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Math Problems

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  • Abdul Muntakim Rafi
    Dear Group, I couldn t connect with the group for my exam.However,today I am posting some questions.I found them pretty much interesting.I hope you will find
    Message 1 of 20 , May 2, 2010
      Dear Group,

      I couldn't connect with the group for my exam.However,today I am posting some questions.I found them pretty much interesting.I hope you will find them interesting too.

      1.A series is formed in the following manner:
      A(1) = 1;
      A(n) = f(m) numbers of f(m) followed by f(m) numbers of 0;
      m is the number of digits in A(n-1)
      Find A(30). Here f(m) is the remainder when m is divided by 9.

      2.Triangle ABC is right angled at B. The bisector of BAC meets BC at D. Let G denote the
      centroid (common point of the medians) of the triangle ABC. Suppose that GD is parallel to
      AB. Find C.

      And another one I will give you.Please don't laugh seeing the question.I wanna be sure if it's answer is 10.

      3.There are 20 people in a party excluding you. It is known that you know the same number
      of people as you don't know. How many of them do you know?

      Let me know what is your answer.Post interesting questions.

      Best Wishes
      Rafi



    • Kamruzzaman Kamrul
      Post your approaches first! ... From: Abdul Muntakim Rafi Subject: [math_club] Math Problems To: math_club@yahoogroups.com Date:
      Message 2 of 20 , May 2, 2010
        Post your approaches first!

        --- On Sun, 5/2/10, Abdul Muntakim Rafi <abdulmuntakim_rafi@...> wrote:

        From: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
        Subject: [math_club] Math Problems
        To: math_club@yahoogroups.com
        Date: Sunday, May 2, 2010, 10:02 AM

         

        Dear Group,

        I couldn't connect with the group for my exam.However, today I am posting some questions.I found them pretty much interesting. I hope you will find them interesting too.

        1.A series is formed in the following manner:
        A(1) = 1;
        A(n) = f(m) numbers of f(m) followed by f(m) numbers of 0;
        m is the number of digits in A(n-1)
        Find A(30). Here f(m) is the remainder when m is divided by 9.

        2.Triangle ABC is right angled at B. The bisector of BAC meets BC at D. Let G denote the
        centroid (common point of the medians) of the triangle ABC. Suppose that GD is parallel to
        AB. Find C.

        And another one I will give you.Please don't laugh seeing the question.I wanna be sure if it's answer is 10.

        3.There are 20 people in a party excluding you. It is known that you know the same number
        of people as you don't know. How many of them do you know?

        Let me know what is your answer.Post interesting questions.

        Best Wishes
        Rafi




      • kamallohia
        Hi Rafi [:)] My answer to first question is: A(30) = A(24) = A(18) = A(12) = A(6) = 77777770000000. Please confirm. ... posting some questions.I found them
        Message 3 of 20 , May 2, 2010

          Hi Rafi:)

          My answer to first question is:

          A(30) = A(24) = A(18) = A(12) = A(6) = 77777770000000.

          Please confirm.

          --- In math_club@yahoogroups.com, Abdul Muntakim Rafi <abdulmuntakim_rafi@...> wrote:
          >
          > Dear Group,
          >
          > I couldn't connect with the group for my exam.However,today I am posting some questions.I found them pretty much interesting.I hope you will find them interesting too.
          >
          > 1.A series is formed in the following manner:
          > A(1) = 1;
          > A(n) =
          > f(m) numbers of f(m) followed by f(m) numbers of 0;
          > m is the number
          > of digits in A(n-1)
          > Find A(30). Here f(m) is the remainder when m is
          > divided by 9.
          >
          > 2.Triangle ABC is right angled at B. The bisector
          > of BAC meets BC at D. Let G denote the
          > centroid (common point of the
          > medians) of the triangle ABC. Suppose that GD is parallel to
          > AB. Find C.
          >
          > And another one I will give you.Please don't laugh seeing the question.I wanna be sure if it's answer is 10.
          >
          > 3.There are 20
          > people in a party excluding you. It is known that you know the same
          > number
          > of people as you don't know. How many of them do you know?
          >
          > Let me know what is your answer.Post interesting questions.
          >
          > Best Wishes
          > Rafi
          >

        • Abdul Muntakim Rafi
          Dear Kamal, Your answer is absolutely right.The sequence cycles after each six words.Thus we can find that A(30)=77777770000000 Actually the questions I posted
          Message 4 of 20 , May 7, 2010
            Dear Kamal,

