Rudi Mathematici x4-8180x3+25090190x2-34200948100x+17481136677369=0 Rudi Mathematici January 1 1 W (1803) Guglielmo LIBRI Carucci dalla Somaja APMO 1989 [1] (1878) Agner Krarup ERLANG (1894) Satyendranath BOSE Let x1 , x2 , , xn be positive real numbers, (1912) Boris GNEDENKO n 2 T (1822) Rudolf Julius Emmanuel CLAUSIUS (1905) Lev Genrichovich SHNIRELMAN = and let S xi . (1938) Anatoly SAMOILENKO i=1 3 F (1917) Yuri Alexeievich MITROPOLSHY Prove that 4 S (1643) Isaac NEWTON i (1838) Marie Ennemond Camille JORDAN n n 5 S S (1871) Federigo ENRIQUES ()+ ≤ ∏ 1 xi (1871) Gino FANO i=1 i=0 i! 2 6 M (1807) Jozeph Mitza PETZVAL (1841) Rudolf STURM The Dictionary 7 T (1871) Felix Edouard Justin Emile BOREL (1907) Raymond Edward Alan Christopher PALEY Clearly: I don't want to write down (1888) Richard COURANT 8 W (1924) Paul Moritz COHN all the "in-between" steps. (1942) Stephen William HAWKING The First Law of Applied 9 T (1864) Vladimir Adreievich STELKOV Mathematics: All infinite series 10 F (1875) Issai SCHUR (1905) Ruth MOUFANG converge, and moreover converge to 11 S (1545) Guidobaldo DEL MONTE the first term. (1707) Vincenzo RICCATI (1734) Achille Pierre Dionis DU SEJOUR A mathematician's reputation rests 12 S (1906) Kurt August HIRSCH on the number of bad proofs he has 3 13 M (1864) Wilhelm Karl Werner Otto Fritz Franz WIEN (1876) Luther Pfahler EISENHART given. (1876) Erhard SCHMIDT Abram BESICOVITCH 14 T (1902) Alfred TARSKI 15 W (1704) Johann CASTILLON Probabilities must be regarded as (1717) Mattew STEWART analogous to the measurements of (1850) Sofia Vasilievna KOVALEVSKAJA physical magnitudes; that is to say, 16 T (1801) Thomas KLAUSEN they can never be known exactly, but 17 F (1847) Nikolay Egorovich ZUKOWSKY (1858) Gabriel KOENIGS only within certain approximation. (1856) Luigi BIANCHI 18 S (1880) Paul EHRENFEST Emile BOREL 19 S (1813) Rudolf Friedrich Alfred CLEBSCH I have no certainties, at most (1879) Guido FUBINI (1908) Aleksandr Gennadievich KUROS probabilities. 4 20 M (1775) Andre` Marie AMPERE Renato CACCIOPPOLI (1895) Gabor SZEGO (1904) Renato CACCIOPPOLI What I tell you three times is true. 21 T (1846) Pieter Hendrik SCHOUTE (1915) Yuri Vladimirovich LINNIK Charles DODGSON (1592) Pierre GASSENDI 22 W (1908) Lev Davidovich LANDAU The proof of the Hilbert Basis 23 T (1840) Ernst ABBE Theorem is not mathematics: it is (1862) David HILBERT theology. 24 F (1891) Abram Samoilovitch BESICOVITCH (1914) Vladimir Petrovich POTAPOV Camille JORDAN (1627) Robert BOYLE 25 S (1736) Joseph-Louis LAGRANGE Probabilities must be regarded as (1843) Karl Herman Amandus SCHWARTZ analogous to the measurement of 26 S (1799) Benoit Paul Emile CLAPEYRON physical magnitudes: they can never 5 27 M (1832) Charles Lutwidge DODGSON be known exactly, but only within 28 T (1701) Charles Marie de LA CONDAMINE certain approximation. (1892) Carlo Emilio BONFERRONI (1817) William FERREL 29 W Emile BOREL (1888) Sidney CHAPMAN God not only plays dice. He also 30 T (1619) Michelangelo RICCI sometimes throws the dice where they 31 F (1715) Giovanni Francesco FAGNANO dei Toschi (1841) Samuel LOYD cannot be seen. (1896) Sofia Alexandrovna JANOWSKAJA