BLOG DO ENG. ARMANDO CAVERO MIRANDA -BRASIL


MACHUPICHU MARAVILHA DO MUNDO

"Two things are infinite: the universe and human stupidity; and I'm not sure about the the universe." ALBERT EINSTEIN - “SE SEUS PROJETOS FOREM PARA UM ANO,SEMEIE O GRÂO.SE FOREM PARA DEZ ANOS,PLANTE UMA ÁRVORE.SE FOREM PARA CEM ANOS,EDUQUE O POVO.” "MATH IS POWER TO CHANGE THE WORLD AND THE KEY TO THE FUTURE" 'OBRIGADO DEUS PELA VIDA,PELA MINHA FAMILIA,PELO TRABALHO,PELO PÃO DE CADA DIA,PROTEGENOS E GUARDANOS DE TODO MAL"

quinta-feira, 26 de dezembro de 2013

ARITMETICA DE BALDOR TEORIA Y PRACTICA


ARITMETICA DE AURELIO BALDOR ES UN LIBRO CLASICO DE LAS MATEMATICAS BASICAS,YO APRENDI MUCHO CON ESTE LIBRO,EL ME TRAE MUCHOS RECUERDOS DE MI  JUVENTUD,AÑOS GLORIOSOS DE ESTUDIO DE SECUNDARIA EN LA GRAN UNIDAD ESCOLAR CARLOS WIESSE COMAS LIMA PERU,LO ENCONTRE HOY NAVEGANDO EN LA WEB LES PASO EL LINK
LINK1:
http://matematica1.com/aritmetica-de-baldor-ejercicios-con-respuestas-en-texto-pdf/
LINK2
http://www.mediafire.com/view/?025ojguhrbfk3wu
LINK3
https://www.mediafire.com/?ezdewl9g9i9ylpt

terça-feira, 24 de dezembro de 2013

segunda-feira, 23 de dezembro de 2013

A Practical Text-Book on Plane and Spherical Trigonometry by Webster Wells

This work is intended as a practical text-book on Plane and Spherical Trigonometry, adapted to the needs of High Schools and Colleges. While considerable space has been devoted to the development of the theoretical portions of the subject, the practical applications of the formulæ have not been neglected, and in the chapters on the Solution of Triangles will be found a carefully selected assortment of exercises and examples, arranged with the view of giving the student a thorough drill in numerical computation.
LINK
http://www.forgottenbooks.org/books/A_Practical_Text-Book_on_Plane_and_Spherical_Trigonometry_1000116548 

Higher Algebra by Herbert E. Hawkes

Higher Algebra
by Herbert E. Hawkes
This text is prepared to meet the needs of the student who will continue his mathematics as far as the calculus, and is written in the spirit of applied mathematics.
LINK
http://www.forgottenbooks.org/books/Higher_Algebra_1000002647 

sábado, 7 de dezembro de 2013

BRILHANTE ATUAÇÃO DO EQUIPE PERUANO NA OLIMPIADA RIOPLATENSE 2013 ARGENTINA


MEDALHAS DE OURO----02
MEDALHAS DE PRATA--- 02
MEDALHA DE BRONÇE--02

MESMO COM DIFICULDADES ECONOMICAS PARA VIAJAR O EQUIPE PERUANO MOSTRA COM RESULTADOS SER ALTAMENTE COMPETITIVO,BRASIL Y ARGENTINA TAMBEM DESTACAROM,DEBO RESALTAR QUE PAISES COMO BRASIL APOIAM AS OLIMPIADAS DE MATEMATICAS,PENSO QUE O MINIMO QUE PODE FAZER O GOBERNO PERUANO E DAR APOIO FINANCIERO E LOGISTICO A OS PROFESSORES   ALUNOS QUE LUTAM POR MELHORAR E DIFUNDIR A CIENCIA NO PAIS.

