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RM 2009 Calendar Rudi Mathematici x4-8 204x3+25237646x2-34502914684x+17687247380985=0 Rudi Mathematici January 1 1 T (1803) Guglielmo LIBRI Carucci dalla Sommaja USAMO 1999 – Pr. 1 (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE Some checkers placed on an n× n (1912) Boris GNEDENKO checkerboard satisfy the following conditions: (1822) Rudolf Julius Emmanuel CLAUSIUS 2 F (1905) Lev Genrichovich SHNIRELMAN (a) every square that does not contain a (1938) Anatoly SAMOILENKO checker shares a side with one that 3 S (1917) Yuri Alexeievich MITROPOLSHY does; 4 S (1643) Isaac NEWTON (b) given any pair of squares that contain checkers, there is a sequence of 2 5 M (1838) Marie Ennemond Camille JORDAN (1871) Federigo ENRIQUES squares containing checkers, starting (1871) Gino FANO and ending with the given squares, such that every two consecutive 6 T (1807) Jozeph Mitza PETZVAL (1841) Rudolf STURM squares of the sequence share a side. 7 W (1871) Felix Edouard Justin Emile BOREL ()2 − (1907) Raymond Edward Alan Christopher PALEY Prove that at least n 2 / 3 checkers have (1888) Richard COURANT 8 T been placed on the board. (1924) Paul Moritz COHN (1942) Stephen William HAWKING Mathematics Terms (1864) Vladimir Adreievich STELKOV 9 F CLEARLY: I don’t want to write down all the (1875) Issai SCHUR 10 S “in- between” steps. (1905) Ruth MOUFANG (1545) Guidobaldo DEL MONTE TRIVIAL: If I have to show you how to do 11 S (1707) Vincenzo RICCATI this, you’re in the wrong class. (1734) Achille Pierre Dionis DU SEJOUR Calculus! 3 12 M (1906) Kurt August HIRSCH (1864) Wilhelm Karl Werner Otto Fritz Franz WIEN 2 13 T They chose an ε that was so small that ε (1876) Luther Pfahler EISENHART (1876) Erhard SCHMIDT was negative. (1902) Alfred TARSKI 14 W “The future science of government should be (1704) Johann CASTILLON 15 T called ‘la cybernetique’.” (1717) Mattew STEWART (1850) Sofia Vasilievna KOVALEVSKAJA Andrè Marie AMPERE 16 F (1801) Thomas KLAUSEN [Asked for a testimony to the fact that Emmy Noether was a great woman mathematician, 17 S (1847) Nikolay Egorovich ZUKOWSKY (1858) Gabriel KOENIGS he said:] “I can testify that she is a great 18 S (1856) Luigi BIANCHI mathematician, but that she is a woman, I (1880) Paul EHRENFEST cannot swear.” 4 19 M (1813) Rudolf Friedrich Alfred CLEBSCH (1879) Guido FUBINI Edmund LANDAU (1908) Aleksandr Gennadievich KUROS “Physics is becoming too difficult for the (1775) Andrè Marie AMPERE 20 T physicists.” (1895) Gabor SZEGO (1904) Renato CACCIOPPOLI David HILBERT 21 W (1846) Pieter Hendrik SCHOUTE “Father of Chemistry and Uncle of the Earl of (1915) Yuri Vladimirovich LINNIK Cork.” (1592) Pierre GASSENDI 22 T (1908) Lev Davidovich LANDAU Robert BOYLE [On his tombstone] (1840) Ernst ABBE 23 F “What I tell you three times is true.” (1862) David HILBERT Charles Lutwidge DOGSON 24 S (1891) Abram Samoilovitch BESICOVITCH (1914) Vladimir Petrovich POTAPOV “If you are afraid of something, measure it, (1627) Robert BOYLE 25 S and you will realize it is a mere triple.” (1736) Joseph-Louis LAGRANGE Renato CACCIOPPOLI (1843) Karl Herman Amandus SCHWARTZ 5 26 M (1799) Benoit Paul Emile CLAPEYRON 27 T (1832) Charles Lutwidge DODGSON (1701) Charles Marie de LA CONDAMINE 28 W (1892) Carlo Emilio BONFERRONI (1817) William FERREL 29 T (1888) Sidney CHAPMAN 30 F (1619) Michelangelo RICCI (1715) Giovanni Francesco FAGNANO dei Toschi 31 S (1841) Samuel LOYD (1896) Sofia Alexandrovna JANOWSKAJA www.rudimathematici.com Rudi Mathematici February 1 S (1900) John Charles BURKILL USAMO 1999 – Pr. 2 6 2 M (1522) Lodovico FERRARI Let ABCD be a cyclic quadrilateral. Prove (1893) Gaston Maurice JULIA 3 T that 4 W (1905) Eric Cristopher ZEEMAN AB − CD + AD − BC ≥ 2 AC − BD . 