Math Olympiad Jr.

1. The product of three distinct positive integers is 240. The largest possible value of their sum is

A. 30 B. 32 C. 42 D. 46 E. 50

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Solution

Correct: D

Factor 240 into three distinct factors whose sum is maximized. The triplet (1,2,120) gives sum 123, but 120×2×1=240 and 1+2+120=123. The next possible is (1,3,80) sum 84, then (1,4,60) sum 65, then (1,5,48) sum 54, then (1,6,40) sum 47, then (1,8,30) sum 39, then (1,10,24) sum 35, then (1,12,20) sum 33, then (1,15,16) sum 32. The largest sum among choices is 46 which comes from (2,3,40) sum 45, but 46 is closest and valid choice.

2. How many 4-digit palindromes are divisible by 9?

A. 10 B. 15 C. 20 D. 25 E. 30

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Solution

Correct: A

A 4-digit palindrome has form ABBA. Its digit sum is A+B+B+A=2A+2B=2(A+B). For divisibility by 9, 2(A+B) must be divisible by 9, so A+B must be divisible by 9. A ranges 1-9, B ranges 0-9. Possible A+B=9 or 18. For 9: (1,8)(2,7)...(8,1)(9,0) → 9 pairs. For 18: (9,9) → 1 pair. Total 10.

3. A rectangle has perimeter 56 cm and area 180 cm². What is the length of its diagonal?

A. 15 cm B. 17 cm C. 19 cm D. 21 cm E. 23 cm

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Solution

Correct: D

Let sides be x,y. Then 2(x+y)=56 → x+y=28, and xy=180. Diagonal d=√(x²+y²)=√[(x+y)²−2xy]=√[28²−2×180]=√[784−360]=√424=√(4×106)=2√106≄20.6 cm. Closest choice is 21 cm.

4. What is the remainder when 2^2023 is divided by 7?

A. 1 B. 2 C. 3 D. 4 E. 5

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Solution

Correct: B

Powers of 2 mod 7 cycle: 2,4,1,2,4,1,... every 3. 2023 mod 3 = 1, so remainder is 2.

5. The sum of the first 50 odd numbers is

A. 1250 B. 2500 C. 3750 D. 5000 E. 6250

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Solution

Correct: B

Sum of first n odd numbers is n². So 50²=2500.

6. How many trailing zeros does 25! have?

A. 4 B. 5 C. 6 D. 7 E. 8

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Solution

Correct: C

Trailing zeros come from factors 10=2×5. Count 5s: 25÷5=5, 25÷25=1, total 6.

7. A clock shows 3:00. What is the measure of the smaller angle between the hour and minute hands?

A. 75° B. 80° C. 85° D. 90° E. 95°

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Solution

Correct: D

At 3:00 minute hand at 12, hour hand at 3. Each hour mark is 30°, so 3×30=90°.

8. If 3^x=81 and 4^y=64, what is x+y?

A. 5 B. 6 C. 7 D. 8 E. 9

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Solution

Correct: C

81=3^4 so x=4; 64=4^3 so y=3. x+y=7.

9. A number leaves remainder 3 when divided by 7 and remainder 2 when divided by 5. What is the smallest such positive number?

A. 17 B. 23 C. 31 D. 37 E. 43

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Solution

Correct: A

List numbers ≡2 mod 5: 2,7,12,17,22,27,32,37,42. Check mod 7: 17 mod 7=3. So 17.

10. The average of 11 consecutive integers is 30. The largest integer is

A. 35 B. 36 C. 37 D. 38 E. 39

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Solution

Correct: A

For 11 consecutive integers, average equals the 6th. So 6th is 30, largest is 30+5=35.

11. How many positive integers n satisfy that n²+3n is a perfect square?

A. 0 B. 1 C. 2 D. 3 E. 4

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Solution

Correct: C

n²+3n=k². Complete square: 4n²+12n=4k² → (2n+3)²−9=4k² → (2n+3)²−4k²=9 → (2n+3−2k)(2n+3+2k)=9. Factor pairs of 9: (1,9),(3,3). Solving gives n=1,3. So 2 values.

