Divisional MO Secondary 2011

Problem for Secondary Group from Divisional Mathematical Olympiad will be solved here.
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Divisional MO Secondary 2011

Unread post by 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. $A \times B$ is a perfect even square. What is the minimum number of prime factors of $B$?
  3. $|x – 2| \leq 5$ and $y \leq |x – 4|$. Find the minimum and maximum values of $xy$.
  4. \[\begin{align*}a + b + c + d + e =& 1\\ a - 2b + c -2d + e =& 2\\ a + b -3c + d -3e =& 3\\ -4a + b + c - 4d + e =& 4\\ a -5b + c + d - 5e =& 5\end{align*}\]
    উপরের সমীকরণগুলো থেকে $a$ এর মান বের কর।
    Find $a$ from the above set of equations.
  5. Find the range of the function \[f(x)=\frac{\lceil 2x\rceil-2 \lfloor x \rfloor}{\lfloor 2x \rfloor-2\lceil x\rceil} \]
    Here, $\lceil x\rceil$ represents the minimum integer greater than $x$ and $\lfloor 2x \rfloor$ 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 $2^x$ and $x^2$ 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 $A, B, C$ the following relations hold:
    $B \cap (C – A) = \{ \} $
    $B \cap C \not = \{ \}$
    $A,B,C \subset N$ ($N$ is the set of natural numbers)
    Given that $A = \{1,2,3,4\}, C = \{3,4,5,6\}$ and no element of $B$ is greater than the largest element of $C$ how many possible options are there for $B$?
Problem Set in pdf: viewtopic.php?f=8&t=469

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Re: Divisional MO Secondary 2011

Unread post by Moon » Wed Feb 02, 2011 6:50 pm

Rangpur Divisional Mathematical Olympiad 2011 : Secondary

Problem 1:
The sum of $81$ consecutive integers is $9^5$. What is their median?

Problem 2:
If $2<f<3$ and $-9<g<-1$, then what is the range of possible values of $fg$?

Problem 3:
If $2048$ is written as a product of two positive integers whose difference is as small as possible, then what is the difference?

Problem 4:
In Quadrilateral $ABCD$, $AB = 14$, $BC = 6$, $CD = 8$ and $AD=AC=X$. Find the range of possible values of $X$.

Problem 5:
$PQRS$ is a cyclic quadrilateral, where $PS = SR$. $PR$ and $QS$ intersect each other at point $O$. If $PS = 12$ and $OS = 6$. Find $OQ$.

Problem 6:
When $N$ is divided by $7$, the quotient is twice the remainder. What is the number greater than $7$ that must divide $N$?

Problem 7:
Let $f(x)$ be a function with the two properties
(a) for any two real numbers $x$ and $y$, $f (x+ y)= x+ f (y)$ and
(b) $f (0)= 19$
What is the value of $f (1952)$?

Problem 8:
Four points are chosen in the order $A$, $B$, $C$, $D$ on a line such that there is a point $X$, not on that line, so that triangles $XAB$ and $XCD$ have the same area. If $AB = 8$ and $BC = 5$, find the length $AD$.

Problem 9:
At a convention, flags of $4$ countries are to be hoisted on $2$ poles so that no pole is left empty and all the flags get hoisted. More than one flag must not be placed at the same height of the same pole. Rather, they can be placed above or below an already placed flag. In this manner, more than one flag can be placed serially on a single pole. How many ways can the flags be hoisted?

Problem 10:
$ABCD$ is a rectangle where $AB = 3$, $BC = 6$ and $CD=CE$. The area of the part of the quadrangle $CDGF$ that lies outside the circular arc can be expressed as $a-b \sqrt{3}-\frac{cx}{4}$ where $a$, $b$, $c$ are integers. Find $a+b+c$.

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Ragib Farhat Hasan
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Re: Divisional MO Secondary 2011

Unread post by Ragib Farhat Hasan » Sat Jan 17, 2015 4:30 pm

Answer to prob.no.3 is 32.

Ragib Farhat Hasan
Posts: 62
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Re: Divisional MO Secondary 2011

Unread post by Ragib Farhat Hasan » Sat Jan 31, 2015 9:16 pm

Answer to prob.no.9 is 60.

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