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Divisional MO Secondary 2010

Posted: Fri Jan 21, 2011 7:11 pm
by BdMO
Dhaka Divisional Mathematical Olympiad 2010 : Secondary
  1. Sum of two numbers is $2$ and their product is $3$. Find the sum of the reciprocal of the numbers.
    [Hint: Reciprocal of $x$ is $\frac 1x$]
  2. What is the remainder when $2^{1024} + 5^{1024} +1$ is divided by $9$?
  3. If $N$ and $P$ are integers greater than $1$ and if $P$ is a factor of both $N+4$ and $N+14$, what are the values of $P$?
    Similar: viewtopic.php?f=42&t=382
  4. In how many ways can four different numbers be arranged so that they are not arranged in increasing or decreasing order?
  5. If $x$ is very very very small $\sin x \approx x$. An operator $S_n$ is defined such that $ S_n(x)= \sin \sin \sin \cdots \sin x$ (a total of $n$ $\sin$ operators are included here). For sufficiently large $n$, $S_n(x) \approx S_{n-1}(x)$. In that case, express $\cos (S_n(x))$ as the nearest rational value.
  6. For how many prime numbers $N$ for which $N+1$ is a perfect square.
  7. In the figure above $AD = 4, AB = 3$ and $CD = 9$. What is the area of triangle $\triangle AEC$?
  8. Assume, $\Phi : A \to A, A=\{0,1,2,\cdots\}$ is a function, which is defined as,
    \[\Phi(x) = \begin{cases}
    0 \quad \text{if } x \text{ is a prime}\\
    \Phi(x - 1) \quad \text{if } x \text{ is not a prime} \end{cases} \]
    Find \[ \sum_{x=2}^{2010} \Phi(x)\]
  9. As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$.
  10. Three points are taken on each of any three sides of a square. What is the total number of points taken on the other side given that a total of $45$ distinct straight lines can be drawn using these points?
  11. The diagram above shows the various paths along which Mr. Ibrahim Khalilullah Nobi can travel from point Teknaf, where it is released, to point Tetulia, where it is rewarded with a food pellet. How many different paths from Teknaf to Tetulia can Nobi take if it goes directly from Teknaf to Tetulia without retracting any point along a path?
  12. From $1$ to $300$, how many integers are multiples of $2$ or $3$ but not of $8$?
    similar: viewtopic.php?f=42&t=378

Re: Divisional MO Secondary 2010

Posted: Sat Jan 17, 2015 4:26 pm
by Ragib Farhat Hasan
Answer to ques. no.4 is 22.

Re: Divisional MO Secondary 2010

Posted: Mon Sep 03, 2018 1:42 am
by Ragib Farhat Hasan
Answer to ques. no. 12 is 201.