### Divisional MO Secondary 2011

Posted:

**Fri Jan 28, 2011 10:14 pm**Dhaka Divisional Mathematical Olympiad 2011 : Secondary

LaTeXed by: Moon

- 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?

viewtopic.php?f=41&t=472 - $A$ 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 $B$?

viewtopic.php?f=41&t=473 - $|x – 2| \leq 5$ and $y \leq |x – 4|$. Find the minimum and maximum values of $xy$.

viewtopic.php?f=41&t=474 - \[\begin{align*}a + b + c + d + e =& 1\\ a - 2b + c -2d + e =& 2\\ a + b -3c + d -3e =& 3\\ -4a + b + c - 4d + e =& 4\\ a -5b + c + d - 5e =& 5\end{align*}\]

উপরের সমীকরণগুলো থেকে $a$ এর মান বের কর।

Find $a$ from the above set of equations.

viewtopic.php?f=41&t=475 - 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 $x$.

viewtopic.php?f=41&t=476 - 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 $BD$.

viewtopic.php?f=41&t=477 - 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?

viewtopic.php?f=41&t=478 - 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?

viewtopic.php?f=41&t=479 - $x$ 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 $1000$?

viewtopic.php?f=41&t=480 - 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 $B$?

viewtopic.php?f=41&t=481

LaTeXed by: Moon