### Divisional MO Secondary 2010

Posted:

**Fri Jan 21, 2011 7:11 pm**Dhaka Divisional Mathematical Olympiad 2010 : Secondary

- 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 $\frac 1x$]

viewtopic.php?f=41&t=423 - What is the remainder when $2^{1024} + 5^{1024} +1$ is divided by $9$?

viewtopic.php?f=41&t=424 - 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 $P$?

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viewtopic.php?f=41&t=425 - In how many ways can four different numbers be arranged so that they are not arranged in increasing or decreasing order?

viewtopic.php?f=41&t=426 - 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.

viewtopic.php?f=42&t=377 - For how many prime numbers $N$ for which $N+1$ is a perfect square.

viewtopic.php?f=41&t=427 - In the figure above $AD = 4, AB = 3$ and $CD = 9$. What is the area of triangle $\triangle AEC$?

viewtopic.php?f=41&t=428 - 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 \[ \sum_{x=2}^{2010} \Phi(x)\]

viewtopic.php?f=42&t=374 - 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 $ABC$.

viewtopic.php?f=42&t=376 - 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?

viewtopic.php?f=41&t=429 - 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?

viewtopic.php?f=42&t=384 - From $1$ to $300$, how many integers are multiples of $2$ or $3$ but not of $8$?

similar: viewtopic.php?f=42&t=378

viewtopic.php?f=41&t=430