            Your answer is absolutely right.The sequence cycles after each six words.Thus we can find that A(30)=77777770000000
            Actually the questions I posted were from National Olympiad 2010.It took me 20 to 30 minutes to solve the problem there.
            I did in the following way-
            A(1)=1
            In case of A(2)
            m=number of digit of A(2-1)=A(1)=1
            f(m) is the remainder when m is divided by 9=1
            Therefore A(2)=10
            In case of A(3)
            m=2
            f(m)=2
            A(3)=2200
            In case of A(4)
            m=4
            f(m)=4
            A(4)=44440000
            In case of A(5)
            m=8
            f(m)=8
            A(5)=8888888800000000
            In case of A(6)
            m=16
            f(m)=7[f(m) is the remainder when m is divided by 9]
            A(6)=77777770000000
            In case of A(7)
            m=14
            f(m)=5
            A(7)=5555500000
            In case of A(8)
            m=10
            f(m)=1
            A(8)=10
            Then A(9)=2200 A(10)=....................................
            What is the value of A(30)
            We have found that these values will come one after another.
            from A(2) to A(7)
            These six values will come again and again.
            Now except A(1),This series recycle after each six terms.
            Then A(30) is the 29th one.Divide 29 with 6.5 is the remainder.The value of 5th one would be the same as A(6)=77777770000000

            Answer the other questions and post questions to the group.

            Best Regards
            Rafi
             


            From: kamallohia <kamallohia@...>
            To: math_club@yahoogroups.com
            Sent: Mon, May 3, 2010 12:39:13 PM
            Subject: [math_club] Re: Math Problems

             

            Hi Rafi:)

            My answer to first question is:

            A(30) = A(24) = A(18) = A(12) = A(6) = 77777770000000.

            Please confirm.

            --- In math_club@yahoogrou ps.com, Abdul Muntakim Rafi <abdulmuntakim_ rafi@...> wrote:
            >
            > Dear Group,
            >
            > I couldn't connect with the group for my exam.However, today I am posting some questions.I found them pretty much interesting. I hope you will find them interesting too.
            >
            > 1.A series is formed in the following manner:
            > A(1) = 1;
            > A(n) =
            > f(m) numbers of f(m) followed by f(m) numbers of 0;
            > m is the number
            > of digits in A(n-1)
            > Find A(30). Here f(m) is the remainder when m is
            > divided by 9.
            >
            > 2.Triangle ABC is right angled at B. The bisector
            > of BAC meets BC at D. Let G denote the
            > centroid (common point of the
            > medians) of the triangle ABC. Suppose that GD is parallel to
            > AB. Find C.
            >
            > And another one I will give you.Please don't laugh seeing the question.I wanna be sure if it's answer is 10.
            >
            > 3.There are 20
            > people in a party excluding you. It is known that you know the same
            > number
            > of people as you don't know. How many of them do you know?
            >
            > Let me know what is your answer.Post interesting questions.
            >
            > Best Wishes
            > Rafi
            >


          • Jon Tavasanis
            Dear Rafi, I m very glad to have you back again. I hope you have passed all of your exams. Now I m attaching my solution of our old triangle problem. Tell me
            Message 5 of 20 , May 8, 2010

            Dear Rafi,

            I'm very glad to have you back again. I hope you have passed all of your exams.

            Now I'm attaching my solution of our old triangle problem. Tell me if you have problems in reading it, since it's a standard PDF file. Sometimes I experienced problems when downloading attached files from the Yahoo e-mail service through Microsoft Internet Explorer. A good choice may be saving the file onto disk before opening it, or even going over to Mozilla Firefox.

            I encourage you to take your first steps in the "maths-for-fun" activity. Technically, how will it be? Have you considered to start a blog, or are you planning to go Facebook instead?

            There you have my contribution. It's a bunch of eight problems of varying difficulty. I hope they will do.

            (1) Give an equation from which an egg-shaped curve can be plotted.

            (2) Three parallel strips have been painted on the floor of an airport lobby. Their lengths are 8, 11, and 13 meters. An architect decided later that the strips should all have the same length. Repainting and/or unpainting the strips cost the same. In order to minimize costs, what length will the strips finally have?

            (3) The perimeter of a right-angle triangle is 96. The altitude on the hypothenuse is 19.2. What are the lengths of its sides?

            (4) The sum of all positive multiples of three that are less than 1000 is multiplied by six and added to four times the sum of the positive non-multiples of three that are less than 1000. What is the resulting sum?

            (5) For a given arithmetic series, the sum of the first 50 terms is 200 and the sum of the next 50 terms is 2700. What is the first term in the series?

            (6) Does equation 1 / (x - 1) = x / (x^2 - 1) have any solution?

            (7) What are the real solutions of x = sqrt(2 * x + 3)?

