Math Problems

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These were taken from a math conference in Boston. Try them on for size...

  1. Two circles are externally tangent. A diameter of the larger circle is 6. A tangent from the center of the smaller circle to a point on the larger circle is 4. What is the area of the smaller circle?
  2. A powderman set a fuse for an explosion to take place in 10 seconds, then ran away at the rate of 30 feet per second. If sound travels at the rate of 1080 feet per second, how far, in feet, had the powderman run when he heard the blast?
  3. Sixteen recruits were standing in a long column, one behind the other. At a signal from the sergeant, the recruits standing in places 8 and 16 step forward into places 1 and 2 respectively, and the others step back. What is the least number of these maneuvers that must occur before the recruit originally in front will be in front once again?
  4. The lengths of the sides of a triangle are 4, 13, and 15. What is the value of the sine of the angle opposite the side whose length is 13?
  5. In a circle whose radius is 7 cm long, a point 3 cm from on endpoint of a chord is 5 cm from the center of the circle. How long is the chord, in cm?
  6. The vertices of equilateral triangle DEF lie on the sides of rectangle ABCD, with E on line segment AB and F on line segment BC. If tan FDC = 1/4, then for what real number k does tan ADE = k/47?
  7. What are all ordered triples of numbers (x,y,z) which simultaneously satisfy xz + yz = 13, xy + yz = 20, and xy + xz = 25?
  8. Altogether, 9 girls, 9 women, 14 Bostonians, 7 out-of-town grils, 5 children from Boston, 6 females from Boston, and 7 out-of-town males signed up to take a math contest. How many different people signed up for that math contest?
  9. A function f satisfies f(3) = 1 and f(3x) = x + f(3x-3) for every integer x > 1. Write the value of f(300) as an integer in standard form.
  10. A train normally goes from Here to There in 2 hours. Once, when the train was delayed 16 minutes, it made up for lost time on an 30 mile stretch of track by exceeding its normal rate of speed by 10 mph. How many miles does the train travel from Here to There?
  11. If log(10)2 = a, then, in terms of a, what is the value of log(10)25?
  12. Of all the points on the circle (x-6)(x-6) + (y-5)(y-5) = 25, what are the coordinates of the one that's nearest to (-2,11)?
  13. What is the larger of the two positive prime factors of 1,000,001?
  14. How long is the line segment whose endpoints have polar coordinates in degrees of (7,40) and (15,100)?
Answers coming soon...
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