3,822 research outputs found

    On the grasshopper problem with signed jumps

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    The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, was the following. Let a1,a2,...,ana_1,a_2,...,a_n be distinct positive integers and let MM be a set of n1n-1 positive integers not containing s=a1+a2+...+ans=a_1+a_2+...+a_n. A grasshopper is to jump along the real axis, starting at the point 0 and making nn jumps to the right with lengths a1,a2,...,ana_1,a_2,...,a_n in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in MM. 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 a1,...ana_1,...a_n 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

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    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)

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    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

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    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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    © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information

    3-5-man mutual zugzwangs in chess

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    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

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    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

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    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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