ACTUALES CAMPEONES DE OLIMPIADAS MATEMATICAS EN ARGENTINA TUVIERON SERIAS DIFICULTADES ECONOMICAS PARA VIAJAR



 REPORTAJE EMITIDO POR RPP PERU Luego de pasar un calvario para poder conseguir los pasajes hacia Argentina, el equipo de jóvenes escolarescampeones de matemáticas por fin podrá decir que estarán presentes en la XXII Olimpiada Rioplatense que se desarrollará del 1 al 15 de diciembre en ese país. En total, son nueve jóvenes que vienen de distintas partes del país y que ganaron un concurso organizado por el Ministerio de Educación y la Asociación de Matemáticos del país y el premio consistía en el costo del pasaje y los viáticos. Esta mañana, el congresista José Luna Vargas hizo entrega de un cheque con una cantidad de dinero que permite que la delegación pueda viajar a competir. “Estos jóvenes son un ejemplo para toda la juventud y por eso mi colaboración para que cumplan con el sueño de viajar”, dijo Luna al entregar el cheque. Más temprano, Elena Burga de Suyo, representante de los padres de familia, declaró que el Ministerio de Educación no tenía recursos para costear el viaje y que cada padre debería "correr con los gastos de viaje". Burga había informado que el año pasado ocurrió algo similar, pero les comunicaron con anticipación; entonces, tuvieron tiempo para realizar actividades con el fin de conseguir los pasajes. A través de un comunicado, el Ministerio de Educación sostuvo en la mañana que el viaje de la delegación peruana se encontraba plenamente financiado y garantizado.
LINK
http://www.rpp.com.pe/2013-11-26-campeones-de-matematicas-pasaron-dificultades-por-viaje-a-olimpiadas-noticia_650492.html

segunda-feira, 2 de dezembro de 2013

exercícios de geometria plana comentados-02-10-12-01_wmv.wmv

Trigonometria e Geometria Plana - Resolução de Questões 4

Resoluções de Provas - ITA - ITA 2012 - Matemática - (Parte 1 de 3)

Matemática Zero 2.0 - Aula 11 - MMC e MDC - (parte 1 de 1)

Matemática Zero 2.0 - Aula 19 - Frações Algébricas - (parte 1 de 2)

Matemática Zero 2.0 - Aula 17 - Fatoração - (parte 1 de 3)

sábado, 30 de novembro de 2013

Electricidad y Magnetismo - Matveev



Mechanics and Theory of Relativity – Matveev

Author: A. N. Matveev
Keywords: physics; mechanics; theory of relativity; special relativity; newtons laws; conservation laws; angular momentum
Language: English
Collection: opensource

About the book:

This is the first volume (second Russian edition) of a course on general physics (the second, third and fourth volumes were published in 1985 (Molecular Physics), 1986 (Electricity and Magnetism) and 1988 (Optics) respectively; the fifth volume (Atomic Physics) is under preparation).

This book provides the theoretical background. Of course, it also contains a description and analysis of physical phenomena, measurement of physical quantities, experimental methods of investigation, and other allied problems, but only from the point of view of theoretical understanding.
LINK 1

LINK2
Password, if needed: mirtitles
 
 

terça-feira, 26 de novembro de 2013

Trigonometric Functions (Problem Solving Approach) by A. Panchishkin and E. Shavgulidze.

The book has some interesting things, for example it introduces the trigonometric functions as a feature of a trigonometric circle instead of a right-angled triangle as many books do. Also the second chapter (Identical Transformations of Trigonometric Expressions) will help you understand many combination formulae (which we learn by heart in many cases without understanding). Chapter 4 (Investigating Trigonometric Functions) treats the trigonometric functions graphically and analyzes the meaning of graphs, also with help of their derivatives.
LINK1
https://archive.org/details/TrigonometricFunctions-ProblemSolvingApproach
LINK2
http://mirtitles.org/2012/07/04/trigonometric-functions-problem-solving-approach/

Problems in Physics V. G. Zubov, V. P. Shalnov

CONTINUANDO SURFANDO EN LA PAGINA WEB ENCONTRE EN STE WEBSITE
http://mirtitles.org/2012/09/10/problems-in-physics-zubov-shalnov/
You can get the book here.
Password, if needed: mirtitles



