I assigned math problems to select students to represent the United States

Last month, the math problems I assigned (made up) were used to select the students to represent the United States in the next year’s International Mathematical Olympiad IMO (www.imo-official.org) competition 2013. This problem below is one of them:

It’s posted at this link

http://www.artofproblemsolving.com/Forum/resources.php?c=182&cid=210&year=2012&

Let  be a quadrilateral with . Diagonals  and  meet at . Let  and  denote the circumcircle and the circumcenter of triangle . Let  and  denote the circumcircle and circumcenter of triangle . Segment  meets  and  again at  and  (other than  and ), respectively. Let  and  be the midpoints of minor arcs  (not including ) and (not including ). Prove that .

USA TSTST stands for The Team Selection for the Team Selection Test. See definition here http://amc.maa.org/imo/2011imo.shtml

I suggested this problem in my book titled Narrative Approaches to the International Mathematical Problems on page 583 that was published by Authorhouse in April, 2012

http://bookstore.authorhouse.com/Products/SKU-000559720/Narrative-Approaches-to-the-International-Mathematical-Problems.aspx

Read page 583 in my book at a university library in the Philippines

http://rizal.lib.admu.edu.ph/rldigital/Narrative_Approaches_to_the_International_Mathematical_problems.pdf

and the problem was also suggested by me in December 2011 at this web link

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2254293&sid=e64c90bb9136105d280c9760a3b436c1#p2254293

The date of the test was July 18th, 2012

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2745851&sid=e64c90bb9136105d280c9760a3b436c1#p2745851

which was after the dates of my suggestions as you can verify.

I have also assigned math problems to select the students at Stanford, Harvard, MIT universities. See here

http://sumo.stanford.edu/smt/2012/tests/algebra-solutions.pdf

The problem used by Stanford is the same as the one on my book cover. See my book cover

http://www.amazon.com/gp/product/images/1452051771/ref=dp_image_z_0?ie=UTF8&n=283155&s=books

I have also assigned math problems to select the most talented students of the twenty countries in South America, Spain and Portugal (IberoAmerican Mathematical Olympiad), and I was told that my problem will be submitted for selection for the International Mathematical Olympiad competition 2013 http://www.uan.edu.co/imo2013/en/.

Stay tuned! There will be more.

Find the least number

Problem 4 of the International Zhautykov Olympiad 2010

Positive integers 1, 2, . . ., n are written on а blackboard (n > 2). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least n for which it is possible.

Solution

Noting that any prime divisor not equal to 2 is an odd number, the sum of two such prime numbers is an even number, and the prime divisor that is also an even number is number 2.

To find the least possible n we should keep a free adder which is a number originally on the board in the series 1, 2, . . ., n that is free from any calculations until the end to add to the last prime divisor. Since adding two odd numbers or even numbers creates a new even number that has 2 as its least prime divisor, the problem requires the addition of a prime number to number 2 to become another prime number.

Furthermore, unlike the additions of the even numbers, the total odd numbers in the series 1, 2, . . ., n have to be in pairs plus one more. The pairs of the odd numbers help adding up to exact even numbers in order for us to get 2 as their prime divisors. The two consecutive prime numbers under 97 that satisfy these conditions are 41 and 43, 59 and 61. We pick the pair 41 and 43 to give us the least n and n = 57.

Erasing the numbers this way: first erase all the odd numbers in pairs from 1 to 39, their pairings are of no significance, to get 2 at the end, leave number 41 alone. Next erase the rest of the odd numbers from 43 to 57 to get another number 2 on the board (we should have a total of three numbers 2 now.) Then erase all the even numbers except number 54 (which is the free adder.) The remaining numbers on the board are 2, 41, and 54. Now erase 41 and 2 to get 43. Then erase 43 and 54 to get 97.

Recent difficult math problem not solved

This is the most recent difficult problem not solved by anyone after being posted at http://www.mathlinks.ro – the most popular website that has been used to post most of the Mathematical Olympiad problems in the world. And it has been solved by our school.

Problem description:

Problem 6 of Korea Mathematical Olympiad 2012

Let O be the incircle of triangle ABC. Segments BC, CA meet with O at D, E. A line passing through B and parallel to DE meets O at F and G. (F is nearer to B than G.) Line CG meets O at H (≠ G). A line passing through G and parallel to EH meets with line AC at I. Line IF meets with circle O at J (≠ F). Lines CJ and EG meets at K. Let l be the line passing through K and parallel to JD. Prove that l, IF, ED meet at one point.

Solution

Click on the bitmaps to enlarge.

