math Olympiad – 97 Practice Problems & Printable PDF

1. If 3x + 5 = 20, what is the value of x?

A. 5 B. 6 C. 7 D. 8

Solution

Correct: A

Subtract 5 from both sides: 3x = 15. Divide by 3: x = 5.

2. Simplify (2x - 3)(x + 4).

A. 2x² + 5x - 12 B. 2x² - 12 C. 2x² + 8x - 12 D. 2x² + x - 12

Solution

Correct: A

Use FOIL: 2x·x + 2x·4 - 3·x - 3·4 = 2x² + 8x - 3x - 12 = 2x² + 5x - 12.

3. Solve x² - 9x + 18 = 0.

A. x = 3 or 6 B. x = 2 or 9 C. x = -3 or -6 D. x = 1 or 18

Solution

Correct: A

Factor: (x - 3)(x - 6) = 0 ⇒ x = 3 or 6.

4. If f(x) = 4x - 7, find f(5).

A. 13 B. 20 C. 27 D. 33

Solution

Correct: A

Substitute: f(5) = 4·5 - 7 = 20 - 7 = 13.

5. What is the slope of the line 3x - 2y = 12?

A. 3/2 B. -3/2 C. 2/3 D. -2/3

Solution

Correct: A

Rearrange to y = mx + b: 2y = 3x - 12 ⇒ y = (3/2)x - 6 ⇒ slope = 3/2.

6. Evaluate 5a² - 2a when a = 3.

A. 39 B. 41 C. 43 D. 45

Solution

Correct: A

Substitute: 5·3² - 2·3 = 5·9 - 6 = 45 - 6 = 39.

7. Factor completely x² - 16.

A. (x - 4)(x + 4) B. (x - 4)² C. (x + 4)² D. (x - 8)(x + 8)

Solution

Correct: A

Difference of squares: x² - 16 = (x - 4)(x + 4).

8. Solve 2(x - 3) + 4 = 3x - 5.

A. x = 3 B. x = 5 C. x = 7 D. x = 9

Solution

Correct: A

Expand: 2x - 6 + 4 = 3x - 5 ⇒ 2x - 2 = 3x - 5 ⇒ -2 + 5 = x ⇒ x = 3.

9. If y = 2x + 1 and x = 4, what is y?

A. 9 B. 10 C. 11 D. 12

Solution

Correct: A

Substitute: y = 2·4 + 1 = 8 + 1 = 9.

10. Simplify (x²)³.

A. x⁵ B. x⁶ C. x⁸ D. x⁹

Solution

Correct: B

Power of a power: (x²)³ = x^(2·3) = x⁶.

11. Find the x-intercept of 2x + 3y = 12.

A. 6 B. 4 C. 3 D. 2

Solution

Correct: A

Set y = 0: 2x = 12 ⇒ x = 6.

12. Expand (x + 5)².

A. x² + 25 B. x² + 10x + 25 C. x² + 5x + 25 D. x² + 10x + 5

Solution

Correct: B

Perfect square: (x + 5)² = x² + 2·5x + 25 = x² + 10x + 25.

13. If 4x - 7 = 2x + 11, what is x?

A. 9 B. 8 C. 7 D. 6

Solution

Correct: A

Subtract 2x: 2x - 7 = 11. Add 7: 2x = 18 ⇒ x = 9.

14. Simplify (3x²y)(2xy³).

A. 6x³y⁴ B. 5x³y⁴ C. 6x²y³ D. 6x²y⁴

Solution

Correct: A

Multiply coefficients and add exponents: 3·2 x^(2+1) y^(1+3) = 6x³y⁴.

15. Solve x/3 + 2 = 5.

A. x = 9 B. x = 12 C. x = 15 D. x = 18

Solution

Correct: A

Subtract 2: x/3 = 3. Multiply by 3: x = 9.

16. What is the y-intercept of y = -3x + 7?

A. 7 B. -3 C. 3 D. -7

Solution

Correct: A

Slope-intercept form gives y-intercept as 7.

17. Factor x² + 7x + 12.

A. (x + 3)(x + 4) B. (x + 2)(x + 6) C. (x + 1)(x + 12) D. (x + 5)(x + 2)

Solution

Correct: A

Find factors of 12 that add to 7: 3 and 4 ⇒ (x + 3)(x + 4).

18. If 2x² = 50, what is the positive value of x?

A. 5 B. 10 C. 25 D. 50

Solution

Correct: A

Divide by 2: x² = 25 ⇒ x = 5 (positive root).