            (8) A five-legged Martian has a drawer full of socks, each of which is red, white, or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. What is the least number of socks that the Martian must remove from the drawer to be certain there will be 5 socks of the same color?

            That's all by now.

            Best regards,

            Jon Tavasanis

          • Abdul Muntakim Rafi
            Dear Group, We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what
            Message 6 of 20 , May 18, 2010
              Dear Group,

              We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

              Best Wishes
              Rafi

            • Tarik Adnan
              Hi, It is really great to hear that you have started a science club in your school. I wrote some notes and guidelines on BdMO. This is available in the website
              Message 7 of 20 , May 20, 2010
                Hi,
                It is really great to hear that you have started a science club in your school.

                I wrote some notes and guidelines on BdMO. This is available in the website of Kushtia Math Circle.
                Just check in the document section.

                Regards
                Moon


                From: "math_club@yahoogroups.com" <math_club@yahoogroups.com>
                To: math_club@yahoogroups.com
                Sent: Wednesday, May 19, 2010 20:10:31
                Subject: [math_club] Digest Number 1408

                math_club

                Messages In This Digest (1 Message)

                1a.
                Math Problems From: Abdul Muntakim Rafi

                Message

                1a.

                Math Problems

                Posted by: "Abdul Muntakim Rafi" abdulmuntakim_rafi@...   abdulmuntakim_rafi

                Tue May 18, 2010 7:58 am (PDT)



                Dear Group,

                We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

                Best Wishes
                Rafi

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                • Abdul Muntakim Rafi
                  Dear Moon, Thanks for the response.I have downloaded almost all document from your site.But I think at first the students should build up their basic in
                  Message 8 of 20 , May 20, 2010
                    Dear Moon,

                    Thanks for the response.I have downloaded almost all document from your site.But I think at first the students should build up their basic in maths.So we will discuss with them on the basic rules of maths.Then I will try to give them the pages for studying.We can solve the problems in the Science Club.

                    Best Wishes
                    Rafi

                    From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5?
                    You wrote that the answer is 378.Is it right?



                    From: Tarik Adnan <moonmath420@...>
                    To: math_club@yahoogroups.com
                    Sent: Thu, May 20, 2010 5:24:57 PM
                    Subject: [math_club] Re: Math Problems

                     

                    Hi,
                    It is really great to hear that you have started a science club in your school.

                    I wrote some notes and guidelines on BdMO. This is available in the website of Kushtia Math Circle.
                    Just check in the document section.

                    Regards
                    Moon


                    From: "math_club@yahoogro ups.com" <math_club@yahoogrou ps.com>
                    To: math_club@yahoogrou ps.com
                    Sent: Wednesday, May 19, 2010 20:10:31
                    Subject: [math_club] Digest Number 1408

                    math_club

                    Messages In This Digest (1 Message)

                    1a.
                    Math Problems From: Abdul Muntakim Rafi

                    Message

                    1a.

                    Math Problems

                    Posted by: "Abdul Muntakim Rafi" abdulmuntakim_ rafi@yahoo. com   abdulmuntakim_ rafi

                    Tue May 18, 2010 7:58 am (PDT)



                    Dear Group,

                    We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

                    Best Wishes
                    Rafi

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                    • Abdul Muntakim Rafi
                      Hello Group, Here are some problems for you. 1.At a party, there girls and three boys sit at a round table. What is the probability that no two persons of the
                      Message 9 of 20 , May 20, 2010
                        Hello Group,
                        Here are some problems for you.

                        1.At a party, there girls and three boys sit at a round table. What
                        is the probability that no two persons of the same sex sit at
                        adjacent places?

                        2.The base 10 representation of 6^4 is 1296 . What is the base-6
                        representation of 1295 ?

                        There is an interesting thing about the second problem.I didn't know anything about base representation of anything before.But I solved the problem.Please tell me how you solve it.

                        Best Wishes
                        Rafi

                      • S. M. Mamun Ar Rashid
                        Dear Rafi:   When teaching mathematics, I always remind myself what Sofia Kovalevskaya once said: It is impossible to be a mathematician without being a poet
                        Message 10 of 20 , May 20, 2010
                          Dear Rafi:
                           
                          When teaching mathematics, I always remind myself what Sofia Kovalevskaya once said: "It is impossible to be a mathematician without being a poet in soul." Therefore, the first thing I suggest you to do is make your class interactive and poetic, not prosaic like 2 × 3 = 6. Tell your students story, tell them history, tell them about great persons like Omar Khyaam, who was one of World's finest poets and mathematicians. And construct your problems and solutions in storylines, so mathematics would be lively and concrete to them, rather than abstract and formidable.  
                           
