Divisional MO Higher Secondary 2011
Posted: Fri Jan 28, 2011 10:36 pm
Dhaka Divisional Mathematical Olympiad 2011 : Higher Secondary
Question 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?
viewtopic.php?f=41&t=472
Question 2:
$A$ is three digit number all of whose digits are different. $B$ is a three digit number all of whose digits are same. Find the minimum difference between $A$ and $B$.
viewtopic.php?f=42&t=483
Question 3:
If $A = \{\Phi\}$, $A \cup B = P(A)$ and $A \cap B = \Phi$, find $B$. $\Phi$ represents empty set.
viewtopic.php?f=42&t=484
Question 4:
\[\begin{align*}
a + b + c + d + e =& 1\\ a - b + c + d + e =& 2 \\ a + b - c + d + e =& 3\\ a + b + c - d + e =& 4\\ a + b + c + d - e =& 5\end{align*} \]
Find $a$ from the above set of equations.
viewtopic.php?f=42&t=485
Question 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 $x$.
viewtopic.php?f=41&t=476
Question 6:
One circle is touching another circle internally. The inner circle is also tangent to a diameter of the outer circle which makes an angle of $60 {\circ}$ with the common tangent of the circles. Radius of the outer circle is 6, what is the radius of the inner circle?
viewtopic.php?f=42&t=486
Question 7:
In a game Arjun has to throw a bow towards a target and then Karna has to throw a bow toeards the target. One who hits the target first wins. The game continues with Karna trying after Arjun and Arjun trying after Karna until someone wins. The probability of Arjun hitting the target with a single shot is $\frac{2}{5}$ and the probability that Arjun will win the game is the same as that of Karna. What is the probability of Karna hitting the target with a single shot.
viewtopic.php?f=42&t=487
Question 8:
$N$ represents a nine digit number each of whose digits are different and nonzero. The number formed by its leftmost three digits is divisible by $3$ and the number formed by its leftmost six digits is divisible by $6$. It is found that $N$ can have $2^k3^l$ different values. Find the value of $k + l$.
viewtopic.php?f=42&t=488
Question 9:
Consider a function $f: \mathbb N$ $\to$ $\mathbb Z$ is so defined that the following relations hold:
\[f(2^n)=f(2^{n+2})\text{ and } f\left (\sum_{n\in X}^{} 2^n\right)=\sum_{n\in X}^{} f(2^n)\]
where $X$ is some finite subset of $\mathbb{N} \cup \{0\}$.
Find $f(1971)$ if it is known that $f(2011) = 1$ and $f(1952) = -1$.
viewtopic.php?f=42&t=489
Question 10:
A point $P$ is chosen inside a right angled triangle $ABC$, perpendicular lines $PS$, $PQ$ and $PR$ are drawn from $P$ on $AB, BC$ and $AC$. $PR = 1, PQ = 2$ and $PS = 3$ and $\angle RPS = 150^{\circ} $. The length of $ AB$ can be written in the form $x \sqrt y +z$ where $x, y, z$ are integers. Find $ x + y + z$.
viewtopic.php?f=42&t=490
Problem set (pdf): viewtopic.php?f=8&t=468
LaTeXed by: Zzzz
Question 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?
viewtopic.php?f=41&t=472
Question 2:
$A$ is three digit number all of whose digits are different. $B$ is a three digit number all of whose digits are same. Find the minimum difference between $A$ and $B$.
viewtopic.php?f=42&t=483
Question 3:
If $A = \{\Phi\}$, $A \cup B = P(A)$ and $A \cap B = \Phi$, find $B$. $\Phi$ represents empty set.
viewtopic.php?f=42&t=484
Question 4:
\[\begin{align*}
a + b + c + d + e =& 1\\ a - b + c + d + e =& 2 \\ a + b - c + d + e =& 3\\ a + b + c - d + e =& 4\\ a + b + c + d - e =& 5\end{align*} \]
Find $a$ from the above set of equations.
viewtopic.php?f=42&t=485
Question 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 $x$.
viewtopic.php?f=41&t=476
Question 6:
One circle is touching another circle internally. The inner circle is also tangent to a diameter of the outer circle which makes an angle of $60 {\circ}$ with the common tangent of the circles. Radius of the outer circle is 6, what is the radius of the inner circle?
viewtopic.php?f=42&t=486
Question 7:
In a game Arjun has to throw a bow towards a target and then Karna has to throw a bow toeards the target. One who hits the target first wins. The game continues with Karna trying after Arjun and Arjun trying after Karna until someone wins. The probability of Arjun hitting the target with a single shot is $\frac{2}{5}$ and the probability that Arjun will win the game is the same as that of Karna. What is the probability of Karna hitting the target with a single shot.
viewtopic.php?f=42&t=487
Question 8:
$N$ represents a nine digit number each of whose digits are different and nonzero. The number formed by its leftmost three digits is divisible by $3$ and the number formed by its leftmost six digits is divisible by $6$. It is found that $N$ can have $2^k3^l$ different values. Find the value of $k + l$.
viewtopic.php?f=42&t=488
Question 9:
Consider a function $f: \mathbb N$ $\to$ $\mathbb Z$ is so defined that the following relations hold:
\[f(2^n)=f(2^{n+2})\text{ and } f\left (\sum_{n\in X}^{} 2^n\right)=\sum_{n\in X}^{} f(2^n)\]
where $X$ is some finite subset of $\mathbb{N} \cup \{0\}$.
Find $f(1971)$ if it is known that $f(2011) = 1$ and $f(1952) = -1$.
viewtopic.php?f=42&t=489
Question 10:
A point $P$ is chosen inside a right angled triangle $ABC$, perpendicular lines $PS$, $PQ$ and $PR$ are drawn from $P$ on $AB, BC$ and $AC$. $PR = 1, PQ = 2$ and $PS = 3$ and $\angle RPS = 150^{\circ} $. The length of $ AB$ can be written in the form $x \sqrt y +z$ where $x, y, z$ are integers. Find $ x + y + z$.
viewtopic.php?f=42&t=490
Problem set (pdf): viewtopic.php?f=8&t=468
LaTeXed by: Zzzz