Divisional MO Higher Secondary 2011

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
Forum rules
Please don't post problems (by starting a topic) in the "Higher Secondary: Solved" forum. This forum is only for showcasing the problems for the convenience of the users. You can post the problems in the main Divisional Math Olympiad forum. Later we shall move that topic with proper formatting, and post in the resource section.
Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

Divisional MO Higher Secondary 2011

Unread post by BdMO » 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?

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 $B$.

Question 3:
If $A = \{\Phi\}$, $A \cup B = P(A)$ and $A \cap B = \Phi$, find $B$. $\Phi$ represents empty set.

Question 4:
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.

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 $x$.

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?

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.

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 $k + l$.

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 $f(1952) = -1$.

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$.

Problem set (pdf): viewtopic.php?f=8&t=468

LaTeXed by: Zzzz

Posts: 134
Joined: Tue Jan 18, 2011 1:31 pm

Re: Divisional MO Higher Secondary 2011

Unread post by BdMO » Wed Feb 02, 2011 9:10 pm

Rangpur Divisional Mathematical Olympiad 2011 : Higher Secondary

Problem 1:
After tossing some coins, it is observed that the probability of obtaining $10$ heads and $15$ tails is the same as the probability of obtaining $15$ heads and $10$ tails. Find the least number of coins tossed.

Problem 2:
If $9^{x+18} = 16^x$ and $b^x = 9^9$, what is the value of $b$?

Problem 3:
If $-3<f<4$ and $-2<g<1$, then what is the range of possible values of $fg$?

Problem 4:
\[(1)_2 + (.1)_2+ (.01)_2+...=?\]
$(A)_b$ signifies that the number $A$ is represented in base $b$.

Problem 5:
The equation $x^3 +3xy + y^3 = 1$ is solved in nonnegative integers. Find the possible values of $x-y$.

Problem 6:
In the figure $AB = 12$ is the diameter of the circle. $MN||AB$ and $\angle BAN = 15^{\circ}$. Find the length of the arc $MN$.

Problem 7:
If $A + B =1$, $B + C = 2$, $C + D = 3$, ..., $X + Y = 25$, $Y + Z = 26$, find $A – Z$.

Problem 8:
At a conference, flags of $5$ 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 9:
Sum of three positive integers is $2^{2011} + 1$, and the product of two of them is $2^{2011}$. How many values can the third integer take?

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 $a-b \sqrt{3}-\frac{c\cdot \pi}{4}$ where $a$, $b$, $c$ are integers. Find $a+b+c$.

LaTeXed by: HandaramTheGreat
(92.08 KiB) Downloaded 315 times

Post Reply