3,822 research outputs found
On the grasshopper problem with signed jumps
The 6th problem of the 50th International Mathematical Olympiad (IMO), held
in Germany, 2009, was the following.
Let be distinct positive integers and let be a set of
positive integers not containing . A grasshopper is to
jump along the real axis, starting at the point 0 and making jumps to the
right with lengths in some order. Prove that the order can be
chosen in such a way that the grasshopper never lands on any point in .
The problem was discussed in many on-line forums, as well by communities of
students as by senior mathematicians. Though there have been attempts to solve
the problem using Noga Alon's famous Combinatorial Nullstellensatz, up to now
all known solutions to the IMO problem are elementary and inductive. In this
paper we show that if the condition that the numbers are positive
is omitted, it allows us to apply the polynomial method to solve the modified
problem.Comment: Submitted to AMS Monthly on 1st March, 2010; last revision in August,
201
Balanced factorisations
Any rational number can be factored into a product of several rationals whose
sum vanishes. This simple but nontrivial fact was suggested as a problem on a
maths olympiad for high-school students. We completely solve similar questions
in all finite fields and in some other rings, e.g., in the complex and real
matrix algebras. Also, we state several open questions.Comment: 7 pages. A Russian version of this paper is at
http://halgebra.math.msu.su/staff/klyachko/papers.ht
Menjana pemodulatan lebar denyut (PWM) penyongsang tiga fasa menggunakan pemproses isyarat digital (DSP)
Baru-baru ini, penyongsang digunakan secara meluas dalam aplikasi industri.
Walaubagaimanapun, teknik Pemodulatan Lebar Denyut (PWM) diperlukan untuk
mengawal voltan keluaran dan frekuensi penyongsang. Dalam tesis ini, untuk
Pemodulatan Lebar Denyut Sinus Unipolar (SPWM) penyongsang tiga fasa adalah
dicadang menggunakan Pemproses Isyarat Digital (DSP). Satu model simulasi
menggunakan MATLAB Simulink dibangunkan untuk menentukan program
Pemodulatan Lebar Denyut Sinus Unipolar (SPWM) Program ini kemudian
dibangunkan dalam Pemproses Isyarat Digital (DSP) TMS320f28335. Hasilnya
menunjukkan bahawa voltan keluaran penyongsang tiga fasa boleh dikendalikan
Baron Munchhausen Redeems Himself: Bounds for a Coin-Weighing Puzzle
We investigate a coin-weighing puzzle that appeared in the Moscow Math
Olympiad in 1991. We generalize the puzzle by varying the number of
participating coins, and deduce an upper bound on the number of weighings
needed to solve the puzzle that is noticeably better than the trivial upper
bound. In particular, we show that logarithmically-many weighings on a balance
suffice.Comment: 19 page
Enhancing Practice and Achievement in Introductory Programming With a Robot Olympics
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3-5-man mutual zugzwangs in chess
This note reports the work of Wirth and Karrer in twin-sourcing all mutual zugzwang positions, mzugs, in 2-5-man endgames. This paper tabulates the mzug statistical data, gives examples of maximal mzugs and refers to a chess endgame website where further data is to be found
An Integrated Subharmonic Coupled-Oscillator Scheme for a 60-GHz Phased-Array Transmitter
This paper describes the design of an integrated coupled-oscillator array in SiGe for millimeter-wave applications. The design focuses on a scalable radio architecture where multiple dies are tiled to form larger arrays. A 2 × 2 oscillator array for a 60-GHz transmitter is fabricated with integrated power amplifiers and on-chip antennas. To lock between multiple dies, an injection-locking scheme appropriate for wire-bond interconnects is described. The 2 × 2 array demonstrates a 200–MHz locking range and 1 × 4 array formed by two adjacent chips has a 60-MHz locking range. The phase noise of the coupled oscillators is below 100 dBc/Hz at a 1-MHz offset when locked to an external reference. To the best of the authors’ knowledge, this is the highest frequency demonstration of coupled oscillators fabricated in a conventional silicon integrated-circuit process
3-5-Man chess data
This reports the work of Karrer and Wirth in identifying percentage results and, respectively, the Depth to Mate (DTM) and Depth to Conversion (DTC) data in all 2-5-man chess endgames
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