19. Simplify (x² - 4x)/(x - 4).

A. x B. x - 4 C. x + 4 D. x²

Solution

Correct: A

Factor numerator: x(x - 4)/(x - 4) = x (for x ≠ 4).

20. Solve 5 - 2x = 3x + 15.

A. x = -2 B. x = -3 C. x = -4 D. x = -5

Solution

Correct: A

Add 2x: 5 = 5x + 15. Subtract 15: -10 = 5x ⇒ x = -2.

21. Evaluate 2a + 3b when a = 4 and b = -2.

A. 2 B. 4 C. 6 D. 8

Solution

Correct: A

Substitute: 2·4 + 3·(-2) = 8 - 6 = 2.

22. What is the degree of the polynomial 3x³ + 2x² - 5x + 1?

A. 3 B. 2 C. 1 D. 0

Solution

Correct: A

The highest exponent is 3, so degree = 3.

23. If 3(x - 2) = 2(x + 1), what is x?

A. 8 B. 7 C. 6 D. 5

Solution

Correct: A

Expand: 3x - 6 = 2x + 2. Subtract 2x: x - 6 = 2 ⇒ x = 8.

24. Simplify (x + 3)(x - 3).

A. x² - 9 B. x² + 9 C. x² - 6x + 9 D. x² + 6x - 9

Solution

Correct: A

Difference of squares: (x + 3)(x - 3) = x² - 9.

25. Find the vertex of the parabola y = x² - 4x + 5.

A. (2, 1) B. (1, 2) C. (-2, 1) D. (2, -1)

Solution

Correct: A

x = -b/(2a) = 4/2 = 2. y = 2² - 4·2 + 5 = 4 - 8 + 5 = 1 ⇒ vertex (2, 1).

26. Solve 7x - 14 = 0.

A. x = 2 B. x = 7 C. x = 14 D. x = -2

Solution

Correct: A

Add 14: 7x = 14 ⇒ x = 2.

27. If g(x) = x² - 2x, find g(3).

A. 3 B. 6 C. 9 D. 12

Solution

Correct: A

Substitute: g(3) = 3² - 2·3 = 9 - 6 = 3.

28. Simplify (2x²)³.

A. 8x⁵ B. 6x⁶ C. 8x⁶ D. 6x⁵

Solution

Correct: C

Power of product: 2³(x²)³ = 8x⁶.

29. What is the sum of roots of x² - 5x + 6 = 0?

A. 5 B. 6 C. -5 D. -6

Solution

Correct: A

For ax² + bx + c = 0, sum of roots = -b/a = 5/1 = 5.

30. Solve 4x + 7 = 3x - 2.

A. x = -9 B. x = -5 C. x = 5 D. x = 9

Solution

Correct: A

Subtract 3x: x + 7 = -2. Subtract 7: x = -9.

31. Evaluate (x + 2)² when x = 3.

A. 25 B. 23 C. 21 D. 19

Solution

Correct: A

Substitute: (3 + 2)² = 5² = 25.

32. Factor 2x² - 8.

A. 2(x - 2)(x + 2) B. 2(x - 2)² C. 2(x + 2)² D. 2(x² - 4)

Solution

Correct: A

Factor out 2: 2(x² - 4) = 2(x - 2)(x + 2).

33. If 1/x = 2/5, what is x?

A. 5/2 B. 2/5 C. 10 D. 1/10

Solution

Correct: A

Cross-multiply: 5 = 2x ⇒ x = 5/2.

34. Simplify (x² + 3x + 2)/(x + 1).

A. x + 2 B. x + 1 C. x - 2 D. x - 1

Solution

Correct: A

Factor numerator: (x + 1)(x + 2)/(x + 1) = x + 2 (x ≠ -1).

35. Find the constant term in the expansion of (x + 4)².

A. 16 B. 8 C. 4 D. 2

Solution

Correct: A

(x + 4)² = x² + 8x + 16 ⇒ constant term = 16.

36. Solve x² = 49.

A. x = ±7 B. x = 7 only C. x = -7 only D. x = ±14

Solution

Correct: A

Take square root: x = ±√49 = ±7.

37. If 2x - 3y = 12 and y = 2, find x.

A. 9 B. 6 C. 3 D. 0

Solution

Correct: A

Substitute y = 2: 2x - 6 = 12 ⇒ 2x = 18 ⇒ x = 9.

38. Simplify (x³)(x⁴).

A. x⁷ B. x¹² C. x¹ D. x⁻¹

Solution

Correct: A

Product rule: x³·x⁴ = x^(3+4) = x⁷.