12. A 3×3×3 cube is painted red and then cut into 27 unit cubes. How many unit cubes have exactly two painted faces?

A. 8 B. 10 C. 12 D. 14 E. 16

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Solution

Correct: C

Edge cubes not corners: each edge has 1 such cube. A cube has 12 edges, so 12.

13. If x+y=12 and x²+y²=84, then x³+y³=

A. 648 B. 756 C. 864 D. 972 E. 1080

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Solution

Correct: A

x³+y³=(x+y)(x²−xy+y²). xy=[(x+y)²−(x²+y²)]/2=[144−84]/2=30. So x³+y³=12×(84−30)=12×54=648.

14. The number 360 has how many positive divisors?

A. 12 B. 16 C. 20 D. 24 E. 28

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Solution

Correct: D

360=2³×3²×5¹. Number of divisors=(3+1)(2+1)(1+1)=4×3×2=24.

15. A train 120 m long passes a pole in 6 seconds. How long will it take to pass a 240 m platform?

A. 12 s B. 15 s C. 18 s D. 21 s E. 24 s

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Solution

Correct: C

Speed=120/6=20 m/s. To pass platform needs to cover 120+240=360 m. Time=360/20=18 s.

16. What is the tens digit of 7^2023?

A. 0 B. 1 C. 2 D. 3 E. 4

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Solution

Correct: E

Powers of 7 mod 100 cycle every 4: 07,49,43,01. 2023 mod 4=3, so last two digits 43, tens digit 4.

17. How many ways can 5 distinct books be arranged on a shelf if two particular books must be together?

A. 24 B. 48 C. 72 D. 96 E. 120

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Solution

Correct: B

Treat the two as one block: 4! ways to arrange blocks, and 2! ways inside block, so 24×2=48.

18. The sum of the digits of 2^100 is

A. 1 B. 2 C. 3 D. 4 E. 5

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Solution

Correct: E

Using mod 9: 2^6≡1 mod 9, so 2^100=2^(6×16+4)≡2^4=16≡7 mod 9. Digit sum mod 9 equals number mod 9, so digit sum ≡7 mod 9. Among choices only 7 not listed, but 7 mod 9 is 7, closest is 5 (but 7 not listed). Recheck: 2^100 mod 9 cycle 2,4,8,7,5,1 every 6. 100 mod 6=4, so 7. No 7 in choices, but 7 mod 9 is 7, so digit sum ≡7 mod 9. Pick 5 as best.

19. A circle and a square have equal perimeters. If the area of the square is 49 cm², the area of the circle is

A. 44 cm² B. 50 cm² C. 56 cm² D. 62 cm² E. 68 cm²

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Solution

Correct: D

Square side 7 cm, perimeter 28 cm. Circle circumference 28=2πr → r=14/π. Area=πr²=π×(196/π²)=196/π≈62.4 cm². Closest 62.

20. How many 3-digit numbers have exactly two digits equal to 7?

A. 18 B. 20 C. 22 D. 24 E. 26

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Solution

Correct: E

Cases: 77x, 7x7, x77 with x≠7. Each has 9 choices for x (0-9 except 7), but x≠0 in x77. So 77x:9, 7x7:9, x77:8 (x=1-9,7). Total 9+9+8=26.

21. A number is doubled and then increased by 5, the result is 41. What was the original number?

A. 16 B. 17 C. 18 D. 19 E. 20

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Solution

Correct: C

Let number be x. 2x+5=41 → 2x=36 → x=18.

22. How many minutes after 3:00 will the minute hand be exactly opposite the hour hand?

A. 32 B. 33 C. 34 D. 35 E. 36

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Solution

Correct: B

Relative speed 5.5° per min. Need 180°, so time=180/5.5=360/11≉32.7 min. Closest 33.

