Divisional MO Secondary 2009

Problem for Secondary Group from Divisional Mathematical Olympiad will be solved here.
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BdMO
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Joined: Tue Jan 18, 2011 1:31 pm

Divisional MO Secondary 2009

Unread post by BdMO » Fri Jan 21, 2011 6:27 pm

Dhaka Divisional Mathematical Olympiad 2009 : Secondary
  1. $x$ and $y$ are two digits and $[x][y]$ represents the number $10x+y$. If $[x][y]$ and $[y][x]$ are both primes and $[x][y]-[y][x]=[\frac{x-y}{2}][2(x+y)]$ find $x+y$.
    viewtopic.php?f=41&t=410
  2. If $(DBC)^2=BCABC$ find the value of $D$?
    viewtopic.php?f=41&t=411
  3. $ABC$ is a right angle triangle where $A$ is the right angle. $D$ is a point on $AC$ so
    that $AB=AD$. $E$ & $F$ bisect $BD$ & $AD$ respectively. $DH \bot BC$ & $DH=\sqrt{2}$. Find the $\angle DCH$ when $EF=1$.
    viewtopic.php?f=41&t=412
  4. \[( 1-\frac{1}{2^2} ) ( 1- \frac{1}{3^2} ) ( 1- \frac{1}{4^2} ) ( 1- \frac{1}{5^2} ) (1- \frac{1}{6^2} ) \cdots (1- \frac{1}{1000^2} ) =?\]
    viewtopic.php?f=41&t=413
  5. Sequence $(a_n) \; ( n \geq 0 )$is defined recursively by $a_0=3$, $a_n=2+a_0 \cdot a_1 \cdot a_2 \cdot a_3 \cdots a_{n-1}, n\geq 1$. Determine $a_{2009}$.
    viewtopic.php?f=41&t=414
  6. Determine the unit’s digit (one’s digit) of the sum of the expression: \[ (1!)^3 + (2!)^3 + (3!)^3 + \cdots + (13!)^3 + (14!)^3 + (15!)^3 \]
    viewtopic.php?f=41&t=415
  7. Among the increasingly ordered permutations of the digits $1,2, \cdots ,7$ find the $2009^{th}$ integer.
    viewtopic.php?f=41&t=416
  8. For triangle $ABC$ $ \angle C$ is $90^{\circ}$. $\angle BAC$ is $30^{\circ}$ & $AB$ is $1$cm. $D$ is a point within $ABC$ so that angle $\angle BDC$ is $90^{\circ}$ & $\angle ACD = \angle DBA$. $AB$ & $CD$ meets at $E$. Find $AE$.
    viewtopic.php?f=41&t=417
  9. In a chess competition there are $101$ players and each of them is fixture to play one match with each of the rest. But one player got sick and could not play any match. At most how many players could play all the matches?
    viewtopic.php?f=41&t=418
  10. Find three consecutive odd whole numbers such that the sum of their squares is a four digit number whose digits are all the same.
    viewtopic.php?f=41&t=419
  11. \[ ( 4^{502} + 2^{1004} ) ^2+(4^{502}-2^{1004} )^2-4^{ \frac{2009}{2} }=2^k\]
    Find $k$.
    viewtopic.php?f=41&t=420
  12. $ABCD$ is a $4 \times 4$ square. $E$ lies on $AB$; $AE=1$. $F$ lies on $AD$ & $AF=AE$. $EFG$ is a right angled triangle where $F$ is the right angle. Find the radius of the circumcirle of the triangle $EFG$.
    viewtopic.php?f=41&t=421
Credit: HandaramTheGreat
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