39. What is the axis of symmetry of y = x² - 6x + 8?

A. x = 3 B. x = -3 C. x = 6 D. x = -6

Solution

Correct: A

x = -b/(2a) = 6/2 = 3.

40. Evaluate 3x² - 2x + 1 for x = 2.

A. 9 B. 10 C. 11 D. 12

Solution

Correct: A

Substitute: 3·4 - 2·2 + 1 = 12 - 4 + 1 = 9.

41. Factor x² - x - 6.

A. (x - 3)(x + 2) B. (x + 3)(x - 2) C. (x - 6)(x + 1) D. (x + 6)(x - 1)

Solution

Correct: A

Find factors of -6 that add to -1: -3 and 2 ⇒ (x - 3)(x + 2).

42. Solve 3x + 4 = 2x + 10.

A. x = 6 B. x = 5 C. x = 4 D. x = 3

Solution

Correct: A

Subtract 2x: x + 4 = 10 ⇒ x = 6.

43. If h(x) = -x² + 4x, find the maximum value.

A. 4 B. 3 C. 2 D. 1

Solution

Correct: A

Vertex at x = -b/(2a) = -4/(-2) = 2. h(2) = -4 + 8 = 4.

44. Simplify (x² - 9)/(x - 3).

A. x + 3 B. x - 3 C. x² D. x

Solution

Correct: A

Factor numerator: (x - 3)(x + 3)/(x - 3) = x + 3 (x ≠ 3).

45. Find the y-value when x = 1 on the line 2x + 3y = 11.

A. 3 B. 4 C. 5 D. 6

Solution

Correct: A

Substitute x = 1: 2 + 3y = 11 ⇒ 3y = 9 ⇒ y = 3.

46. Expand (2x - 1)².

A. 4x² - 4x + 1 B. 4x² - 1 C. 4x² + 1 D. 4x² - 2x + 1

Solution

Correct: A

Perfect square: (2x)² - 2·2x·1 + 1² = 4x² - 4x + 1.

47. Solve 5x - 3 = 2x + 12.

A. x = 5 B. x = 6 C. x = 7 D. x = 8

Solution

Correct: A

Subtract 2x: 3x - 3 = 12. Add 3: 3x = 15 ⇒ x = 5.

48. Evaluate (x - 4)² when x = 1.

A. 9 B. -9 C. 16 D. -16

Solution

Correct: A

Substitute: (1 - 4)² = (-3)² = 9.

49. Factor 3x² + 6x.

A. 3x(x + 2) B. 3x(x - 2) C. x(3x + 6) D. 3(x² + 2x)

Solution

Correct: A

Factor out 3x: 3x(x + 2).

50. If 2x = 8, what is x²?

A. 16 B. 64 C. 4 D. 32

Solution

Correct: A

Divide by 2: x = 4 ⇒ x² = 16.

51. Simplify (x⁵)/(x²).

A. x³ B. x⁷ C. x¹⁰ D. x⁻³

Solution

Correct: A

Quotient rule: x^(5-2) = x³.

52. Find the product of roots of x² - 8x + 15 = 0.

A. 15 B. 8 C. -15 D. -8

Solution

Correct: A

For ax² + bx + c = 0, product of roots = c/a = 15/1 = 15.

53. Solve 9 - x = 2x + 3.

A. x = 2 B. x = 3 C. x = 4 D. x = 5

Solution

Correct: A

Add x: 9 = 3x + 3. Subtract 3: 6 = 3x ⇒ x = 2.

54. If f(x) = x² + 1, find f(-3).

A. 10 B. 9 C. 8 D. 7

Solution

Correct: A

Substitute: f(-3) = (-3)² + 1 = 9 + 1 = 10.

55. Simplify (x + 1)² - (x - 1)².

A. 4x B. 2x C. x D. 0

Solution

Correct: A

Expand: (x² + 2x + 1) - (x² - 2x + 1) = 4x.

56. Find the x-coordinate of the vertex of y = 2x² - 8x + 5.

A. 2 B. -2 C. 4 D. -4

Solution

Correct: A

x = -b/(2a) = 8/4 = 2.

57. Evaluate 2x + 3y when x = 5 and y = -1.

A. 7 B. 8 C. 9 D. 10

Solution

Correct: A

Substitute: 2·5 + 3·(-1) = 10 - 3 = 7.

58. Factor x² + 2x - 15.

A. (x + 5)(x - 3) B. (x - 5)(x + 3) C. (x + 1)(x - 15) D. (x - 1)(x + 15)

Solution

Correct: A

Find factors of -15 that add to 2: 5 and -3 ⇒ (x + 5)(x - 3).

