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Contest 2

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math contest problems competition high school
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  1. A ball is dropped from a height of 5 ft on a surface that causes it tobounce up 10% of the distance that it fell. If the probability that the ball ison its way up at any point in time is represented by the fraction ( m/n ) inlowest terms, where m and n are positive integers, what is m + n ?2. A circle on the coordinate plane is centered at the srcin and has radius r . What proportion of the circle is located below the line y = r/ 2?3. A recursive sequence has terms t n in the form a n b n and is defined such thatall even numbered terms are the reciprocal of the immediately preceding termand in all odd numbered terms, a n and b n are equal to a n − 2 +1 and b n − 2 +1respectively. If  t 1 = 23 what is:( a 2005 + a 2006 + a 2007 + a 2008 )( b 2005 + b 2006 + b 2007 + b 2008 )4. Isosceles triangle ABC  is inscribed in a circle with radius r 1 such thatside AB is a diameter of the circle. If a circle inscribed within triangle ABC  has radius r 2 , what is r 2 r 1 ?5. Let F  , T  , and W  be digits with (100 F  + 10 T  + W  )( F  + T  + W  ) = 2008.What is F  · T  · W  ?6. How many positive 3 digit integers satisfy the property that the tens digitis the geometric mean of the hundreds and units digits?7. Let E  and F  be the respective midpoints of  BC  and CD on unit square ABCD . If lines DE  and BF  intersect at G , what is the area of triangle BDG ?1  8. Let n represent the number of 16 digit palindromes divisible by 11. Whatis the remainder when n is divided by 11?9. What is the sum of the tens and units digits of 9 99 − 11 11 ?10. If tan θ > 1 and ∞  n =1 1tan n θ = sin θ +cos θ , find θ in the interval 0 ≤ θ ≤ π 11. The function f  ( x ) = ax 3 + bx 2 + cx + d has 3 positive real roots. If thesum of the roots of  f  ( x ) is 4, what is the largest possible value of  ca ?12. Two circles are located on the coordinate plane such that the distancebetween their centers is √ 2 units. If both circles have a radius of 1 unit,what is the area of the region enclosed by their intersection?13. If  r is equal to the sum of the first n positive integers and s is equal tothe sum of the first n positive integer squares, for how many values of  n inthe interval 0 < n < 2008 will sr be an integer?14. If cos2 θ = 14 , what is sin 4 θ + cos 4 θ ?15. What is smallest positive integer m for which the expression ( √ 3 + 3 i 2) m is an integer?16. If the function f  ( x ) = x 2 − bx + 9999 has positive integer roots, what isthe difference between the minimum and maximum possible values of  b ?2  17. If  f  ( x ) = ( x 2 + 4 x + 3) 2 + 4( x 2 + 4 x + 3) + 3, what is the sum of itscomplex roots?18. If  x 2 + y 2 + z 2 = 22 x +10 y +2 z , what is the maximum possible value of  x + y + z ?19. A right triangle has perimeter of length 7 and hypotenuse of length 3. If  θ is the largest non-right angle in the triangle, what is the value of cos θ ?20. The function f  ( x ) is a quadratic such that f  (0) = − f   (0). If  g ( x ) = f  ( x ) − f   ( x ) and the sum of the roots of  g ( x ) is 54 , find the sum of the roots of  f  ( x ).3
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