                          About chosing your lecture contents, you could wish to consult this website:  http://www.aaamath.com/grade8.html
                           
                          Wish you good luck!
                           
                          Kind regards,
                           
                          S. M. Mamun Ar Rashid
                           

                          --- On Tue, 5/18/10, Abdul Muntakim Rafi <abdulmuntakim_rafi@...> wrote:

                          From: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
                          Subject: [math_club] Math Problems
                          To: math_club@yahoogroups.com
                          Date: Tuesday, May 18, 2010, 7:58 AM

                           
                          Dear Group,

                          We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

                          Best Wishes
                          Rafi


                        • Tamim Shahriar
                          You can also browse through the archive: http://mathforum.org/dr/math/ It will surely help to prepare for classes. regards, Subeen.
                          Message 11 of 20 , May 20, 2010
                            You can also browse through the archive: http://mathforum.org/dr/math/
                            It will surely help to prepare for classes.


                            regards,
                            Subeen.
                          • S. M. Mamun Ar Rashid
                            (1) Problem type: Circular Permutation, Probability   In a circular arrangement of people, you must keep one person fixed, and then work on arranging the
                            Message 12 of 20 , May 20, 2010
                              (1) Problem type: Circular Permutation, Probability
                               
                              In a circular arrangement of people, you must keep one person fixed, and then work on arranging the remaining people. This is because 123 & 231 are different permutations in line arrangement, but the same permutation in circular/round arrangement. So, when one person is fixed in round arrangement, any change in arrangements of the rest will create a new permutation. This way, duplicity is avoided.
                              Therefore, let's fix a girl in a position. Now we have 2 other girls and 3 boys to arrange. 5 people can be arranged in total 5! or (5.4.3.2.1) or 120 ways.
                               
                              But if we want to keep boys and girls in alternate positions, first we need to place one group, and then place the other group in gaps of the first group. So, after fixing one girl in her position, the 2 other girls can be placed in 2! ways and the 3 boys can be placed in 3! ways in the girls' gaps. Therefore, total no. of ways here is 2! × 3! or (2.1) × (3.2.1) or 12 ways.
                               
                              Probability = 12/120 or 1/10. 
                               
                              Note: There is another issue. Some problems might specify that clock-wise and anti-clock-wise arrangements are to be considered the same. In such case, you would divide the total no. of arrangements by 2.
                               
                              (2) Problem type: Number System, Base, Conversion Of Base
                               
                              Suppose, you have bought an ice-cream the price of which is 31 Bangladeshi Taka (BDT). How will you pay the seller the money? Perhaps you would give her one 1-BDT note and three 10-BDT notes? Why? Because we know that 31 amounts to one 1 plus three 10s.
                               
                              But if you lived in ancient Babylon (present-day Iraq) and bought the ice cream there for 31 Babylonian Taka (BBT), then you would pay the seller one 1-BBT note but three 60-BBT notes. Why is the difference between modern Bangladesh and ancient Babylon? Because in Bangladesh, and all most all over the world today, we use 10-base number system, but the ancient Babylonians used 60-base number system.
                               
                              The scenario would be yet different in ancient far-way Maya Kingdom in Central America that used 20-base number system. For 31, the May would pay one 1-Maya Taka note but three 20-Maya Taka notes.
                               
                              Although it may sound puzzling, the matter is simple. In our 10-base system,
                               
                              5327 (decimal or 10-base)
                              = (7×1) + (2×10) + (3×100) + (5×1000)
                              =   (7×10^0) + (2×10^1) + (3×10^2) + (5×10^3)
                               
                               
                              In ancient Maya system,
                               
                              5327 (Maya)
                              =   (7×20^0) + (2×20^1) + (3×20^2) + (5×20^3)
                              = 7 + 40 + 1,200 + 40,000 (now converted into decimal)
                              = 41,247 (in decimal)
                               
                              In 6-base system,
                               
                              5327 (base 6)
                              =   (7×6^0) + (2×6^1) + (3×6^2) + (5×6^3)
                              = 7 + 12 + 108 + 1080 (now converted into decimal)
                              = 1207 (in decimal)
                               
                              So, here we are going from a particular base to decimal. We can also convert a decimal number to particular base number. To do this quickly, first we need to arrange powers of that base, and then chose suitable combinations from those powers (starting from the highest) to form the desired number:
                               
                              So, what is the base-6 representation of 1295 (which is in base-10, i.e. decimal)?
                               