59. Solve x/2 - 3 = 1.

A. x = 8 B. x = 6 C. x = 4 D. x = 2

Solution

Correct: A

Add 3: x/2 = 4. Multiply by 2: x = 8.

60. If 3x - 2y = 7 and x = 3, find y.

A. 1 B. 2 C. 3 D. 4

Solution

Correct: A

Substitute x = 3: 9 - 2y = 7 ⇒ -2y = -2 ⇒ y = 1.

61. Simplify (2x)³.

A. 8x³ B. 6x³ C. 2x³ D. 8x

Solution

Correct: A

Power of product: 2³x³ = 8x³.

62. What is the discriminant of x² - 4x + 4 = 0?

A. 0 B. 16 C. -16 D. 4

Solution

Correct: A

Discriminant = b² - 4ac = (-4)² - 4·1·4 = 16 - 16 = 0.

63. Solve 4x + 5 = 2x + 13.

A. x = 4 B. x = 3 C. x = 2 D. x = 1

Solution

Correct: A

Subtract 2x: 2x + 5 = 13. Subtract 5: 2x = 8 ⇒ x = 4.

64. Evaluate x² - 5x + 6 for x = 4.

A. 2 B. 3 C. 4 D. 5

Solution

Correct: A

Substitute: 16 - 20 + 6 = 2.

65. Factor 5x² - 5.

A. 5(x - 1)(x + 1) B. 5(x - 1)² C. 5(x + 1)² D. 5x(x - 1)

Solution

Correct: A

Factor out 5: 5(x² - 1) = 5(x - 1)(x + 1).

66. If 1/(x + 2) = 1/5, what is x?

A. 3 B. 2 C. 1 D. 0

Solution

Correct: A

Cross-multiply: x + 2 = 5 ⇒ x = 3.

67. Simplify (x² - 4x + 4)/(x - 2).

A. x - 2 B. x + 2 C. x² D. x

Solution

Correct: A

Factor numerator: (x - 2)²/(x - 2) = x - 2 (x ≠ 2).

68. Find the slope of the line through (2, 3) and (4, 7).

A. 2 B. 1 C. 3 D. 4

Solution

Correct: A

Slope = (7 - 3)/(4 - 2) = 4/2 = 2.

69. Expand (x - 5)(x + 2).

A. x² - 3x - 10 B. x² + 3x - 10 C. x² - 7x - 10 D. x² + 7x - 10

Solution

Correct: A

FOIL: x² + 2x - 5x - 10 = x² - 3x - 10.

70. Solve 6x - 7 = 3x + 8.

A. x = 5 B. x = 6 C. x = 7 D. x = 8

Solution

Correct: A

Subtract 3x: 3x - 7 = 8. Add 7: 3x = 15 ⇒ x = 5.

71. If p(x) = x³ - x, find p(2).

A. 6 B. 7 C. 8 D. 9

Solution

Correct: A

Substitute: 8 - 2 = 6.

72. Simplify (x⁻²)(x⁵).

A. x³ B. x⁻¹⁰ C. x⁷ D. x⁻³

Solution

Correct: A

Product rule: x^(-2+5) = x³.

73. Find the x-intercept of y = 3x - 9.

A. 3 B. -3 C. 9 D. -9

Solution

Correct: A

Set y = 0: 3x = 9 ⇒ x = 3.

74. Evaluate (x + 1)/(x - 1) when x = 3.

A. 2 B. 1 C. 3 D. 4

Solution

Correct: A

Substitute: (3 + 1)/(3 - 1) = 4/2 = 2.

75. Factor x² - 10x + 25.

A. (x - 5)² B. (x + 5)² C. (x - 10)(x + 2) D. (x - 25)(x + 1)

Solution

Correct: A

Perfect square: x² - 10x + 25 = (x - 5)².

76. Solve 2x² - 8 = 0.

A. x = ±2 B. x = ±4 C. x = ±8 D. x = ±16

Solution

Correct: A

Add 8: 2x² = 8 ⇒ x² = 4 ⇒ x = ±2.

77. If 4x - 3y = 5 and y = 1, find x.

A. 2 B. 3 C. 4 D. 5

Solution

Correct: A

Substitute y = 1: 4x - 3 = 5 ⇒ 4x = 8 ⇒ x = 2.

78. Simplify (x²y³)².

A. x⁴y⁶ B. x²y⁵ C. x⁴y⁵ D. x²y⁶

Solution

Correct: A

Power of product: x^(2·2) y^(3·2) = x⁴y⁶.