5 T (1757) Jean Marie Constant DUHAMEL (1612) Antoine ARNAULD 6 F (1695) Nicolaus (II) BERNOULLI Mathematic Terms 7 S (1877) Godfried Harold HARDY OBVIOUSLY: I hope you weren’t sleeping when (1883) Eric Temple BELL we discussed this earlier, because I refuse to 8 S (1700) Daniel BERNOULLI repeat it. (1875) Francis Ysidro EDGEWORTH 7 9 M (1775) Farkas Wolfgang BOLYAI RECALL: I shouldn’t have to tell you this, but (1907) Harod Scott MacDonald COXETER for those of you who erase your memory tapes 10 T (1747) Aida YASUAKI after every test... 11 W (1800) William Henry Fox TALBOT Mathematical Psychology (1839) Josiah Willard GIBBS (1915) Richard Wesley HAMMING Zenophobia: the irrational fear of convergent 12 T (1914) Hanna CAEMMERER NEUMANN sequences. 13 F (1805) Johann Peter Gustav Lejeune DIRICHLET “Common sense is not really so common.” 14 S (1468) Johann WERNER Antoine ARNAULD (1849) Hermann HANKEL (1896) Edward Artur MILNE “Technical skill is mastery of complexity while 15 S (1564) Galileo GALILEI creativity is mastery of simplicity.” (1861) Alfred North WHITEHEAD Eric Cristopher ZEEMAN (1946) Douglas HOFSTADTER 8 16 M (1822) Francis GALTON “Reductio ad absurdum, which Euclid loved so (1853) Georgorio RICCI-CURBASTRO much, is one of a mathematician’s finest (1903) Beniamino SEGRE weapons. It is a far finer gambit than any chess 17 T (1890) Sir Ronald Aymler FISHER play: a chess player may offer the sacrifice of a (1891) Adolf Abraham Halevi FRAENKEL pawn or even a piece, but a mathematician offers 18 W (1404) Leon Battista ALBERTI the game.” 19 T (1473) Nicolaus COPERNICUS Godfried HARDY 20 F (1844) Ludwig BOLTZMANN “It would be better for the true physics if there 21 S (1591) Girard DESARGUES were no mathematicians on earth.” (1915) Evgenni Michailovitch LIFSHITZ Daniel BERNOULLI 22 S (1903) Frank Plumpton RAMSEY “Epur si muove!” 9 23 M (1583) Jean-Baptiste MORIN (1951) Shigefumi MORI Galileo GALILEI 24 T (1871) Felix BERNSTEIN “Connaitre, decouvrir, communiquer... telle est 25 W (1827) Henry WATSON la destineè d’un savant.” Dominique ARAGO 26 T (1786) Dominique Francois Jean ARAGO “A mathematician will recognize Cauchy, Jacobi 27 F (1881) Luitzen Egbertus Jan BROUWER or Helmholtz after reading a few pages, just as a 28 S (1735) Alexandre Theophile VANDERMONDE musician recognize, from the first few bars, (1860) Herman HOLLERITH Mozart, Beethoven or Schubert.” Ludwig BOLTZMANN “Whenever you can, count.” Francis GALTON www.rudimathematici.com Rudi Mathematici March 1 S (1611) John PELL USAMO 1999 – Pr. 3 10 2 M (1836) Julius WEINGARTEN Let p > 2 be a prime and let a,b,c,d be 3 T (1838) George William HILL (1845) Georg CANTOR integers not divisible by p , such that 4 W (1822) Jules Antoine LISSAJUS (1512) Gerardus MERCATOR ⎧ra ⎫ ⎧rb⎫ ⎧rc⎫ ⎧rd ⎫ 5 T + + + = (1759) Benjamin GOMPERTZ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ 2 (1817) Angelo GENOCCHI ⎩ p ⎭ ⎩ p ⎭ ⎩ p ⎭ ⎩ p ⎭ 6 F (1866) Ettore BORTOLOTTI for any integer r not divisible by p . Prove (1792) William HERSCHEL 7 S that at least two of the numbers a+b, a+c, (1824) Delfino CODAZZI a+d,b+c, b+d, c+d 8 S (1851) George CHRYSTAL are divisible by p . 