                              Step 1: Arrange “Powers of 6”:
                               
                              6^0, 6^1, 6^2, 6^3, 6^4…
                               
                              =1, 6, 36, 216, 1296…
                               
                              Step 2: Choose suitable combinations of these powers 6, 36, 216, 1296… to form 1295.
                              We cannot choose 1296; it is beyond limit. So, chose the next power as many times as possible. We must choose 216 five times.
                              5×216 (=1080)
                               
                              We now have 1295-1080=215 remaining. Go to 36 maximum possible, i.e., five times.
                              5×36 (=180)
                               
                              Now we have 215-180=35. Go next to 6 five times: 5×6 (=30). Finally, we have 5.
                               
                              Therefore, 1295 (in decimal)
                              =1080+180+30+5
                              = (5×216) + (5×36) + (5×6) + (5×1)
                              = (5×6^3) + (5×6^2) + (5×6^1) + (5×6^0)
                              = 5555 (in base-6)
                               
                              Regards,
                               
                              S. M. Mamun Ar Rashid

                              --- On Thu, 5/20/10, Abdul Muntakim Rafi <abdulmuntakim_rafi@...> wrote:

                              From: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
                              Subject: [math_club] Math Problems
                              To: math_club@yahoogroups.com
                              Date: Thursday, May 20, 2010, 9:34 AM

                               
                              Hello Group,
                              Here are some problems for you.

                              1.At a party, there girls and three boys sit at a round table. What
                              is the probability that no two persons of the same sex sit at
                              adjacent places?

                              2.The base 10 representation of 6^4 is 1296 . What is the base-6
                              representation of 1295 ?

                              There is an interesting thing about the second problem.I didn't know anything about base representation of anything before.But I solved the problem.Please tell me how you solve it.

                              Best Wishes
                              Rafi


                            • kamal lohia
                              From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5? You wrote that the answer is 378.Is it right? My Answer to your question is 267.
                              Message 13 of 20 , May 21, 2010
                                From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5?
                                You wrote that the answer is 378.Is it right?
                                My Answer to your question is 267.


                                From: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
                                To: math_club@yahoogroups.com
                                Sent: Thu, May 20, 2010 8:41:08 PM
                                Subject: Re: [math_club] Re: Math Problems

                                 

                                Dear Moon,

                                Thanks for the response.I have downloaded almost all document from your site.But I think at first the students should build up their basic in maths.So we will discuss with them on the basic rules of maths.Then I will try to give them the pages for studying.We can solve the problems in the Science Club.

                                Best Wishes
                                Rafi

                                From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5?
                                You wrote that the answer is 378.Is it right?



                                From: Tarik Adnan <moonmath420@ yahoo.com>
                                To: math_club@yahoogrou ps.com
                                Sent: Thu, May 20, 2010 5:24:57 PM
                                Subject: [math_club] Re: Math Problems

                                 

                                Hi,
                                It is really great to hear that you have started a science club in your school.

                                I wrote some notes and guidelines on BdMO. This is available in the website of Kushtia Math Circle.
                                Just check in the document section.

                                Regards
                                Moon


                                From: "math_club@yahoogro ups.com" <math_club@yahoogrou ps.com>
                                To: math_club@yahoogrou ps.com
                                Sent: Wednesday, May 19, 2010 20:10:31
                                Subject: [math_club] Digest Number 1408

                                Messages In This Digest (1 Message)

                                1a.
                                Math Problems From: Abdul Muntakim Rafi

                                Message

                                1a.

                                Math Problems

                                Posted by: "Abdul Muntakim Rafi" abdulmuntakim_ rafi@yahoo. com   abdulmuntakim_ rafi

                                Tue May 18, 2010 7:58 am (PDT)



                                Dear Group,

                                We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

                                Best Wishes
                                Rafi

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                                • Abdul Muntakim Rafi
                                  Dear Kamal, My answer is also 267.It was a question from divisional Olympiad 2010.I found it in KMC s site.There the given answer was 378.It was a mistake.I
                                  Message 14 of 20 , May 21, 2010
                                    Dear Kamal,

                                    My answer is also 267.It was a question from divisional Olympiad 2010.I found it in KMC's site.There the given answer was 378.It was a mistake.I wrote to Moon about it so that he could see the mistake.He is the administrator of KMC.However,thanks for the response.

                                    Best Wishes
                                    Rafi



                                    From: kamal lohia <kamallohia@...>
                                    To: math_club@yahoogroups.com
                                    Cc: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
                                    Sent: Fri, May 21, 2010 3:55:45 PM
                                    Subject: Re: [math_club] Re: Math Problems

                                     

                                    From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5?
                                    You wrote that the answer is 378.Is it right?
                                    My Answer to your question is 267.


