks attacking each other in a 8*8 chessboard
In a 8×8 chessboard two different squares are chosen at random.If two rooks are placed in these two squares , what is the probability that they will attack each other.(Remember that the rooks can move horizontally or vertically from their position)
think that one rook is at corner . now it has actually 14 options where another rook can be placed
if u take it forword one row it has actually 13 options . note that u can take it upto 4 row.
here (14+13+12+11) case arose .
now start it to move column wise u get same result
now take it up (1,1) position it also has (14 ) option and then it redused to 13,12,11 respectively
it can move in ( 4*4) squere
n(a) = 16(50)
n(s) = 64c2 = 63*32
p(a) = 25/63
You have a land in side of a river which is flowing straight.You want to take a rectangular shape of the land taking the bank of the river as one of its side.So, you can fence only three sides of the land.If you have only 60m of fence then what is the maximum area of your land?
Thank you.
Consider the function f:R×R-->R defined by f((a,b))=lim_n-->inf (ln(e^(na)+e^(nb)))/n.For each of the following statements state sequentially whether they are true or false (write T for true and F for false,your answer should be in the form like TTF)
1.f(0,x)=x for all x>=0
2.f is not onto on R
3.For every b the function f(x,b) is continuous on R
4.For every a the function f(a,x) is differentiable on R
Thank you
Please upload the solutions