Partial Differential Equations – Mikhailov


SURFANDO EN LA WEB ENCONTRE ESTE LIBRO INTERESANTE PARA LOS ESTUDIANTES DE INGENIERIA,CIENCIAS Y RAMAS AFINES,EN LA PAGINA WEB SIGUIENTE
http://mirtitles.org/
You can get the book here (IA) and here (filecloud).
Password, if needed: mirtitles






Problems in Higher Mathematics – Minorsky


About the book: The list of topics covered is quite exhaustive and the book has over 2500 problems and solutions. The topics covered are plane and solid analytic geometry, vector algebra, analysis, derivatives, integrals, series, differential equations etc. A good reference for those looking for many problems to solve. The book was translated from the Russian by Yuri Ermolyev and was first published by Mir Publishers in 1975.

You can get the book here (IA) and here (filecloud).
Password, if needed: mirtitles

EMCONTREI ESTE LIVRO NA PAGINA WEB
http://mirtitles.org/ 

terça-feira, 19 de novembro de 2013

Cálculo II - Aula 16 - Parte 2 - Exercícios sobre diferenciabilidade, plano tangente ao gráfico, regra da cadeia e relação gradiente-curva de nível.

Geometria Analítica e Vetores - Aula 1 - Operações com Matrizes UNICAMP BRASIL





Nesta primeira aula de Geometria Analítica e Vetores, o prof. Ricardo Miranda Martins, do Departamento de Matemática da UNICAMP, explica o que são matrizes, vetores, sistemas lineares, coordenadas e seções cônicas.
Bibliografia

1. Reginaldo J. Santos, Matrizes, Vetores e Geometria Analítica, Imprensa Universitária da UFMG (LIVRO TEXTO).
O livro está a venda pela Internet na livraria Leitura, www.livrarialeitura.com.br

Uma versão online está disponível: http://www.mat.ufmg.br/~regi/gaalt/ga....

2. J. M. Martínez, Notas de Geometria Analítica, Versão online do livro: http://arquivoescolar.org/bitstream/a... .

segunda-feira, 18 de novembro de 2013

Algebra By Boundless

Algebra
By Boundless
Description
The Boundless Algebra textbook is a college-level, introductory textbook that covers the important subject of Algebra -- one of the basic building blocks of studies in higher mathematics. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.

Table of Contents
1) The Building Blocks of Algebra 2) Graphs, Functions, and Models 3) Functions, Equations, and Inequalities 4) Polynomial and Rational Functions 5) Exponents and Logarithms
6) Systems of Equations and Matrices 7) Conic Sections 8) Sequences, Series and Combinatorics
LINK PROJECT GUTEMBERG
http://self.gutenberg.org/eBooks/WPLBN0002828782-Algebra-by-Boundless.aspx?
LINK BOOK
http://self.gutenberg.org/Get956uFile.aspx?&bookid=2828782

Unfolding the Labyrinth : Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science By Smarandache, Florentin

Unfolding the Labyrinth : Open Problems in Physics, Mathematics, Astrophysics, and Other Areas of Science
By Smarandache, Florentin
Summary
The reader will find herein a collection of unsolved problems in mathematics and the physical sciences. Theoretical and experimental domains have each been given consideration. The authors have taken a liberal approach in their selection of problems and questions, and have not shied away from what might otherwise be called speculative, in order to enhance the opportunities for scientific discovery.
LINK PROJECT GUTEMBERG
http://self.gutenberg.org/eBooks/WPLBN0002828491-Unfolding-the-Labyrinth---Open-Problems-in-Physics--Mathematics--Astrophysics--and-Other-Areas--by-Smarandache--Florentin.aspx?
LINK BOOK
http://uploads.worldlibrary.net/uploads/pdf/20130821172116unfoldinglabyrinth_pdf.pdf

Probleme Compilate Si Rezolvate de Geometrie Si Trigonometrie By Florentin Smarandache