                                    From: Abdul Muntakim Rafi <abdulmuntakim_ rafi@yahoo. com>
                                    To: math_club@yahoogrou ps.com
                                    Sent: Thu, May 20, 2010 8:41:08 PM
                                    Subject: Re: [math_club] Re: Math Problems

                                     

                                    Dear Moon,

                                    Thanks for the response.I have downloaded almost all document from your site.But I think at first the students should build up their basic in maths.So we will discuss with them on the basic rules of maths.Then I will try to give them the pages for studying.We can solve the problems in the Science Club.

                                    Best Wishes
                                    Rafi

                                    From 1 to 1000, how many integers are multiples of 3 or 6 but not of 5?
                                    You wrote that the answer is 378.Is it right?



                                    From: Tarik Adnan <moonmath420@ yahoo.com>
                                    To: math_club@yahoogrou ps.com
                                    Sent: Thu, May 20, 2010 5:24:57 PM
                                    Subject: [math_club] Re: Math Problems

                                     

                                    Hi,
                                    It is really great to hear that you have started a science club in your school.

                                    I wrote some notes and guidelines on BdMO. This is available in the website of Kushtia Math Circle.
                                    Just check in the document section.

                                    Regards
                                    Moon


                                    From: "math_club@yahoogro ups.com" <math_club@yahoogrou ps.com>
                                    To: math_club@yahoogrou ps.com
                                    Sent: Wednesday, May 19, 2010 20:10:31
                                    Subject: [math_club] Digest Number 1408

                                    Messages In This Digest (1 Message)

                                    1a.
                                    Math Problems From: Abdul Muntakim Rafi

                                    Message

                                    1a.

                                    Math Problems

                                    Posted by: "Abdul Muntakim Rafi" abdulmuntakim_ rafi@yahoo. com   abdulmuntakim_ rafi

                                    Tue May 18, 2010 7:58 am (PDT)



                                    Dear Group,

                                    We opened a Science Club in our school.I am taking the Maths classes there.I have taken only one class.Can you give me any suggestion on what should I discuss with the students of 8 and 9.I am in class 10.Which questions can ensure the increase of their ability of doing maths.Tomorrow I will discuss with them about the basic knowledge of maths.Can advise me anything more?

                                    Best Wishes
                                    Rafi

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                                    • Abdul Muntakim Rafi
                                      Dear Group, Here are some problems for u.They aren t very hard.But you will pass your time well with them because they are very interesting.They seemed very
                                      Message 15 of 20 , Jun 3, 2010
                                        Dear Group,

                                        Here are some problems for u.They aren't very hard.But you will pass your time well with them because they are very interesting.They seemed very interesting to me.You don't need to apply anything here.It's a lot like puzzle.

                                        1.Five men are wearing either blue or red hats. A man wearing a blue hat always tells
                                        the truth. A man wearing a red hat always lies. They look at each other’s hat.
                                        A Says “I see 4 blue hats.”
                                        B Says “I see 3 blue hats and 1 red hat.”
                                        C Says “I see 1 blue hat and 3 red hats.”
                                        D says “I see 4 red hats.”
                                        E Does not say anything.
                                        Find the color of each person’s hat.

                                        2.Asmaa, and her brother Ahmed are chess players. Asmaa's son Shamim and her daughter Sharmeen are also chess players. The worst player's twin (who
                                        is one of the 4 chess players) and best player are of the opposite sex. The
                                        worst player and the best player are the same age. Who is the worst player?

                                        Best Wishes
                                        Rafi

                                      • Abdul Muntakim Rafi
                                        Dear Group, Here are some problems.They are not easy. 1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an
                                        Message 16 of 20 , Jun 6, 2010
                                          Dear Group,

                                          Here are some problems.They are not easy.

                                          1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                          2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                          If you can't solve it,please discuss it with other members of the group.
                                          I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                          I will be waiting for ur answer.

                                          Best Wishes
                                          Rafi

                                        • kamal lohia
                                          Hi Rafi 1) if a = b, then p^a + p^b = 2*p^a . Now p must be 2 only and a can be any odd positive integer. If a
                                          Message 17 of 20 , Jun 6, 2010
                                            Hi Rafi
                                            1) if a = b, then p^a + p^b = 2*p^a .
                                            Now p must be 2 only and a can be any odd positive integer.

                                            If a < b, then p^a + p^b = p^a(1 + p^b-a) can never be perfect square.



                                            From: Abdul Muntakim Rafi <abdulmuntakim_rafi@...>
                                            To: math_club@yahoogroups.com
                                            Sent: Mon, June 7, 2010 12:13:02 AM
                                            Subject: [math_club] Math Problems

                                             

                                            Dear Group,

                                            Here are some problems.They are not easy.

                                            1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                            2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                            If you can't solve it,please discuss it with other members of the group.
                                            I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                            I will be waiting for ur answer.