Problem with no solution in 27 years is now solved

Problem with no solution in 27 years is now solved. See here

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=329710&p=2596257#p2596257

Problem 2 of the Vietnamese MO Team Selection Test 1985

Let ABC be a triangle with AB = AC. A ray Ax is constructed in space such that the three planar angles of the trihedral angle ABCx at its vertex A are equal. If a point S moves on Ax, find the locus of the incenter of triangle SBC.

Solution

Find two positive integers

Find two positive integers a and b such that b is smaller than 200 and also satisfying the condition that the ratio a/b is smaller than the ratio 45/61 and greater than the ratio 59/80.

CD with my math solutions being sold by Mathematical Association of America

This CD being sold to raise fund by the Mathematical Association of America has the solutions to many difficult math problems donated by me.

You can buy it from the Mathematical Association of America.

Math problem not solved in 18 years

Problem 5 of the Vietnamese Mathematical Olympiad 1994

S is a sphere with center O. G and G’ are two perpendicular great circles on S. Take A, B, C on G and D on G’ such that the altitudes of the tetrahedron ABCD intersect at a point. Find the locus of the intersection.

Solution (Click on graphics to enlarge details)

Note: Please buy my books for more solutions of the many wondrous / difficult problems.

New mathematical theorem

I have found a new mathematical phenomenon that I dubbed it as a mathematical theorem. These days there is no clear process on how to propose a mathematical theorem and thus I just post it here.

Dinh’s mathematical theorem of 2012:

The segment connecting the midpoints of opposite sides of a quadrilateral is orthogonal to the segment connecting the two orthocenters of the triangles forming by the quadrilateral’s diagonals that share the other two sides of the quadrilateral.

List of libraries in the world that have my math books

This is a partial list of the libraries in the world that have my math books

The World Cat library

http://www.worldcat.org/title/narrative-approaches-to-the-international-mathematical-problems/oclc/794042414&referer=brief_results

http://www.worldcat.org/title/how-to-solve-the-worlds-mathematical-olympiad-problems-volume-1/oclc/693533166&referer=brief_results

http://www.worldcat.org/title/hard-mathematical-olympiad-problems-and-their-solutions/oclc/747808929&referer=brief_results

The UCLA library, U.S.A.

http://ucla.worldcat.org/title/how-to-solve-the-worlds-mathematical-olympiad-problems-volume-1/oclc/693533166&referer=brief_results

http://ucla.worldcat.org/title/narrative-approaches-to-the-international-mathematical-problems/oclc/794042414&referer=brief_results

http://ucla.worldcat.org/title/hard-mathematical-olympiad-problems-and-their-solutions/oclc/747808929&referer=brief_results

The University of Arizona library, U.S.A.

http://universityofarizona.worldcat.org.ezproxy1.library.arizona.edu/title/narrative-approaches-to-the-international-mathematical-problems/oclc/794042414&referer=brief_results

The Texas Tech University library, U.S.A.

http://texastech-aleph.hosted.exlibrisgroup.com/F/ITTSSUYR6NGAQSNR2DPA1RYI12T5FU2QC6JC6CH11PABFXL2M7-55228?func=full-set-set&set_number=004472&set_entry=000001&format=999

The University of Denver library, Colorado, U.S.A.

http://bianca.penlib.du.edu/search/?searchscope=2&searchtype=o&searcharg=794042414

http://lib-lakshmi.cair.du.edu/iii/encore/record/C__Rb5401068__SNarrative+Approaches+to+the+International+Mathematical+Problems__Orightresult__X5?lang=eng&suite=cobalt

The Buffalo State College library, New York, U.S.A.

http://bsc.sunyconnect.suny.edu:4380/F?func=find-b&find_code=035&request=747808929

The Cleveland State University library, Ohio, U.S.A.

http://scholar.csuohio.edu/search~/a?searchtype=t&searcharg=how+to+solve+the+world%27s+mathematical+olympiad+problems&SORT=D

The Auburn University Montgomery library, Alabama, U.S.A.

http://ehis.ebscohost.com/eds/results?sid=b606fa0e-5fc4-43f5-a88b-d984fcf2327b%40sessionmgr14&vid=1&hid=5&bquery=((how+AND+to+AND+solve+AND+the+AND+world%26%2339%3bs+AND+mathematical+AND+olympiad+AND+problems))&bdata=JnR5cGU9MCZzaXRlPWVkcy1saXZl

http://aum.lib.auburn.edu/cgi-bin/Pwebrecon.cgi?DB=local&BOOL1=all+of+these&FLD1=Keyword+Anywhere+(GKEY)&CNT=50+records+per+page&SAB1=?693533166