Probleme_Compilate_Si_Rezolvate_de_Geome Probleme Compilate Si Rezolvate de Geometrie Si Trigonometrie
By Florentin Smarandache
Table of Contents
Geometrie (clasa a IX-a). 4 -- Geometrie §i trigonometrie. 27 -- Probleme diverse. 47 -- Probleme recapitulative. 56 -- Geometrie spatiala. 61 -- Drepte §i plane. 85 -- Probleme recapitulative. 97 -- Proectii. 128 -- Geornetrie §i trigonormtrie (clasa a X-a). 144 --
trie_Si_Trigonometrie-by_Florentin_Smarandache.aspx
LINK1
http://uploads.worldlibrary.net/uploads/pdf/20110903013643problemegeomtrig.pdf
LINK2
http://self.gutenberg.org/eBooks/WPLBN0002097120-Probleme_Compilate_Si_Rezolvate_de_Geometrie_Si_Trigonometrie-by_Florentin_Smarandache.aspx

Destino: Educação - Episódio 7 - Geral



O GRANDE DESAFIO EM AMERICA LATINA E A QUALIDADE DA EDUCAÇAO QUE PODEMOS APRENDER DOS PAISES QUE SÃO EXEMPLO EM EDUCAÇÃO,SEMPRE E BOM VER AS EXPERIENCIAS DOS OUTROS PAISES,OUTRAS REALIDADES E VERIFICAR QUE PODE SER APLICAVEL A NOSSO REALIDADE,NÃO E SOMENTE COPIAR,E CREAR SOLUÇOES CREATIVAS.
O PROBLEMA FUNDAMENTAL E DEFINIR QUE TIPO DE SOCIEDADE NOS QUEREMOS,QUE TIPO DE CIDADAO VAMOS FORMAR,ISTO SIGNIFICA VALORES ETICOS FUNDAMENTAIS,HOJE ESQUECIDOS.

domingo, 17 de novembro de 2013

Solid Geometry with Problems and Applications (Revised edition) by N. J. Lennes et al.

The Project Gutenberg EBook of Solid Geometry with Problems and
Applications (Revised edition), by H. E. Slaught and N. J. Lennes
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: Solid Geometry with Problems and Applications (Revised edition)
Author: H. E. Slaught
N. J. Lennes
Release Date: August 26, 2009 [EBook #29807]
Language: English
Character set encoding: ISO-8859-1
*** START OF THIS PROJECT GUTENBERG EBOOK SOLID GEOMETRY ***
LINK DOWNLOAD
http://www.gutenberg.org/files/29807/29807-pdf.pdf

PLANE GEOMETRY BY G.A. WENTWORTH

The Project Gutenberg EBook of Plane Geometry, by George Albert Wentworth
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org
Title: Plane Geometry
Author: George Albert Wentworth
Release Date: July 3, 2010 [EBook #33063]
Language: English
Character set encoding: ISO-8859-1
*** START OF THIS PROJECT GUTENBERG EBOOK PLANE GEOMETRY ***
LINK DOWNLOAD
http://www.gutenberg.org/files/33063/33063-pdf.pdf

PROBLEMS IN PLANE AND SOLID GEOMETRY v.1 Plane Geometry VIKTOR PRASOLOV


Abstract. This book has no equal. The priceless treasures of elementary geometry are
nowhere else exposed in so complete and at the same time transparent form. The short
solutions take barely 1.5 − 2 times more space than the formulations, while still remaining
complete, with no gaps whatsoever, although many of the problems are quite difficult. Only
this enabled the author to squeeze about 2000 problems on plane geometry in the book of
volume of ca 600 pages thus embracing practically all the known problems and theorems of
elementary geometry.
The book contains non-standard geometric problems of a level higher than that of the
problems usually offered at high school. The collection consists of two parts. It is based on
three Russian editions of Prasolov’s books on plane geometry.
The text is considerably modified for the English edition. Many new problems are added
and detailed structuring in accordance with the methods of solution is adopted.
The book is addressed to high school students, teachers of mathematics, mathematical
clubs, and college students.
LINK DOWNLOAD
http://students.imsa.edu/~tliu/Math/planegeo.pdf

sábado, 16 de novembro de 2013

Algebra Autor: Zvavich LI

Algebra e iniciar a análise. Conjunto de tarefas para se preparar para o exame escrito em álgebra eo início da análise para o curso colegial. 11 cl
Autor: Zvavich Li