                                            Best Wishes
                                            Rafi


                                          • Tarik Adnan
                                            ... If a How do you know??? Take a=2k, b=2k+3, p=2 Solution: When a=b, we get, p=2,
                                            Message 18 of 20 , Jun 7, 2010
                                              >>>
                                              If a < b, then p^a + p^b = p^a(1 + p^b-a) can never be perfect square.>>>>

                                              How do you know??? Take a=2k, b=2k+3, p=2

                                              Solution:

                                              When a=b, we get, p=2, a=2k+1

                                              When WLOG a<b

                                              As are relatively prime, they are both squares, i.e. a=2k

                                              Let b-a=x (just for convenience, btw x is odd)

                                              So,

                                              Now we consider two cases:

                                              1. p odd
                                              Then ; which implies, m-1=1, so

                                              2.p=2
                                              That is
                                              Here , which implies


                                              Problem 2 is a folklore (classic) problem; IMO 1985/4, APMO 2007/1 and many more problems including this one use the same idea.
                                              List the primes less than 42, write the  numbers as product of prime powers, and argue with pigeon-hole principle.
                                              I mean let the 131 numbers be, and the 13 primes less than 42, , Write
                                              By the way, what is the source of the second problem?

                                              Moon


                                              From: "math_club@yahoogroups.com" <math_club@yahoogroups.com>
                                              To: math_club@yahoogroups.com
                                              Sent: Monday, June 7, 2010 19:39:44
                                              Subject: [math_club] Digest Number 1420

                                              math_club

                                              Messages In This Digest (2 Messages)

                                              1a.
                                              Math Problems From: Abdul Muntakim Rafi
                                              1b.
                                              Re: Math Problems From: kamal lohia

                                              Messages

                                              1a.

                                              Math Problems

                                              Posted by: "Abdul Muntakim Rafi" abdulmuntakim_rafi@...   abdulmuntakim_rafi

                                              Sun Jun 6, 2010 11:43 am (PDT)



                                              Dear Group,

                                              Here are some problems.They are not easy.

                                              1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                              2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                              If you can't solve it,please discuss it with other members of the group.
                                              I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                              I will be waiting for ur answer.

                                              Best Wishes
                                              Rafi

                                              1b.

                                              Re: Math Problems

                                              Posted by: "kamal lohia" kamallohia@...   kamallohia

                                              Sun Jun 6, 2010 10:06 pm (PDT)



                                              Hi Rafi
                                              1) if a = b, then p^a + p^b = 2*p^a .
                                              Now p must be 2 only and a can be any odd positive integer.


                                              ____________ _________ _________ __
                                              From: Abdul Muntakim Rafi <abdulmuntakim_ rafi@yahoo. com>
                                              To: math_club@yahoogrou ps.com
                                              Sent: Mon, June 7, 2010 12:13:02 AM
                                              Subject: [math_club] Math Problems

                                              Dear Group,

                                              Here are some problems.They are not easy.

                                              1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                              2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                              If you can't solve it,please discuss it with other members of the group.
                                              I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                              I will be waiting for ur answer.

                                              Best Wishes
                                              Rafi

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                                            • Abdul Muntakim Rafi
                                              Dear moon, Thanks for ur reply. I posted the problems from the National Olympiad BDMO 2010.The 2nd question was from the Secondary group(10th question) and the
                                              Message 19 of 20 , Jun 8, 2010
                                                Dear moon,

                                                Thanks for ur reply.

                                                I posted the problems from the National Olympiad BDMO 2010.The 2nd question was from the Secondary group(10th question) and the 1st question was also from BDMO Secondary and Higher Secondary Group.

                                                Best wishes
                                                Rafi               




                                                From: Tarik Adnan <moonmath420@...>
                                                To: math_club@yahoogroups.com
                                                Sent: Mon, June 7, 2010 10:13:16 PM
                                                Subject: [math_club] Re: Math Problems

                                                 

                                                >>>
                                                If a < b, then p^a + p^b = p^a(1 + p^b-a) can never be perfect square.>>>>

                                                How do you know??? Take a=2k, b=2k+3, p=2

                                                Solution:

                                                When a=b, we get, p=2, a=2k+1

                                                When WLOG a<b

                                                As are relatively prime, they are both squares, i.e. a=2k

                                                Let b-a=x (just for convenience, btw x is odd)

                                                So,

                                                Now we consider two cases:

                                                1. p odd
                                                Then ; which implies, m-1=1, so

                                                2.p=2
                                                That is
                                                Here , which implies


                                                Problem 2 is a folklore (classic) problem; IMO 1985/4, APMO 2007/1 and many more problems including this one use the same idea.
                                                List the primes less than 42, write the  numbers as product of prime powers, and argue with pigeon-hole principle.
                                                I mean let the 131 numbers be, and the 13 primes less than 42, , Write
                                                By the way, what is the source of the second problem?

