Answers to 1999 Lehigh/AT&T High School Math Contest

The number in brackets is the number of people out of 175 who answered the question correctly. If you wish to receive complete solutions, send an e-mail to Don Davis (dmd1@lehigh.edu).
  1. 1/2. [144]
  2. 12/5. [75]
  3. 54. [160]
  4. 52. [153]
  5. 30. [81]
  6. -1. [63]
  7. 1606/4995. [36]
  8. -1. [84]
  9. 6. [55]
  10. 54. [78]
  11. 0.6. [68]
  12. 86. (number of odd integers less or equal 171.) [38]
  13. 48. [106]
  14. 3 sqrt(13). (use law of cosines or coordinate geometry) [49]
  15. (28,21). [49]
  16. 112. (28 pairs of circles, each with four tangents.) [19]
  17. (2, 2 sqrt(2)). [43]
  18. 6. (exponent of 2 is 15, exponent of 3 is 6) [30]
  19. (pi - 3 sqrt(3)/4)/3. ((area of circle minus area of triangle of side sqrt(3))/3) [37]
  20. 1. (equate coefficients of various powers of x) [39]
  21. 5. (1,2,-1,-1+sqrt2, -1-sqrt2) [9]
  22. 8/49. (Square the expansion of 1/(1-x). Get 1/(8 7/8 7/8)). [11]
  23. 7. [55]
  24. -31.5. (Write two eqs for a and d. Get d=2) [14]
  25. 4. (It is where x^2 + 8x + 12 has a minimum) [34]
  26. -2i, -2, -3. [55]
  27. 0. (Use that number is congruent mod 9 to its digital sum) [17]
  28. 31. (1+2(4+6+4+1)) [45]
  29. 50. (He saved 10 minutes of driving in each direction) [29]
  30. 180. (242 circles added, 62 of which touch C) [19]
  31. 35/46. (12 times 23 ways to choose the second and third persons. # of ways with none adjacent is 10 times 21) [7]
  32. 3. ((4,3),(8,4),(20,5)) [22]
  33. (8,27) (Write equations involving x^(2/3) and y^(2/3). Divide the equations.) [10]
  34. 35. (Expand (7-1)^83 + (7+1)^83.) [4]
  35. x1 x2 = 1. (Get a right triangle with base x1-x2, height 2, and hypotenuse x1+x2) [7]
  36. 2. (It is sum of powers of .0005. Only the 9th, 10th, and 11th powers contribute to the 36th decimal place.) [6]
  37. 11/26. (It is 9/26 + 4/13 1/4) [3]
  38. (1,1,sqrt3) (Intersection of sphere with triangle is inscribed circle of the triangle) [4]
  39. 3. (3025, 2025, 9801. To get this, start by considering 100a+b = (a+b)^2) [10]
  40. 12/13. (Use Hero's formula. Get 3 = sqrt((3-r)/(3-3r)).) [2]