Link 
http://eknigi.org/engine/download.php?id=51881 
LINK DOWNLOAD
http://www.mediafire.com/?vpjo8ah83k8vehs 


 

segunda-feira, 11 de novembro de 2013

Traité d'algèbre élémentaire, à l'usage des lycées, des collèges et des candidats à l'école militaire de Saint-Cyr, par E. Burat,... -E. Belin (Paris)-1876


LINK1
http://gallica.bnf.fr/ark:/12148/bpt6k6531600m/f7.image.r=Alg%C3%A8bre.langPT

Problèmes d'algèbre et exercices de calcul algébrique avec les solutions (5e éd.) / par G. Ritt,... -L. Hachette (Paris)-1860


LINK1
http://gallica.bnf.fr/ark:/12148/bpt6k207316v.r=Alg%C3%A8bre.langPT.swf
LINK2
http://gallica.bnf.fr/Search?ArianeWireIndex=index&q=Alg%C3%A8bre&lang=PT&n=15&p=2&pageNumber=610

Géométrie algébrique/Perrin, Daniel -EDP Sciences (COURTABOEUF)-1994



LINK
http://flipbook.cantook.net/?d=http%3A%2F%2Fwww.edenlivres.fr%2Fflipbook%2Fpublications%2F24057.js&oid=356&c=&m=&l=&r=EXERCISES%20%20geometrie%20solide&f=pdf

Exercices de géométrie Editor : A. Mame (Tours)

quinta-feira, 24 de outubro de 2013

domingo, 20 de outubro de 2013

JEE Main 2013 Maths Video Solution by Plancess Team

JEE Advanced 2013 Paper 2 Solutions - Maths (Code 4)



http://www.plancessiit.com - Learn the most easiest technique to solve "JEE Advanced 2013 Maths Paper 2" explained by the IIT Maths Expert -
Ram Charan Teja :
IIT-JEE '11 AIR : 1102

Adolescente que ocupó primer puesto en admisión a la UNI aspira a trabaj...

A História da Matemática 4 - Além do Infinito


A quarta e última parte da série mostra que, para muitas pessoas, o prazer da Matemática está no entendimento do problema, e não simplesmente na solução correta. Em 1900, o matemático francês David Hilbert identificou os mais importantes enigmas não resolvidos que desafiavam os matemáticos, definindo o roteiro de pesquisas para a Matemática no século XX. 15 dos 23 problemas já foram pelo menos parcialmente resolvidos. Os restantes continuam dando trabalho a quem persegue uma resposta.

Hoje, contamos com o computador, que revolucionou a Matemática ao permitir cálculos ultrarrápidos e ao ajudar os matemáticos a ver o caos. Só que provar sem entender ainda é uma questão que perturba os matemáticos.

Física Geral III - Aula 11 - Indução - Parte 1

quinta-feira, 17 de outubro de 2013

WCAS'13 - "Electrónica de Potencia" - Conferencia Nº 5


Publicado em 10/10/2013
"Convertidor Electrónico de Potencia para fuente de Energía Renovable"
Ponente: Dr. Rodolfo Moreno.
UNIVERSIDAD NACIONAL DE INGENIERIA LIMA PERU

Facebook: https://www.facebook.com/pages/Electr...
IEEE CAS UNI: https://www.facebook.com/ieeecasuni

terça-feira, 15 de outubro de 2013

Matriz inversa, traspuesta y adjunta 2ºBACHI unicoos

Sistema de ecuaciones - Metodo de CRAMER 2ºBACHI unicoos SCD determinant...

