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Rudi Mathematici Rudi Mathematici x4-8200x3+25213040x2-34452464000x+17652769695744=0 Rudi Mathematici January 1 1 T (1803) Guglielmo LIBRI Carucci dalla Sommaja USAMO 1997 – Problem 1 (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE Let ppp 321 ,,, be the prime numbers (1912) Boris GNEDENKO K (1822) Rudolf Julius Emmanuel CLAUSIUS 2 W listed in increasing order, and let x0 be a (1905) Lev Genrichovich SHNIRELMAN (1938) Anatoly SAMOILENKO real number between 0 and 1. For positive (1917) Yuri Alexeievich MITROPOLSHY 3 T integer k , define: 4 F (1643) Isaac NEWTON (1838) Marie Ennemond Camille JORDAN ⎧ if0 xk−1 = ,0 5 S (1871) Federigo ENRIQUES ⎪ (1871) Gino FANO xk = ⎨⎧ pk ⎫ ⎨ ⎬ if xk−1 ≠ ,0 6 S (1807) Jozeph Mitza PETZVAL ⎪ x (1841) Rudolf STURM ⎩⎩ k −1 ⎭ 2 7 M (1871) Felix Edouard Justin Emile BOREL (1907) Raymond Edward Alan Christopher PALEY where {x} denotes the fractional part of x . (1888) Richard COURANT 8 T (1924) Paul Moritz COHN Find, with proof, all x0 satisfying (1942) Stephen William HAWKING < x < 10 for which the sequence 9 W (1864) Vladimir Adreievich STELKOV 0 10 T (1875) Issai SCHUR xxx ,,, eventually becomes 0 . (1905) Ruth MOUFANG 210 K (1545) Guidobaldo DEL MONTE 11 F Why Astronomy is better than Sex: (1707) Vincenzo RICCATI (1734) Achille Pierre Dionis DU SEJOUR If you get tired, wait ten minutes and try it again. 12 S (1906) Kurt August HIRSCH (1864) Wilhelm Karl Werner Otto Fritz Franz WIEN Statisticians 13 S (1876) Luther Pfahler EISENHART (1876) Erhard SCHMIDT What do you get when you cross a statistician with a chiropractor? 3 14 M (1902) Alfred TARSKI (1704) Johann CASTILLON You get an adjusted R squared from 15 T (1717) Mattew STEWART a BACKward regression problem. (1850) Sofia Vasilievna KOVALEVSKAJA “I know not what I appear to the world, but to (1801) Thomas KLAUSEN 16 W myself I seem to have been only like a boy 17 T (1847) Nikolay Egorovich ZUKOWSKY playing on the sea-shore, and diverting myself (1858) Gabriel KOENIGS in now and then finding a smoother pebble or (1856) Luigi BIANCHI 18 F (1880) Paul EHRENFEST a prettier shell, whilst the great ocean of truth lay all undiscovered before me” 19 S (1813) Rudolf Friedrich Alfred CLEBSCH (1879) Guido FUBINI Isaac NEWTON (1908) Aleksandr Gennadievich KUROS (1775) Andre’ Marie AMPERE “The proof of the Hilbert Basis Theorem is not 20 S (1895) Gabor SZEGO mathematics; it is theology.” (1904) Renato CACCIOPPOLI Camille JORDAN 4 21 M (1846) Pieter Hendrik SCHOUTE (1915) Yuri Vladimirovich LINNIK “It's very good jam,” said the Queen. (1592) Pierre GASSENDI “Well, I don't want any to-day, at any rate.” 22 T (1908) Lev Davidovich LANDAU “You couldn't have it if you did want it,” the Queen said. “The rule is jam tomorrow and 23 W (1840) Ernst ABBE (1862) David HILBERT jam yesterday but never jam to-day.” 24 T (1891) Abram Samoilovitch BESICOVITCH “It must come sometimes to “jam to-day,” Alice (1914) Vladimir Petrovich POTAPOV objected. 25 F (1627) Robert BOYLE “No it can't,” said the Queen. “It's jam every (1736) Joseph-Louis LAGRANGE other day; to-day isn't any other day, you (1843) Karl Herman Amandus SCHWARTZ know.” 26 S (1799) Benoit Paul Emile CLAPEYRON “I don't understand you,” said Alice. “It's 27 S (1832) Charles Lutwidge DODGSON dreadfully confusing.” 