The University of Iowa library, U.S.A.

http://smartsearch.uiowa.edu/primo_library/libweb/action/search.do?&dscnt=2&frbg=&tab=default_tab&dstmp=1339730289272&srt=rank&ct=search&mode=Basic&dum=true&indx=1&tb=t&fromLogin=true&vl(freeText0)=dinh%2C+steve&vid=uiowa&fn=search

The University of Massachusetts library, U.S.A.

http://umass.worldcat.org/title/how-to-solve-the-worlds-mathematical-olympiad-problems-volume-1/oclc/693533166&referer=brief_results

http://umass.worldcat.org/title/narrative-approaches-to-the-international-mathematical-problems/oclc/794042414&referer=brief_results

http://umass.worldcat.org/title/hard-mathematical-olympiad-problems-and-their-solutions/oclc/747808929&referer=brief_results

The University of California Davis library, U.S.A.

http://ucdavis.worldcat.org/title/narrative-approaches-to-the-international-mathematical-problems/oclc/794042414&referer=brief_results

The Technical University of Munich library, Germany

Universitätsbibliothek TU München University Library TUM

München, D-80333 Germany

https://opac.ub.tum.de/InfoGuideClient.tumsis/start.do?Login=wotum&Query=-1=%22steve%20dinh%22

http://www.worldcat.org/title/hard-mathematical-olympiad-problems-and-their-solutions/oclc/747808929&referer=brief_results

The Indian Institute of Technology Bombay library, India

http://www.library.iitb.ac.in/newsearchbook/q_word.php

http://www.library.iitb.ac.in/newsearchbook/ca_det.php?m_doc_no=299557

The University of Ottawa library, Canada

http://catalogue.bib.uottawa.ca/html/item.jsp?item=b4265507&language=en&trms=narrative%20approaches%20to%20the%20international%20mathematical%20problems&whr=1&ppg=4&loc=0&phrase=false

The University of Alberta library, Canada

https://era.library.ualberta.ca/public/view/item/uuid:30e012be-bec1-4580-acad-46415c0e9714

The University of Sydney library, Australia

http://opac.library.usyd.edu.au/record=b4104614

The University of Canberra library, Australia

http://webpac.canberra.edu.au/record=b1646296~S4

The Monash University library, Australia 

http://search.lib.monash.edu/primo_library/libweb/action/display.do?tabs=detailsTab&ct=display&fn=search&doc=catmua3206314&indx=1&recIds=catmua3206314&recIdxs=0&elementId=0&renderMode=poppedOut&displayMode=full&frbrVersion=&dscnt=0&frbg=&scp.scps=scope%3A%28catmua%29%2Cscope%3A%28arrow%29%2Cscope%3A%28catmsa%29%2Cscope%3A%28exam%29%2Cscope%3A%28catmum%29%2Cprimo_central_multiple_fe&tab=default_tab&dstmp=1346371364706&srt=rank&mode=Basic&dum=true&vl(freeText0)=narrative+approaches+to+the+international+mathematical+problems&vid=MUL

The University of Hong Kong library

http://library.hku.hk/record=b4764536

The Hong Kong Polytechnic University library

http://library.polyu.edu.hk/search/c?SEARCH=QA43+.D565+2012eb+&searchscope=6

http://library.polyu.edu.hk/search~/i1452051771

The University of Macau library

http://umaclib3.umac.mo/search/?searchtype=X&searcharg=steve+dinh&submit.x=6&submit.y=13&submit=Submit

The University of Warwick library, United Kingdom

http://webcat.warwick.ac.uk/record=b2565295

The National Taiwan University library, Taiwan

http://140.112.113.1/record=b4077296

The National University of Singapore library, Singapore
http://linc.nus.edu.sg:2084/search/?searchscope=16&searchtype=o&searcharg=OCN747808929

http://libencore.nus.edu.sg/iii/encore/record/C%7CRb3203091%7CSthe+hard+mathematical+olympiad+problems+and+their+solutions%7COrightresult%7CX5;jsessionid=ABC0EDAB4542457749D7C99BDA19A5E6?lang=eng&suite=def

The University of Guyana library, South America

http://library.uog.edu.gy/cgi-bin/koha/opac-detail.pl?biblionumber=51976

The Ateneo de Manila University library, Philippines

http://rizalls.lib.admu.edu.ph/TLCScripts/interpac.dll?LabelDisplay&Config=SAMPLE&Branch=,0,&FormId=48253629&RecordNumber=449289