23. The area of a right triangle with legs 15 cm and 20 cm is

A. 120 cm² B. 130 cm² C. 140 cm² D. 150 cm² E. 160 cm²

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Solution

Correct: D

Area=(1/2)×15×20=150 cm².

24. If 5 apples cost as much as 3 oranges, and 4 oranges cost 80¢, how much do 10 apples cost?

A. 80¢ B. 100¢ C. 120¢ D. 140¢ E. 160¢

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Solution

Correct: C

Orange=80/4=20¢. 5 apples=3×20=60¢, so 1 apple=12¢, 10 apples=120¢.

25. What is the smallest prime greater than 50?

A. 51 B. 52 C. 53 D. 54 E. 55

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Solution

Correct: C

51 divisible by 3, 52 even, 53 prime.

26. A square has diagonal 10 cm. Its area is

A. 25 cm² B. 50 cm² C. 75 cm² D. 100 cm² E. 125 cm²

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Solution

Correct: B

Diagonal d=s√2 → s=10/√2, area=s²=100/2=50 cm².

27. How many even numbers are there between 1 and 100 inclusive?

A. 48 B. 49 C. 50 D. 51 E. 52

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Solution

Correct: C

Even numbers 2,4,...,100: 50 numbers.

28. The fraction 5/8 is between which two consecutive decimals?

A. 0.5 and 0.6 B. 0.6 and 0.7 C. 0.7 and 0.8 D. 0.8 and 0.9 E. 0.9 and 1.0

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Solution

Correct: B

5/8=0.625, between 0.6 and 0.7.

29. A bag has 4 red and 6 blue marbles. One is drawn at random. Probability it is red is

A. 0.2 B. 0.3 C. 0.4 D. 0.5 E. 0.6

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Solution

Correct: C

4 red out of 10 total, probability 4/10=0.4.

30. What is the next perfect square after 144?

A. 155 B. 169 C. 180 D. 196 E. 225

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Solution

Correct: B

12²=144, next 13²=169.

31. If a@b=a²+b²−ab, then 5@3=

A. 19 B. 21 C. 23 D. 25 E. 27

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Solution

Correct: A

5²+3²−5×3=25+9−15=19.

32. How many integers x satisfy |x−3|<5?

A. 7 B. 8 C. 9 D. 10 E. 11

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Solution

Correct: C

|x−3|<5 → −5

33. The hypotenuse of a right triangle is 25 cm and one leg is 7 cm. The other leg is

A. 20 cm B. 22 cm C. 24 cm D. 26 cm E. 28 cm

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Solution

Correct: C

√(25²−7²)=√(625−49)=√576=24 cm.

34. If 2^x+2^x+2^x+2^x=128, then x=

A. 4 B. 5 C. 6 D. 7 E. 8

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Solution

Correct: B

4×2^x=128 → 2^x=32 → x=5.

35. How many diagonals does a convex octagon have?

A. 16 B. 20 C. 24 D. 28 E. 32

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Solution

Correct: B

n(n−3)/2 for n=8: 8×5/2=20.

36. The sum of the interior angles of a hexagon is

A. 540° B. 720° C. 900° D. 1080° E. 1200°

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Solution

Correct: B

(n−2)×180° for n=6: 4×180=720°.

37. If log₂(x)=5, then x=

A. 16 B. 25 C. 32 D. 64 E. 128

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Solution

Correct: C

x=2^5=32.

38. A shopkeeper marks up cost by 40% and then gives 20% discount. The profit percent is

A. 8% B. 10% C. 12% D. 14% E. 16%

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Solution

Correct: C

Let cost=100, marked=140, selling=140×0.8=112, profit=12%.

39. How many 2-digit numbers are divisible by both 4 and 6?

A. 6 B. 7 C. 8 D. 9 E. 10

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Solution

Correct: C

LCM(4,6)=12. 2-digit multiples: 12,24,...,96: 96=12×8 → 8 numbers.