5 28 M (1701) Charles Marie de LA CONDAMINE Charles DOGSON (1892) Carlo Emilio BONFERRONI (1817) William FERREL “Mathematics is a game played according to 29 T (1888) Sidney CHAPMAN certain simple rules with meaningless marks on paper.” 30 W (1619) Michelangelo RICCI David HILBERT 31 T (1715) Giovanni Francesco FAGNANO dei Toschi (1841) Samuel LOYD “A mathematician's reputation rests on the (1896) Sofia Alexandrovna JANOWSKAJA number of bad proofs he has given” Abram BESICOVITCH www.rudimathematici.com Rudi Mathematici February 5 1 F (1900) John Charles BURKILL USAMO 1997 – Problem 2 2 S (1522) Lodovico FERRARI Let ABC be a triangle, and draw isosceles (1893) Gaston Maurice JULIA 3 S triangles BCD , CAE , ABF externally 6 4 M (1905) Eric Cristopher ZEEMAN to ABC , with BC , CA , AB as their 5 T (1757) Jean Marie Constant DUHAMEL respective bases. Prove that the lines through 6 W (1612) Antoine ARNAULD (1695) Nicolaus (II) BERNOULLI A , B , C perpendicular to the lines EF , 7 T (1877) Godfried Harold HARDY (1883) Eric Temple BELL FD , DE , respectively, are concurrent. (1700) Daniel BERNOULLI 8 F Why Astronomy is better than Sex: (1875) Francis Ysidro EDGEWORTH Nobody cares if you are ugly. 9 S (1775) Farkas Wolfgang BOLYAI (1907) Harod Scott MacDonald COXETER Statisticians (1747) Aida YASUAKI 10 S A statistician is someone who is skilled at 7 11 M (1800) William Henry Fox TALBOT drawing a precise line from an unwarranted (1839) Josiah Willard GIBBS assumption to a foregone conclusion. (1915) Richard Wesley HAMMING 12 T (1914) Hanna CAEMMERER NEUMANN “Common sense is not really so common.” 13 W (1805) Johann Peter Gustav Lejeune DIRICHLET Antoine ARNAUD “Archimedes will be remembered when 14 T (1468) Johann WERNER (1849) Hermann HANKEL Aeschylus is forgotten, because languages die (1896) Edward Artur MILNE and mathematical ideas do not. “Immortality” 15 F (1564) Galileo GALILEI may be a silly word, but probably a (1861) Alfred North WHITEHEAD mathematician has the best chance of (1946) Douglas HOFSTADTER whatever it may mean.” (1822) Francis GALTON 16 S (1853) Georgorio RICCI-CURBASTRO Godfried HARDY (1903) Beniamino SEGRE “it would be better for the true physics if there 17 S (1890) Sir Ronald Aymler FISHER were no mathematicians on earth” (1891) Adolf Abraham Halevi FRAENKEL 8 18 M (1404) Leon Battista ALBERTI Daniel BERNOULLI “Epur si muove” 19 T (1473) Nicolaus COPERNICUS Galileo GALILEI 20 W (1844) Ludwig BOLTZMANN (1591) Girard DESARGUES ...an incorrect theory, even if it cannot be 21 T (1915) Evgenni Michailovitch LIFSHITZ inhibited by any contradiction that would 22 F (1903) Frank Plumpton RAMSEY refute it, is none the less incorrect, just as a (1583) Jean-Baptiste MORIN criminal policy is none the less criminal even if 23 S (1951) Shigefumi MORI it cannot be inhibited by any court that would curb it. 24 S (1871) Felix BERNSTEIN Luitzen BROUWER 9 25 M (1827) Henry WATSON “Euler calculated without effort, just as men 26 T (1786) Dominique Francois Jean ARAGO breathe, as eagles sustain themselves in the 27 W (1881) Luitzen Egbertus Jan BROUWER air” 28 T (1735) Alexandre Theophile VANDERMONDE Dominique ARAGO 29 F (1860) Herman HOLLERITH www.rudimathematici.com Rudi Mathematici March 9 1 S (1611) John PELL USAMO 1997 – Problem 3 2 S (1836) Julius WEINGARTEN Prove that for any integer n , there exists an 10 3 M (1838) George William HILL (1845) Georg CANTOR unique polynomial Q with coefficients in 4 T (1822) Jules Antoine LISSAJUS { K 9,,3,2,1 } such that 5 W (1512) Gerardus MERCATOR (1759) Benjamin GOMPERTZ (− ) = ( 52 ) =− nQQ . (1817) Angelo GENOCCHI Why Astronomy is better than Sex: 6 T (1866) Ettore BORTOLOTTI Forty years from now, you can still participate 7 F (1792) William HERSCHEL (1824) Delfino CODAZZI regularly. 