The Hong Kong City library, Hong Kong

http://libcat.hkpl.gov.hk/webpac_eng/wgbroker.exe?2011101400230500011335+-access+top.all-materials-page+search+open+T+how%20to%20solve%20the%20world’s%20mathematical%20olympiad%20problems%23%23A:NONE%23NONE:NONE::%23%23

The Auckland city library, New Zealand

http://search.aucklandlibraries.govt.nz/?q=how%20to%20solve%20the%20world’s%20mathematical%20olympiad%20problems&refx=&uilang=en

The Dublin city library, Ireland

http://libcat.dublincity.ie/02_Catalogue/02_004_TitleResults.aspx?page=1&searchTerm=How+to+solve+the+world’s+Mathematical+Olympiad+problems%2c+Steve+D&searchType=1&media=&referrer=02_001_Search.aspx

The National library of Australia

http://trove.nla.gov.au/result?q=how+to+solve+the+world%27s+mathematical+olympiad+problems

http://trove.nla.gov.au/work/166492100?q=dinh%2C+steve&c=book&versionId=181448400

The city libraries of Sydney, Australia

http://library.cityofsydney.nsw.gov.au/opac/default.aspx

The Santa Cruz public library, California, U.S.A.

http://aqua.santacruzpl.org/default.ashx?q=How+to+solve+the+world%27s+mathematical+olympiad+problems

The San Jose public libraries, California, U.S.A.

http://catalog.sjlibrary.org/search~/a?searchtype=X&searcharg=How+to+solve+the+world%27s+mathematical+olympiad+problems&search-submit=Go&SORT=D&searchscope=1

The Santa Clara county libraries, California, U.S.A.

http://sccl.bibliocommons.com/search?q=How+to+solve+the+world%27s+mathematical+olympiad+problems&submit=Search&t=keyword

http://sccl.bibliocommons.com/item/show/1475137016_hard_mathematical_olympiad_problems_and_their_solutions

http://sccl.bibliocommons.com/item/show/1503324016_narrative_approaches_to_the_international_mathematical_problems

The Multnomah county library, Portand, Oregon, U.S.A.

http://catalog.multcolib.org/search/a?searchtype=Y&searcharg=how+to+solve+the+world%27s+mathematical+olympiad+problems&SORT=R&searchscope=1&submit=Search+catalog

The Alibris library, U.S.A.

http://library.alibris.com/booksearch?wtit=1468568515

http://library.alibris.com/booksearch?wtit=1452051771

http://library.alibris.com/booksearch?wtit=1463444907

Log of people in the world that follow my blogs

There are an average of four people searching for us everyday. We cannot catch up and post all those information daily.

On 09/08/2012 these people looked up my blogs:

Someone from the United Kingdom found your book ‘Book with title Narrative…‘ on Google with the keyword:”“two touching circles s and t share a common tangent which meets s at a and t at b.let ap be a diameter of s and let the tangent from p to t touch it at q. show that ap = pq.””

Someone from the Philippines found your book ‘Book with title Narrative…‘ on Google with the keyword:”two circles are in contact internally at a point t. let the chord ab of the largest circle be tangent to the smaller circle at point p. then the line tp bisect atb.”

Someone from the United States found your book ‘Book with title Narrative…‘ on Google with the keyword:”“romanian mathematical olympiad” 62nd”

Someone from Vietnam found your book ‘Book with title Narrative…‘ on Google with the keyword:”find the sum of square 1 + 2² + 3² + .. + 1999²”

 

On 09/05/2012 these people looked up my blogs:

Someone from the Philippines found your book ‘Book with title Narrative…‘ on Google with the keyword:”two circles are in contact internally at a point t. let the chord ab of the largest circle be tangent to the smaller circle at point p. then the line tp bisect atb.”

Someone from the United States found your book ‘Book with title Narrative…‘ on Google with the keyword:”“romanian mathematical olympiad” 62nd”

Someone from Vietnam found your book ‘Book with title Narrative…‘ on Google with the keyword:”find the sum of square 1 + 2² + 3² + .. + 1999²“

Someone from Greece found your paper ‘The Hard Mathematical…‘ on Google with the keyword:”the hard mathematical olympiad problems and their solutions“

Today 08/22/2012 these people looked up my blogs:

Someone from India found your paper ‘What day of the week was 01/01/0001‘ on Google

Someone from India found your paper  ‘Solutions in book title…‘ on Google with the keyword:”Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S………? ………………………………… If a , b , c and d denote length of sides of the quadrilateral, then prove that – 2 ≤ a² + b² + c² + d² ≤ 4 

Someone from Turkey found your book ‘Solutions in math book with…‘ on Google with the keyword:” Euler line olympiad problems“