FISICA Leyes de KIRCHHOFF 2ºBACHI unicoos OHM circuito malla

Statik und Festigkeitslehre 2011/2012 1/22 Estática e Resistência dos Materiais

Euler's Method and Heun's Method

Boundary-value Problems and Finite-difference Equations WATERLOO ENGINEERING

terça-feira, 8 de outubro de 2013

Mathematik für Ingenieure B3 27/28 MATEMATICAS PARA INGENIEROS TRANSFORMADA DE LAPLACE

Física Geral III - Aula 14 - Magnetismo - Parte 2 UNICAMP BRASIL

Física Geral III - aula 14 - Magnetismo - Parte 1

Física Geral III - Aula 13 - Correntes Alternadas - Parte 2 PARTE EXPERIMENTAL

Física Geral III - Aula 13 - Correntes Alternadas - Parte 1 UNICAMP BRASIL


Nesta aula, o professor Luiz Marco Brescansin faz uma revisão sobre um circuito LC oscilante e sobre um circuito RLC fracamente amortecido. Introduz as oscilações forçadas e resolve um circuito RLC com uma fonte senoidal, apresentando os conceitos de impedância e de ressonância em um circuito.

Geometria Analítica e Vetores - Aula 19 - Exercícios de retas e planos

Cálculo II - Aula 6 - Parte 2 - Continuidade e cálculo de limites de fun...

Cálculo II - Aula 6 - Parte 1 - Continuidade e cálculo de limites de fun...




Cálculo II - Aula 6 - Parte 1 - Continuidade e cálculo de limites de funções de duas variáveis II
Nesta aula, a professora Martha Salermo Monteiro, do Instituto de Matemática e Estatística da USP, traz alguns objetos da matemática para ilustrar as paraboloides. Ela também fala das curvas em que o limite é igual a zero, das curvas de limite diferente de zero e do teorema para provar que o limite da função não existe

.Cálculo II - Aula 7 - Parte 3 - Resolução de problemas e exercícios da lista 1


Publicado em 04/10/2013

Publicado em 04/10/2013
Cálculo II - Aula 7 - Parte 3 - Resolução de problemas e exercícios da lista 1

A professora Martha Salermo Monteiro, do Instituto de Matemática e Estatística da USP, resolve mais alguns exercícios da Lista 1, disponível no link: http://univesptv.cmais.com.br/calculo-ii

Cálculo II - Aula 5 - Parte 2 - Continuidade e cálculo de limites de funções de 2 variáveis

Cálculo II - Aula 5 - Parte 1 - Continuidade e cálculo de limites de funções de 2 variáveis

Cursos Unicamp - Cálculo III - E.D.O Linear de 2ª Ordem; Wronskiano - Pa...

Cursos Unicamp - Cálculo III - E.D.O Linear de 2ª Ordem; Wronskiano - Pa...

Cursos Unicamp - Cálculo III - E.D.O Linear de 2ª Ordem; Wronskiano - Pa...

An introduction to Riemann integration - Dr Joel Feinstein

MATRIX INEQUALITIES


LINK
 http://keoserey.wordpress.com/page/16/
 http://www.mediafire.com/?eane7ty05qvg22t

sábado, 5 de outubro de 2013

MATHEMATICS ALL THE WAY BY ENGINEERING L.K.SHARMA

WEBSITE ORIGINAL
http://mathematicsgyan.weebly.com/
Er. L.K.Sharma an engineering graduate from NIT, Jaipur (Rajasthan), {Gold medalist, University of Rajasthan} is a well known name among the engineering aspirants for the last 15 years. He has been honored with "BHAMASHAH AWARD" two times for the academic excellence in the state of Rajasthan. He is popular among the student community for possessing the excellent ability to communicate the mathematical concepts in analytical and graphical way.
He has worked with many IIT-JEE coaching institutes of Delhi and Kota, {presently associated with Guidance, Kalu Sarai, New Delhi as senior mathematics faculty}. He has been a senior mathematics {IIT-JEE} faculty at Delhi Public School, RK Puram for five years.
 