8 S (1851) George CHRYSTAL Why pi is inferior to e 9 S (1818) Ferdinand JOACHIMSTHAL You can't confuse e with a food product. (1900) Howard Hathaway AIKEN 11 10 M (1864) William Fogg OSGOOD Why e is inferior to pi e is less challenging to spell than pi. 11 T (1811) Urbain Jean Joseph LE VERRIER (1853) Salvatore PINCHERLE “And what are these fluxions? The velocities of 12 W (1685) George BERKELEY evanescent increments? They are neither finite (1824) Gustav Robert KIRKHHOFF (1859) Ernesto CESARO quantities, nor quantities infinitely small, nor (1861) Jules Joseph DRACH yet nothing. May we not call them ghosts of 13 T (1957) Rudy S'ALEMBERT departed quantities?” (1864) Jozef KURSCHAK 14 F George BERKELEY (1879) Albert EINSTEIN (1860) Walter Frank Raphael WELDON “Common sense is nothing more than a deposit 15 S (1868) Grace CHISOLM YOUNG of prejudices laid down in the mind before you reach eighteen.” 16 S (1750) Caroline HERSCHEL (1789) Georg Simon OHM Albert EINSTEIN (1846) Magnus Gosta MITTAG-LEFFLER “We [he and Halmos] share a philosophy about 12 17 M (1876) Ernest Benjamin ESCLANGON (1897) Charles FOX linear algebra: we think basis-free, we write basis-free, but when the chips are down we 18 T (1640) Philippe de LA HIRE (1690) Christian GOLDBACH close the office door and compute with (1796) Jacob STEINER matrices like fury.” (1862) Adolf KNESER 19 W (1910) Jacob WOLFOWITZ Irving KAPLANSKY 20 T (1840) Franz MERTENS The mathematician's best work is art, a high (1884) Philip FRANCK perfect art, as daring as the most secret (1938) Sergi Petrovich NOVIKOV dreams of imagination, clear and limpid. 21 F (1768) Jean Baptiste Joseph FOURIER Mathematical genius and artistic genius touch (1884) George David BIRKHOFF one another. (1917) Irving KAPLANSKY 22 S Gosta MITTAG-LEFFLER (1754) Georg Freiherr von VEGA 23 S (1882) Emmy Amalie NOETHER “A Mathematician is a machine for turning (1897) John Lighton SYNGE coffee into theorems. “ 13 24 M (1809) Joseph LIOUVILLE Paul ERDOS (1948) Sun-Yung (Alice) CHANG The profound study of nature is the most 25 T (1538) Christopher CLAUSIUS fertile source of mathematical discoveries. (1848) Konstantin ADREEV 26 W (1913) Paul ERDOS Jean Baptiste FOURIER 27 T (1857) Karl PEARSON “What we know is not much. What we do not know is immense.” 28 F (1749) Pierre Simon de LAPLACE (1825) Francesco FAA’ DI BRUNO Pierre Simon de LAPLACE 29 S (1873) Tullio LEVI-CIVITA (1896) Wilhelm ACKERMAN 30 S (1892) Stefan BANACH 14 31 M (1596) Rene’ DESCARTES www.rudimathematici.com Rudi Mathematici April (1640) Georg MOHR 14 1 T USAMO 1997 – Problem 4 (1776) Marie-Sophie GERMAIN (1895) Alexander Craig AITKEN To clip a convex n-agon means to choose a 2 W (1934) Paul Joseph COHEN pair of consecutive sides AB , BC and to (1835) John Howard Van AMRINGE 3 T replace them by the three segments , (1892) Hans RADEMACHER AM (1900) Albert Edward INGHAM MN and NC , where M is the midpoint (1909) Stanislaw Marcin ULAM (1971) Alice RIDDLE of AB and N is the midpoint of BC .
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