Stephen HAWKING www.rudimathematici.com Rudi Mathematici February 5 1 S (1900) John Charles BURKILL APMO 1989 [2] 2 S (1522) Lodovico FERRARI Prove that the equation 6 3 M (1893) Gaston Maurice JULIA 2 2 2 2 6 ∗ ()6a + 3b + c = 5n 4 T (1905) Eric Cristopher ZEEMAN 5 W (1757) Jean Marie Constant DUHAMEL has no solutions in integers except 6 T (1612) Antoine ARNAULD a = b = c = n = 0 (1695) Nicolaus (II) BERNOULLI (1877) Godfried Harold HARDY 7 F (1883) Eric Temple BELL The Dictionary (1700) Daniel BERNOULLI 8 S Trivial: If I have to show you how to do (1875) Francis Ysidro EDGEWORTH (1775) Farkas Wolfgang BOLYAI this, you're in the wrong class 9 S (1907) Harod Scott MacDonald COXETER There are two groups of people in the 7 10 M (1747) Aida YASUAKI world: those who believe that the world 11 T (1800) William Henry Fox TALBOT (1839) Josiah Willard GIBBS can be divided into two groups of (1915) Richard Wesley HAMMING people, and those who don't. 12 W (1914) Hanna CAEMMERER NEUMANN 13 T (1805) Johann Peter Gustav Lejeune DIRICHLET Connaitre, decouvrir, communiquer. (1468) Johann WERNER 14 F Telle est la destinée d'un savant (1849) Hermann HANKEL (1896) Edward Artur MILNE François ARAGO 15 S (1564) Galileo GALILEI Common sense is not really so common (1861) Alfred North WHITEHEAD (1946) Douglas HOFSTADTER Antoine ARNAULD 16 S (1822) Francis GALTON (1853) Georgorio RICCI-CURBASTRO "Obvious" is the most dangerous word (1903) Beniamino SEGRE in mathematics. 8 17 M (1890) Sir Ronald Aymler FISHER (1891) Adolf Abraham Halevi FRAENKEL Eric Temple BELL 18 T (1404) Leon Battista ALBERTI ...it would be better for the true physics 19 W (1473) Nicolaus COPERNICUS if there were no mathematicians on 20 T (1844) Ludwig BOLTZMANN hearth. (1591) Girard DESARGUES 21 F (1915) Evgenni Michailovitch LIFSHITZ Daniel BERNOULLI 22 S (1903) Frank Plumpton RAMSEY ...an incorrect theory, even if it cannot be 23 S (1583) Jean-Baptiste MORIN inhibited bay any contradiction that (1951) Shigefumi MORI would refute it, is none the less 9 24 M (1871) Felix BERNSTEIN incorrect, just as a criminal policy is 25 T (1827) Henry WATSON none the less criminal even if it cannot 26 W (1786) Dominique Francois Jean ARAGO be inhibited by any court that would 27 T (1881) Luitzen Egbertus Jan BROUWER curb it. 28 F (1735) Alexandre Theophile VANDERMONDE Jan BROUWER (1860) Herman HOLLERITH Mathemata mathematici scribuntur Nicolaus COPERNICUS www.rudimathematici.com Rudi Mathematici March 9 1 S (1611) John PELL APMO 1989 [3] (1836) Julius WEINGARTEN 2 S Let A1, A2, A3 be three points in the plane, and (1838) George William HILL 10 3 M for convenience let A4=A1, A5=A2. For n=1, 2, (1845) Georg CANTOR and 3 suppose that Bn is the midpoint of 4 T (1822) Jules Antoine LISSAJUS AnAn+1, and suppose that Cn is the midpoint of (1512) Gerardus MERCATOR AnBn. Suppose that AnCn+1 and BnCn+2 meet at 5 W (1759) Benjamin GOMPERTZ Dn, and that AnBn+1 meet at En. Calculate the (1817) Angelo GENOCCHI ratio of the area of triangle D1D2D3 to the area 6 T (1866) Ettore BORTOLOTTI of triangle E1E2E3. 