 
LINK DOWNLOAD
http://www.mediafire.com/?8j1db6ug0yzin4n

quinta-feira, 3 de outubro de 2013

JEE Advanced 2013 Video Solution Paper-2, Code-3, Q-58

Ashani Dasgupta's Math Blog

EXCELLENT BLOG MATHEMATICS
http://cheentasutra.wordpress.com/




Indian Math Olympiad

Crease of a square paper


A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.
Discussion:
creaseAssuming the side of the square is ‘s’. Let a part of the crease be ‘x’ (hence the remaining part is ‘s-x’). We apply Pythagoras Theorem we solve for x:
x^2 + \frac {s^2 }{4} = (s-x)^2 implies x = \frac {3s}{8} and s-x = \frac {5s}{8}
Hence the ratio is 5:3.
Standard



Indian Math Olympiad

RMO 1990


  1. Two boxes contain between them 65 balls of several different sizes. Each ball is white, black, red or yellow. If you take any five balls of the same colour, at least two of them will always be of the same size (radius). Prove that there are at least three balls which lie in the same box have the same colour and have the same size (radius).
  2. For all positive real numbers a, b, c prove that \frac{a}{b+c} + \frac{b}{a+c}+ \frac{c}{a+b} \ge \frac{3}{2}
  3. A square sheet of paper ABCD is so folded that B falls on the mid-point of M of CD. Prove that the crease will divide BC in the ratio 5:3.

    Discussion

  4. Find the remainder when 2^{1990} is divided by 1990.
  5. P is any point inside a triangle ABC. The perimeter of the triangle AB +BC +CA=2s. Prove that s < AP + BP + CP < 2s.
  6. N is a 50 digit number (in a decimal scale). All digits except the 26^{th} digit (from the left) are 1.If N is divisible by 13, find the 26^{th} digit.
  7. A census-man on duty visited a house in which the lady inmates declined to reveal their individual ages, but said “We do not mind giving you the sum of the ages of any two ladies you may choose”. There upon the census man said, “In that case, please give me the sum of the ages of every possible pair of you”. They gave the sums as follows: 30, 33, 41, 58, 66, 69. The census-man took these figures and happily went away. How did he calculate the individual ages of the ladies from these figures?
  8. If the circumcenter and centroid of a triangle coincide, prove that the triangle must be equilateral.
Standard
Indian Math Olympiad

INMO 2012


  1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB =\sqrt {2 + \sqrt {2} } and AB subtends 1350 at the center of the circle. Find the maximum possible area of ABCD.
  2. Let p_1<p_2< p_3< p_4 and q_1<q_2<q_3<q_4 be two sets of prime numbers such that p_4 - p_1 = 8   and q_4 - q_1 = 8 . Suppose p_1>5 and q_1>5 . Prove that 30 divides p_1 - q_1 .
  3. Define a sequence <fn(x)> n∈N of functions as f_0(x )=1, f_1(x )=x , (f_n(x))^2 - 1 = f_{n-1} (x) f_{n+1} (x) , for n \ge 1 . Prove that each f_n(x ) is a polynomial with integer coefficients.
  4. Let ABC be a triangle. An interior point P of ABC is said to be good if we can find exactly 27 rays emanating from P intersecting the sides of the triangle ABC such that the triangle is divided by these rays into 27 smaller triangles of equal area. Determine the number of good points for a given triangle ABC.
  5. Let ABC be an acute angled triangle. Let D, E, F be points on BC, CA, AB such that AD is the median, BE is the internal bisector and CF is the altitude. Suppose that angle FDE = angle C and angle DEF = angle A and angle EFD = angle B. Show that ABC is equilateral.
  6. Let f :Z \mapsto Z be a function satisfying f(0) \neq 0 , f(1)=0 and
    1. f(xy) + f(x)f(y) = f(x) + f(y) ,
    2. (f(x-y) - f(0) ) f(x) f(y) = 0 for all x , y \in Z simultaneously.
      1. Find the set of all possible values of the function f.
      2. If f(10) \neq 0 and f(2) = 0 , find the set of all integers n such that f(n) \neq 0 .