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### Divisional MO Secondary 2010

Posted: Fri Jan 21, 2011 7:11 pm
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 $\frac 1x$]
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2. What is the remainder when $2^{1024} + 5^{1024} +1$ is divided by $9$?
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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
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4. In how many ways can four different numbers be arranged so that they are not arranged in increasing or decreasing order?
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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.
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6. For how many prime numbers $N$ for which $N+1$ is a perfect square.
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7. In the figure above $AD = 4, AB = 3$ and $CD = 9$. What is the area of triangle $\triangle AEC$?
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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 $\sum_{x=2}^{2010} \Phi(x)$
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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 $ABC$.
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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?
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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?
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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
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### Re: Divisional MO Secondary 2010

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

### Re: Divisional MO Secondary 2010

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