Exam guideGrades 3 to 10
AMC 8 and Math Kangaroo Guide with Practice Questions
The maths competitions students take in the US, Canada and the UK: how each one works, the facts worth knowing by heart, how to guess wisely, and twelve AMC-style questions with answers.
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12 practice questions at the end. Pick an answer for each question, then press Check my answers: your score and an explanation for every question appear at once. Nothing you choose leaves this page.
01The competitions at a glance
| Contest | Who | Format |
|---|---|---|
| AMC 8 | Grade 8 and below | 25 multiple-choice questions in 40 minutes, 1 point each, no penalty. Held in January |
| AMC 10 and AMC 12 | Grade 10 and below; grade 12 and below | 25 questions in 75 minutes: 6 points for a right answer, 1.5 for a blank, 0 for a wrong one. Held in November, on an A date and a B date |
| AIME | By invitation, from AMC 10 and 12 scores | 15 questions with answers from 000 to 999, in 3 hours |
| Math Kangaroo (USA) | Grades 1 to 12 | 24 questions for grades 1–4 and 30 for grades 5–12, in 75 minutes, worth 3, 4 or 5 points; no penalty. 18 March 2027 |
| MATHCOUNTS | Grades 6 to 8 | School, chapter, state and national rounds, alone and in teams |
| CEMC (Canada) | Grades 7 to 12 | Gauss for grades 7 and 8, then Pascal, Cayley and Fermat, and the Euclid with written solutions |
| UKMT (UK) | Years 7 to 13 | The Junior, Intermediate and Senior Mathematical Challenges, 25 multiple-choice questions each, then follow-on rounds |
02Facts worth knowing by heart
- Primes under 50 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
- Squares up to 252 = 625; cubes up to 103 = 1000; powers of 2 up to 210 = 1024
- Right triangles 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, and their multiples
- Angles a polygon with n sides has interior angles adding to (n − 2) × 180°
- Areas triangle ½bh; trapezium ½(a + b)h; circle πr2; circumference 2πr
- Sum 1 to n n(n + 1)/2
- Counting n! orders of n things; C(n, r) ways to choose r of them
- Probability favourable outcomes ÷ all equally likely outcomes
- Units digits powers repeat in cycles of at most 4: 3, 9, 7, 1 for powers of 3
03Strategy
- Use the answer choices try them in the question when the algebra is slow
- Estimate, then eliminate rule out choices that are clearly too big or too small
- Draw a picture for every geometry and counting question
- Small cases try the pattern with 1, 2 and 3 before the big number
- AMC 8 and Math Kangaroo no penalty, so answer every question
- AMC 10 and 12 a blank earns 1.5, a wild guess about 1.2 on average; guess once you can rule out two choices
- Time AMC 8 gives about 1.5 minutes a question: move on and come back
04Practice questions1 mark each
11 + 2 + 3 + … + 20 = ?
2How many prime numbers are there between 30 and 50?
3A rectangle has perimeter 30 and area 56. What is the length of its longer side?
4What is the units digit of 32027?
5Two fair dice are rolled. What is the probability that the total is 7?
6A circle has circumference 10π. What is its area?
7The average of four numbers is 15. Three of them are 10, 12 and 20. What is the fourth?
8In how many ways can the letters of the word MATH be arranged?
9What is the sum of the interior angles of a pentagon?
10A price is raised by 20% and then cut by 20%. Compared with the starting price, the final price is:
11What is the smallest positive integer that leaves a remainder of 1 when divided by 2, 3, 4, 5 or 6?
12A 3 × 3 × 3 cube is painted on the outside and cut into 27 unit cubes. How many unit cubes have exactly one painted face?
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Answers and explanations
- 1Cn(n + 1)/2 = 20 × 21 ÷ 2 = 210.
- 2B31, 37, 41, 43 and 47.
- 3DThe sides add up to 15 and multiply to 56, so they are 7 and 8.
- 4AUnits digits of powers of 3 repeat 3, 9, 7, 1. 2027 leaves remainder 3 when divided by 4, so the units digit is 7.
- 5E6 of the 36 outcomes make 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1). 6/36 = 1/6.
- 6C2πr = 10π gives r = 5, so the area is π × 52 = 25π.
- 7AThe four add up to 60, and 60 − (10 + 12 + 20) = 18.
- 8E4 different letters: 4! = 4 × 3 × 2 × 1 = 24.
- 9B(5 − 2) × 180° = 540°.
- 10D1.2 × 0.8 = 0.96, which is 4% less than the start.
- 11EThe LCM of 2, 3, 4, 5 and 6 is 60, so the answer is 60 + 1 = 61.
- 12BOne in the centre of each of the 6 faces.
How to use this guide
- Past AMC papers and solutions are free online: do one a week, timed.
- After each paper, redo every question you missed without looking at the solution.
- Check the dates and rules each year on the MAA, Math Kangaroo, CEMC or UKMT site.
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