 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 \[ \sum_{x=0}^{2010} \Phi(x)\]
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  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 $n$, $S_n(x) \approx S_{n1}(x)$. In that case, express $\cos (S_n(x))$ as the nearest rational value.
viewtopic.php?f=42&t=377  From $1$ to $1000$, how many integers are multiples of $3$ or $6$ but not of $5$?
viewtopic.php?f=42&t=378  Find primes greater than $5$ satisfying the equation: \[11x^{36}  21x^{10} + 26 x^{2} = 48\]
viewtopic.php?f=42&t=379  What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
viewtopic.php?f=42&t=380  Boomboom joined Scout Jamboree. Every scout was said to handshake with each other. Some of them did not do. The total number of handshakes was $7$. Find the minimum number of handshakes which were not done?
viewtopic.php?f=42&t=381  If $N$ and $P$ are integers greater than $1$ and if $P$ is a factor of both $N+4$ and $N+12$, what are the values of $P$?
viewtopic.php?f=42&t=382  Out of the digits $1, 2, 3, 4, 5$, three are chosen to form numbers so that their digits are either in increasing or decreasing order. What is the total number of numbers formed? If one number is chosen from each category so that the sum of those two numbers is maximum, what is that sum?
viewtopic.php?f=42&t=383  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  $n$ points are taken on each side of a regular m gon. What is the total number of straight lines that can be drawn using all those points?(except the sides of $m$ gon)
viewtopic.php?f=42&t=385  If $ \dfrac{x+2}{8}$ is an integer greater than $2$, find the remainder when $x$ is divided by $8$.
viewtopic.php?f=42&t=386
Divisional MO Higher Secondary 2010
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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.
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.
Divisional MO Higher Secondary 2010
Dhaka Divisional Mathematical Olympiad 2010 : Higher Secondary
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