Divisional MO Higher Secondary 2010

Problem for Higher Secondary Group from Divisional Mathematical Olympiad will be solved here.
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Divisional MO Higher Secondary 2010

Unread post by BdMO » Tue Jan 18, 2011 1:49 pm

Dhaka Divisional Mathematical Olympiad 2010 : Higher Secondary

  1. 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 \[ \sum_{x=0}^{2010} \Phi(x)\]
  2. 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 $ABC$.
  3. 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.
  4. From $1$ to $1000$, how many integers are multiples of $3$ or $6$ but not of $5$?
  5. Find primes greater than $5$ satisfying the equation: \[11x^{36} - 21x^{10} + 26 x^{2} = 48\]
  6. What is the remainder when $2^{1024} + 5^{1024}$ is divided by $3$?
  7. 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?
  8. 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 $P$?
  9. 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?
  10. 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?
  11. $n$ 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)
  12. If $ \dfrac{x+2}{8}$ is an integer greater than $2$, find the remainder when $x$ is divided by $8$.
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