11 9 M (1818) Ferdinand JOACHIMSTHAL (1900) Howard Hathaway AIKEN (Note: {}x = x − ⎣⎦x denotes the fractional 10 T (1864) William Fogg OSGOOD (1811) Urbain Jean Joseph LE VERRIER part of x.) 11 W (1853) Salvatore PINCHERLE Mathematic Terms 12 T (1685) George BERKELEY (1824) Gustav Robert KIRKHHOFF WLOG (Without Loss Of Generality): I’m not (1859) Ernesto CESARO about to do all the possible cases, so I’ll do one 13 F (1861) Jules Joseph DRACH and let you figure out the rest. (1957) Rudy D’ALEMBERT IT CAN EASILY BE SHOWN: Even you, in 14 S (1864) Jozef KURSCHAK (1879) Albert EINSTEIN your finite wisdom, should be able to prove (1860) Walter Frank Raphael WELDON 15 S this without me holding your hand. (1868) Grace CHISOLM YOUNG 12 16 M (1750) Caroline HERSCHEL The Four Operations (1789) Georg Simon OHM Ambition, distraction, uglification and (1846) Magnus Gosta MITTAG-LEFFLER derision. (L. Carroll) 17 T (1876) Ernest Benjamin ESCLANGON (1897) Charles FOX “Common sense is nothing more than a deposit 18 W (1640) Philippe de LA HIRE of prejudices laid down in the mind before you (1690) Christian GOLDBACH (1796) Jacob STEINER reach eighteen.” 19 T (1862) Adolf KNESER Albert EINSTEIN (1910) Jacob WOLFOWITZ “We [he and Halmos] share a philosophy about (1840) Franz MERTENS 20 F linear algebra: we think basis-free, we write (1884) Philip FRANCK (1938) Sergi Petrovich NOVIKOV basis-free, but when the chips are down we (1768) Jean Baptiste Joseph FOURIER close the office door and compute with 21 S (1884) George David BIRKHOFF matrices like fury.” 22 S (1917) Irving KAPLANSKY Irving KAPLANSKY 13 23 M (1754) Georg Freiherr von VEGA “A Mathematician is a machine for turning (1882) Emmy Amalie NOETHER coffee into theorems.” (1897) John Lighton SYNGE Paul ERDOS 24 T (1809) Joseph LIOUVILLE (1948) Sun-Yung (Alice) CHANG “Perfect numbers (like perfect men) are very 25 W (1538) Christopher CLAUSIUS rare.” (1848) Konstantin ADREEV 26 T René DESCARTES (1913) Paul ERDOS “A mathematician is a person who can find 27 F (1857) Karl PEARSON analogies between theorems; a better 28 S (1749) Pierre Simon de LAPLACE mathematician is one who can see analogies 29 S (1825) Francesco FAÀ DI BRUNO between proofs and the best mathematician (1873) Tullio LEVI-CIVITA can notice analogies between theories. One can (1896) Wilhelm ACKERMAN imagine that the ultimate mathematician is 14 30 M (1892) Stefan BANACH one who can see analogies between analogies.” 31 T (1596) René DESCARTES Stefan BANACH www.rudimathematici.com Rudi Mathematici April (1640) Georg MOHR 1 W USAMO 1999 – Pr. 4 (1776) Marie-Sophie GERMAIN (1895) Alexander Craig AITKEN ()> Let a1 ,a2 ,K,an n 3 be real 2 T (1934) Paul Joseph COHEN (1835) John Howard Van AMRINGE numbers such that: 3 F (1892) Hans RADEMACHER + + + ≥ (1900) Albert Edward INGHAM a1 a2 K an n (1909) Stanislaw Marcin ULAM (1971) Alice RIDDLE And (1809) Benjamin PEIRCE 2 2 2 2 4 S + + + ≥ .
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