79. What is the minimum value of y = x² - 2x + 4?

A. 3 B. 4 C. 2 D. 1

Solution

Correct: A

Vertex at x = 1: y = 1 - 2 + 4 = 3.

80. Solve 3(x + 2) = 2(x + 5).

A. x = 4 B. x = 3 C. x = 2 D. x = 1

Solution

Correct: A

Expand: 3x + 6 = 2x + 10. Subtract 2x: x + 6 = 10 ⇒ x = 4.

81. Evaluate 2x² - x when x = -1.

A. 3 B. 2 C. 1 D. 0

Solution

Correct: A

Substitute: 2(-1)² - (-1) = 2 + 1 = 3.

82. Factor 4x² - 1.

A. (2x - 1)(2x + 1) B. (2x - 1)² C. (2x + 1)² D. (4x - 1)(x + 1)

Solution

Correct: A

Difference of squares: (2x)² - 1² = (2x - 1)(2x + 1).

83. If 2x + 3 = 11, what is x²?

A. 16 B. 25 C. 36 D. 49

Solution

Correct: A

Subtract 3: 2x = 8 ⇒ x = 4 ⇒ x² = 16.

84. Simplify (x³ - x)/(x² - 1).

A. x B. x + 1 C. x - 1 D. x²

Solution

Correct: A

Factor: x(x² - 1)/(x² - 1) = x (x² ≠ 1).

85. Find the equation of the line with slope 2 through (1, 3).

A. y = 2x + 1 B. y = 2x - 1 C. y = 2x + 3 D. y = 2x - 3

Solution

Correct: A

Point-slope: y - 3 = 2(x - 1) ⇒ y = 2x + 1.

86. Expand (3x - 2)(x + 4).

A. 3x² + 10x - 8 B. 3x² - 10x - 8 C. 3x² + 10x + 8 D. 3x² - 10x + 8

Solution

Correct: A

FOIL: 3x² + 12x - 2x - 8 = 3x² + 10x - 8.

87. Solve 7x + 1 = 3x + 17.

A. x = 4 B. x = 5 C. x = 6 D. x = 7

Solution

Correct: A

Subtract 3x: 4x + 1 = 17. Subtract 1: 4x = 16 ⇒ x = 4.

88. If q(x) = x² - 4, find q(0).

A. -4 B. 0 C. 4 D. 2

Solution

Correct: A

Substitute: 0 - 4 = -4.

89. Simplify (x⁻³)(x³).

A. 1 B. x⁶ C. x⁻⁹ D. x

Solution

Correct: A

Product rule: x^(-3+3) = x⁰ = 1.

90. Find the y-intercept of 2x - 5y = 10.

A. -2 B. 2 C. 5 D. -5

Solution

Correct: A

Set x = 0: -5y = 10 ⇒ y = -2.

91. Evaluate (x² - 1)/(x - 1) when x = 0.

A. 1 B. -1 C. 0 D. 2

Solution

Correct: A

Substitute x = 0: (0 - 1)/(0 - 1) = (-1)/(-1) = 1.

92. Factor x² + 8x + 16.

A. (x + 4)² B. (x - 4)² C. (x + 8)(x + 2) D. (x + 16)(x + 1)

Solution

Correct: A

Perfect square: x² + 8x + 16 = (x + 4)².

93. Solve x² + 9 = 25.

A. x = ±4 B. x = ±3 C. x = ±5 D. x = ±16

Solution

Correct: A

Subtract 9: x² = 16 ⇒ x = ±4.

94. If 5x - 2y = 8 and y = 4, find x.

A. 16/5 B. 8/5 C. 4/5 D. 2/5

Solution

Correct: A

Substitute y = 4: 5x - 8 = 8 ⇒ 5x = 16 ⇒ x = 16/5.

95. Simplify (2x⁻²)³.

A. 8x⁻⁶ B. 6x⁻⁵ C. 8x⁻⁵ D. 6x⁻⁶

Solution

Correct: A

Power of product: 2³x^(-2·3) = 8x⁻⁶.

96. What is the axis of symmetry of y = -x² + 4x - 3?

A. x = 2 B. x = -2 C. x = 4 D. x = -4

Solution

Correct: A

x = -b/(2a) = -4/(-2) = 2.

97. Solve 2(x - 1) = x + 3.

A. x = 5 B. x = 4 C. x = 3 D. x = 2

Solution

Correct: A

Expand: 2x - 2 = x + 3. Subtract x: x - 2 = 3 ⇒ x = 5.