                                                Moon


                                                From: "math_club@yahoogro ups.com" <math_club@yahoogrou ps.com>
                                                To: math_club@yahoogrou ps.com
                                                Sent: Monday, June 7, 2010 19:39:44
                                                Subject: [math_club] Digest Number 1420

                                                math_club

                                                Messages In This Digest (2 Messages)

                                                1a.
                                                Math Problems From: Abdul Muntakim Rafi
                                                1b.
                                                Re: Math Problems From: kamal lohia

                                                Messages

                                                1a.

                                                Math Problems

                                                Posted by: "Abdul Muntakim Rafi" abdulmuntakim_ rafi@yahoo. com   abdulmuntakim_ rafi

                                                Sun Jun 6, 2010 11:43 am (PDT)



                                                Dear Group,

                                                Here are some problems.They are not easy.

                                                1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                                2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                                If you can't solve it,please discuss it with other members of the group.
                                                I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                                I will be waiting for ur answer.

                                                Best Wishes
                                                Rafi

                                                1b.

                                                Re: Math Problems

                                                Posted by: "kamal lohia" kamallohia@yahoo. com   kamallohia

                                                Sun Jun 6, 2010 10:06 pm (PDT)



                                                Hi Rafi
                                                1) if a = b, then p^a + p^b = 2*p^a .
                                                Now p must be 2 only and a can be any odd positive integer.


                                                ____________ _________ _________ __
                                                From: Abdul Muntakim Rafi <abdulmuntakim_ rafi@yahoo. com>
                                                To: math_club@yahoogrou ps.com
                                                Sent: Mon, June 7, 2010 12:13:02 AM
                                                Subject: [math_club] Math Problems

                                                Dear Group,

                                                Here are some problems.They are not easy.

                                                1.Find all the prime numbers p and positive integers a and b such that p^a + p^b is the square of an integer.

                                                2.In a set of 131 natural numbers, no number has a prime factor greater than 42. Prove that it is possible to choose four numbers from this set such that their product is a perfect square.

                                                If you can't solve it,please discuss it with other members of the group.
                                                I have some questions.In 10th question does the 131 natural numbers have any relation with the problem.if it has,what it is?
                                                I will be waiting for ur answer.

                                                Best Wishes
                                                Rafi

                                                Stay on top

                                                of your group

                                                activity with

                                                Yahoo! Toolbar

                                                Yahoo! Finance

                                                It's Now Personal

                                                Guides, news,

                                                advice & more.

                                                Yahoo! Groups

                                                Mental Health Zone

                                                Schizophrenia groups

                                                Find support

                                                Need to Reply?

                                                Click one of the "Reply" links to respond to a specific message in the Daily Digest.



                                              • Tarik Adnan
                                                Hmm...the second one was a bit hard for secondary, however, it was the last one...so I guess it is OK... I did not look at the problem sets after BdMO, and I
                                                Message 20 of 20 , Jun 9, 2010
                                                  Hmm...the second one was a bit hard for secondary, however, it was the last one...so I guess it is OK...
                                                  I did not look at the problem sets after BdMO, and I did not recognize it as I am solving a lot of problem lately :-s

                                                  btw the first problem you posted was a bit simplified than the original one. We need to find a,b such that p^a+p^b is a square of a RATIONAL number, not just integer. The solution method is same, but a bit more case work and modification is needed.
                                                  I could not complete this one in BdMO because I ran out of time...:(...btw I won the contest...so who cares :p

                                                  Moon


                                                  From: "math_club@yahoogroups.com" <math_club@yahoogroups.com>
                                                  To: math_club@yahoogroups.com
                                                  Sent: Wednesday, June 9, 2010 19:07:15
                                                  Subject: [math_club] Digest Number 1422

                                                  math_club

                                                  Messages In This Digest (1 Message)

                                                  1a.
                                                  Re: Math Problems From: Abdul Muntakim Rafi

                                                  Message

                                                  1a.

                                                  Re: Math Problems

                                                  Posted by: "Abdul Muntakim Rafi" abdulmuntakim_rafi@...   abdulmuntakim_rafi

                                                  Tue Jun 8, 2010 9:26 am (PDT)



                                                  Dear moon,

                                                  Thanks for ur reply.

                                                  I posted the problems from the National Olympiad BDMO 2010.The 2nd question was from the Secondary group(10th question) and the 1st question was also from BDMO Secondary and Higher Secondary Group.

                                                  Best wishes
                                                  Rafi


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