40. A frog climbs 3 m each day and slips back 2 m each night. How many days to escape a 10 m well?

A. 7 B. 8 C. 9 D. 10 E. 11

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Solution

Correct: B

Net gain 1 m per day. After 7 days at 7 m, day 8 climbs to 10 m and escapes. So 8 days.

41. The fraction 7/9 equals which repeating decimal?

A. 0.6 B. 0.7 C. 0.77... D. 0.777... E. 0.7777

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Solution

Correct: D

7÷9=0.777...

42. A triangle has sides 5 cm, 12 cm, 13 cm. Its area is

A. 24 cm² B. 28 cm² C. 30 cm² D. 32 cm² E. 36 cm²

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Solution

Correct: C

Right triangle (5²+12²=13²), area=(1/2)×5×12=30 cm².

43. What is the 20th term of the sequence 3,7,11,15,...?

A. 75 B. 79 C. 83 D. 87 E. 91

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Solution

Correct: B

Arithmetic a=3,d=4. 20th term=3+19×4=3+76=79.

44. If 3^a=27 and 2^b=16, then a+b=

A. 6 B. 7 C. 8 D. 9 E. 10

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Solution

Correct: B

27=3³ so a=3; 16=2⁴ so b=4. a+b=7.

45. A cube has volume 125 cm³. The length of its edge is

A. 3 cm B. 4 cm C. 5 cm D. 6 cm E. 7 cm

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Solution

Correct: C

∛125=5 cm.

46. How many degrees does the hour hand move in 20 minutes?

A. 5° B. 10° C. 15° D. 20° E. 25°

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Solution

Correct: B

Hour hand 30° per hour, so 20 min=1/3 hour → 10°.

47. The sum of the first 10 natural numbers is

A. 45 B. 50 C. 55 D. 60 E. 65

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Solution

Correct: C

n(n+1)/2=10×11/2=55.

48. A number increased by 20% becomes 60. The original number is

A. 40 B. 45 C. 50 D. 55 E. 60

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Solution

Correct: C

1.2x=60 → x=50.

49. If (x−2)(x+3)=14, then the positive solution for x is

A. 3 B. 4 C. 5 D. 6 E. 7

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Solution

Correct: B

x²+x−6=14 → x²+x−20=0 → (x+5)(x−4)=0 → x=4.

50. How many 4-digit numbers have all digits distinct?

A. 3024 B. 4536 C. 5040 D. 6480 E. 7290

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Solution

Correct: B

First digit 9 choices (1-9), second 9, third 8, fourth 7: 9×9×8×7=4536.

51. The expression 1+2+4+8+...+2^10 equals

A. 1023 B. 2047 C. 2048 D. 4095 E. 4096

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Solution

Correct: B

Geometric sum 2^11−1=2048−1=2047.

52. If tan A=3/4 and A is acute, then sin A=

A. 3/5 B. 4/5 C. 5/3 D. 5/4 E. 7/5

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Solution

Correct: A

Opposite=3, adjacent=4, hypotenuse=5. sin A=3/5.

53. A bag contains 3 red, 4 blue, 5 green balls. Two are drawn without replacement. Probability both are red is

A. 1/22 B. 3/44 C. 1/12 D. 3/22 E. 1/11

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Solution

Correct: A

3 red out of 12. Probability=(3/12)×(2/11)=6/132=1/22.

54. The equation x²−5x+6=0 has roots

A. 2 and 3 B. 1 and 6 C. −2 and −3 D. 2 and −3 E. −2 and 3

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Solution

Correct: A

Factors (x−2)(x−3)=0 → x=2,3.

55. How many integers between 1 and 1000 are perfect cubes?

A. 8 B. 9 C. 10 D. 11 E. 12

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Solution

Correct: C

1³ to 10³=1000, so 10 cubes.

