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Divisional MO Secondary 2011

Divisional MO Secondary 2011

Post Number:#1  Unread postby BdMO » Fri Jan 28, 2011 10:14 pm

Dhaka Divisional Mathematical Olympiad 2011 : Secondary

  1. In a box there are 50 gold rings of 10 different sizes and 75 silver rings of 12 different sizes. The size of a gold ring might be the same as that of a silver ring. What is the minimum number of rings one need to pick up to be sure of having at least two rings different both in size and material?
  2. A is the product of seven odd prime numbers. AB is a perfect even square. What is the minimum number of prime factors of B?
  3. x25  and yx4 . Find the minimum and maximum values of xy.
  4. a+b+c+d+e= a2b+c2d+e= a+b3c+d3e= 4a+b+c4d+e= a5b+c+d5e= 1 2 3 4 5  

    উপরের সমীকরণগুলো থেকে a এর মান বের কর।
    Find a from the above set of equations.
  5. Find the range of the function

    Here, x  represents the minimum integer greater than x and 2x  represents the maximum integer less than x.
  6. The diagonal AB in quadrilateral ADBC bisects the angle CBD. DB and DA are tangent to the circumcircle of triangle ABC at points A and B. The perimeter of triangle ABC is 20 and perimeter of triangle ABD is 12. Find the length of BD.
  7. A six headed monster has 120 children. He wants to give a different name to each of his children. The names will consist of 3 English letters and one letter can be used more than once in a single name. What is the minimum number of letters the monster must use?
  8. A palindrome, such as 83438, is a number that remains the same when its digits are reversed. N is a six digit palindrome which is divisible by 6. The number obtained by eliminating its leftmost and rightmost digits is divisible by 4. How many possible values of N are possible?
  9. x is a positive integer so that 2x and x2 leaves same remainder when divided by 3. There are many possible values for x. What will be the remainder when the sum of the first 2011 values of x is divided by 1000?
  10. For three non empty finite sets ABC  the following relations hold:
    ABCN  (N is the set of natural numbers)
    Given that A=1234C=3456  and no element of B is greater than the largest element of C how many possible options are there for B?

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