7 F (1792) William HERSCHEL (1824) Delfino CODAZZI The Dictionary 8 S (1851) George CHRYSTAL It can easily be shown: No more than (1818) Ferdinand JOACHIMSTHAL 9 S (1900) Howard Hathaway AIKEN four hours are needed to prove it 11 10 M (1864) William Fogg OSGOOD Theorem: All the numbers are boring 11 T (1811) Urbain Jean Joseph LE VERRIER (1853) Salvatore PINCHERLE Proof (by contradiction): Suppose x is (1685) George BERKELEY 12 W the first non-boring number. Who (1824) Gustav Robert KIRKHHOFF (1859) Ernesto CESARO cares? (1861) Jules Joseph DRACH 13 T (1957) Rudy D'ALEMBERT Mathematics is the most beautiful 14 F (1864) Jozef KURSCHAK and the most powerful creation of the (1879) Albert EINSTEIN human spirit. Mathematics is as old 15 S (1860) Walter Frank Raphael WELDON (1868) Grace CHISOLM YOUNG as Man. (1750) Caroline HERSCHEL 16 S (1789) Georg Simon OHM Stefan BANACH (1846) Magnus Gosta MITTAG-LEFFLER In mathematics the art of proposing a 12 17 M (1876) Ernest Benjamin ESCLANGON (1897) Charles FOX question must be held on higher value (1640) Philippe de LA HIRE 18 T than solving it. (1690) Christian GOLDBACH (1796) Jacob STEINER Georg CANTOR (1862) Adolf KNESER 19 W (1910) Jacob WOLFOWITZ When writing about transcendental (1840) Franz MERTENS 20 T issues, be transcendentally clear. (1884) Philip FRANCK (1938) Sergi Petrovich NOVIKOV Rene` DESCARTES 21 F (1768) Jean Baptiste Joseph FOURIER The search for truth is more (1884) George David BIRKHOFF important than its possession. 22 S (1917) Irving KAPLANSKY (1754) Georg Freiherr von VEGA 23 S Albert EINSTEIN (1882) Emmy Amalie NOETHER (1897) John Lighton SYNGE Property is a nuisance. 13 24 M (1809) Joseph LIOUVILLE (1948) Sun-Yung (Alice) CHANG Paul ERDOS 25 T (1538) Christopher CLAUSIUS Don't worry about people stealing 26 W (1848) Konstantin ADREEV your ideas. If your ideas are any good, (1913) Paul ERDOS you'll have to ram them down 27 T (1857) Karl PEARSON people's throat. 28 F (1749) Pierre Simon de LAPLACE Howard AIKEN 29 S (1825) Francesco FAA` DI BRUNO (1873) Tullio LEVI-CIVITA Geometry is the noblest branch of (1896) Wilhelm ACKERMAN physics. 30 S (1892) Stefan BANACH 14 31 M (1596) Rene` DESCARTES William OSGOOD www.rudimathematici.com Rudi Mathematici April 14 1 T (1640) Georg MOHR (1776) Marie-Sophie GERMAIN APMO 1989 [4] (1895) Alexander Craig AITKEN Let S be a set consisting of m pairs (a,b) of 2 W (1934) Paul Joseph COHEN positive integers with the property that 3 T (1835) John Howard Van AMRINGE 1 ≤ a < b ≤ n . Show that there are at (1892) Hans RADEMACHER least (1900) Albert Edward INGHAM (1909) Stanislaw Marcin ULAM n 2 (1971) Alice RIDDLE m − 4 F (1809) Benjamin PEIRCE 4 (1842) Francois Edouard Anatole LUCAS 4m ∗ (1949) Shing-Tung YAU 3n (1588) Thomas HOBBES 5 S (1607) Honore` FABRI triples (a,b,c) such that (a,b), (a,c) and (b,c) (1622) Vincenzo VIVIANI belong to S. (1869) Sergi Alexeievich CHAPLYGIN 6 S The Dictionary 15 7 M (1768) Francais Joseph FRANCAIS Check for yourself: This is the boring 8 T (1903) Marshall Harvey STONE part of the proof, so you can do it on (1791) George PEACOCK 9 W your own time (1816) Charles Eugene DELAUNAY (1919) John Presper HECKERT It is proven that celebration of 10 T (1857) Henry Ernest DUDENEY birthdays is healthy.