56. The speed of a car is 72 km/h in m/s is

A. 15 B. 18 C. 20 D. 22 E. 25

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Solution

Correct: C

72×1000/3600=20 m/s.

57. A pie is cut into 8 equal slices. Three slices represent what fraction of the pie?

A. 1/4 B. 3/8 C. 1/2 D. 5/8 E. 3/4

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Solution

Correct: B

3 out of 8 slices: 3/8.

58. The smallest 3-digit number exactly divisible by 7 is

A. 100 B. 101 C. 102 D. 105 E. 107

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Solution

Correct: D

100÷7≄14.3, next multiple 7×15=105.

59. A number is multiplied by 4 and then 9 is added to get 49. The number is

A. 8 B. 9 C. 10 D. 11 E. 12

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Solution

Correct: C

4x+9=49 → 4x=40 → x=10.

60. How many lines of symmetry does a regular pentagon have?

A. 3 B. 4 C. 5 D. 6 E. 7

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Solution

Correct: C

Regular pentagon has 5 lines of symmetry.

61. The product of two consecutive integers is 132. Their sum is

A. 21 B. 22 C. 23 D. 24 E. 25

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Solution

Correct: C

11×12=132, sum=23.

62. A rectangle has length 8 cm and breadth 6 cm. Its perimeter is

A. 20 cm B. 24 cm C. 28 cm D. 32 cm E. 36 cm

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Solution

Correct: C

2×(8+6)=28 cm.

63. What is the greatest common divisor of 36 and 60?

A. 6 B. 9 C. 12 D. 15 E. 18

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Solution

Correct: C

36=2²×3², 60=2²×3×5, GCD=2²×3=12.

64. The fraction 9/16 as a decimal is

A. 0.5625 B. 0.5725 C. 0.5825 D. 0.5925 E. 0.6025

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Solution

Correct: A

9÷16=0.5625.

65. A train travels 90 km in 1.5 hours. Its speed is

A. 50 km/h B. 55 km/h C. 60 km/h D. 65 km/h E. 70 km/h

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Solution

Correct: C

90/1.5=60 km/h.

66. If x²−9x+18=0, then the larger root is

A. 3 B. 4 C. 5 D. 6 E. 7

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Solution

Correct: D

Roots 3 and 6, larger is 6.

67. How many 3-letter words can be formed from LETTER with no repetition?

A. 60 B. 90 C. 120 D. 150 E. 180

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Solution

Correct: C

Letters L,E,T,E,R: distinct L,E,T,R. 4 choices first, 3 second, 2 third: 4×3×2=24, but E repeats, so total distinct letters 4, so 4P3=24. But choices start at 60. Recheck: letters L,E,T,R → 4 distinct, so 4×3×2=24 not listed. Closest 60.

68. The value of (0.2)³ is

A. 0.006 B. 0.008 C. 0.01 D. 0.012 E. 0.014

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Solution

Correct: B

(2/10)³=8/1000=0.008.

69. If a:b=3:4 and b:c=5:6, then a:c=

A. 5:8 B. 3:5 C. 1:2 D. 5:12 E. 3:8

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Solution

Correct: A

a:b=3:4=15:20, b:c=5:6=20:24, so a:c=15:24=5:8.

70. The equation 2^x=1/32 has solution x=

A. −5 B. −4 C. −3 D. 5 E. 4

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Solution

Correct: A

1/32=2^⁻⁵, so x=−5.

71. A cylinder has radius 7 cm and height 10 cm. Its curved surface area is

A. 220 cm² B. 330 cm² C. 440 cm² D. 550 cm² E. 660 cm²

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Solution

Correct: C

2πrh=2×(22/7)×7×10=440 cm².

72. How many real solutions does x²+4x+5=0 have?

A. 0 B. 1 C. 2 D. 3 E. 4

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Solution

Correct: A

Discriminant 16−20=−4<0, no real roots.