# AMS Menger Awards 2018

(shared from the AMS website for motivational purposes)

The AMS presented the Karl Menger Memorial Awards at the 2018 Intel International Science and Engineering Fair (Intel ISEF), May 13-18, 2018 in Pittsburgh, PA. The First Place Award of US$2000 was given to Ryusei Sakai, Sota Kojima, and Yuta Yokohama, Shiga Prefectural Hikone Higashi High School, Japan, for “Extension of Soddy’s Hexlet: Number of Spheres Generated by Nested Hexlets.” [Photo: bottom row (left to right): Dr. Keith Conrad (committee chair), Rachana Madhukara, Yuta Yokohama, Sota Kojima, Ryusei Sakai; top row (left to right): Chavdar Lalov, Gianfranco Cortes-Arroyo, Gopal Goel, Savelii Novikov, Boris Baranov. Not pictured: Muhammad Abdulla. Photo by the Society for Science & the Public.] The Menger Awards Committee also presented the following awards: • Second Award of$1,000: Gopal Krishna Goel (Krishna Homeschool, OR), “Discrete Derivatives of Random Matrix Models and the Gaussian Free Field” and Rachana Madhukara, Canyon Crest Academy, CA, “Asymptotics of Character Sums”
• Third Award of \$500: Chavdar Tsvetanov Lalov, Geo Milev High School of Mathematics, Bulgaria, “Generating Functions of the Free Generators of Some Submagmas of the Free Omega Magma and Planar Trees”; Gianfranco Cortes-Arroyo, West Port High School, FL, Generalized Persistence Parameters for Analyzing Stratified Pseudomanifolds”; Muhammad Ugur Oglu Abdulla, West Shore Junior/Senior High School, FL, “A Fine Classification of Second Minimal Odd Orbits”; Boris Borisovich Baranov and Savelii Novikov, School 564, St. Petersburg, Russian Federation, “On Two Letter Identities in Lie Rings”
• Certificate of Honorable Mention: Dmitrii Mikhailovskii, School 564, St. Petersburg, Russian Federation, “New Explicit Solution to the N-Queens Problem and the Millennium Problem”; Chi-Lung Chiang and Kai Wang, The Affiliated Senior High School of National Taiwan Normal University, Chinese Taipei, “’Equal Powers Turn Out’ – Conics, Quadrics, and Beyond”; Kayson Taka Hansen, Twin Falls High School, ID, “From Lucas Sequences to Lucas Groups”; Gustavo Xavier Santiago-Reyes and Omar Alejandro Santiago-Reyes, Escuela Secundaria Especializada en Ciencias, Matematicas y Tecnología, Puerto Rico, “Mathematics of Gene Regulation: Control Theory for Ternary Monomial Dynamical Systems”; Karthik Yegnesh, Methacton High School, PA, “Braid Groups on Triangulated Surfaces and Singular Homology”

A booklet on Karl Menger was also given to each winner. This is the 28th year of the presentation of the Karl Menger Memorial Awards. The Society’s participation in the Intel ISEF is supported in part by income from the Karl Menger Fund, which was established by the family of the late Karl Menger. For more information about this program or to make contributions to this fund, contact the AMS Development Office.

Cheers to the winners,

Nalin Pithwa.

# A nice analysis question for RMO practice

Actually, this is a famous problem. But, I feel it is important to attempt on one’s own, proofs of famous questions within the scope of RMO and INMO mathematics. And, then compare one’s approach or whole proof with the one suggested by the author or teacher of RMO/INMO.

Problem:

How farthest from the edge of a table can a deck of playing cards be stably overhung if the cards are stacked on top of one another? And, how many of them will be overhanging completely away from the edge of the table?

Reference:

I will post it when I publish the solution lest it might affect your attempt at solving this enticing mathematics question !

Please do not try and get the solution from the internet.

Regards,

Nalin Pithwa.

# Intel Pentium P5 floating point unit error (1994) — an RMO problem !!!

Problem:

Two number theorists, bored in a chemistry lab, played a game with a large flask containing 2 litres of a colourful chemical solution and an ultra-accurate pipette. The game was that they would take turns to recall a prime number p such that $p+2$ is also a prime number. Then, the first number theorist would pipette out 1/p litres of chemical and the second $\frac{1}{(p+2)}$ litres. How many times do they have to play this game to empty the flask completely?

Hint:

A bit of real analysis is required.

Reference:

I will publish the reference when I post the solution. So that all students/readers can curb their impulse to see the solution immediately!!!

I hope you will be hooked to the problem in a second….!!! Here is a beautiful utility of pure math! 🙂

Cheers,

Nalin Pithwa

PS: I do not know if the above problem did (or, will?? )appear as RMO question. It is just my wild fun with math to kindle the intellect of students in analysis !! 🙂

# Pick’s theorem: a geometry problem for RMO practice

Pick’s theorem:

Consider a square lattice of unit side. A simple polygon (with non-intersecting sides) of any shape is drawn with its vertices at the lattice points. The area of the polygon can be simply obtained as $B/2+I-1$ square units, where B is the number of lattice points on the boundary; I = number of lattice points in the interior region of the polygon. Prove this theorem.

Proof:

Refer Wikipedia 🙂 🙂 🙂

https://en.wikipedia.org/wiki/Pick%27s_theorem

Cheers,

Nalin Pithwa.

# Pre-RMO or RMO algebra practice problem: infinite product

Find the product of the following infinite number of terms:

$\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \ldots = \prod_{m=2}^{\infty}\frac{m^{3}-1}{m^{3}+1}$

$m^{3}-1=(m-1)(m^{2}+m+1)$, and also, $m^{3}+1=(m+1)(m^{2}-m+1)=(m-1+2)((m-1)^{2}+(m-1)+1)$

Hence, we get $P_{m}=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \ldots \times \frac{m^{3}-1}{m^{3}+1}$, which in turn, equals

$(\frac{1}{3} \times \frac{7}{3}) \times (\frac{2}{4} \times \frac{13}{7}) \times (\frac{3}{5} \times \frac{21}{13})\times \ldots (\frac{m-1}{m+1} \times \frac{m^{2}+m+1}{m^{2}-m+1})$, that is, in turn equal to

$\frac{2}{3} \times \frac{m^{2}+m+1}{m(m+1)}$, that is, in turn equal to

$\frac{}{} \times (1+ \frac{1}{m(m+1)})$, so that when $m \rightarrow \infty$, and then $P_{m} \rightarrow 2/3$.

personal comment: I did not find this solution within my imagination !!! 🙂 🙂 🙂

The credit for the solution goes to “Popular Problems and Puzzles in Mathematics” by Asok Kumar Mallik, IISc Press, Foundation Books. Thanks Prof. Mallik !!

Cheers,

Nalin Pithwa