The AMC and AIME reward pattern recognition and clever problem-solving far more than raw syllabus knowledge. The way to build that is exposure: work through a large number of varied competition problems, then study clean solutions to absorb the recurring ideas and techniques. Maths Daily Helper generates fresh AMC and AIME-style problems across the core competition areas, each with a full worked solution, so you always have a new challenge and a model answer to learn from.
Areas covered
- Algebra: clever manipulation, functional relationships, systems, polynomials.
- Number theory: divisibility, modular arithmetic, digit problems, factors.
- Combinatorics: counting, casework, probability, pigeonhole.
- Geometry: similar triangles, area ratios, coordinate methods.
Try a few AMC / AIME-style problems
Attempt each one on paper first, then reveal the working.
What is the remainder when 3¹⁰⁰ is divided by 5?
1. The powers of 3 mod 5 cycle: 3¹ ≡ 3, 3² ≡ 4, 3³ ≡ 2, 3⁴ ≡ 1, then repeat every 4.
2. 100 mod 4 = 0, matching the end of the cycle, where 3⁴ ≡ 1.
Answer: 1.
How many ways can the letters of the word "MATH" be arranged?
1. There are 4 distinct letters, so the number of arrangements is 4!.
2. 4! = 4 × 3 × 2 × 1 = 24.
Answer: 24.
If x + 1/x = 3, find x² + 1/x².
1. Square both sides: (x + 1/x)² = x² + 2 + 1/x² = 9.
2. So x² + 1/x² = 9 − 2.
Answer: 7.
Find the sum of the digits of the number 10⁴ − 1.
1. 10⁴ = 10000, so 10⁴ − 1 = 9999.
2. Sum of digits = 9 + 9 + 9 + 9 = 36.
Answer: 36.
Two fair coins are tossed. What is the probability of getting exactly one head?
1. The equally likely outcomes are HH, HT, TH, TT, four in total.
2. Exactly one head occurs in HT and TH, that is 2 outcomes.
3. Probability = 2/4 = 1/2.
Answer: 1/2.
A square has area 49. What is the length of its diagonal?
1. Side length = √49 = 7.
2. The diagonal of a square is side × √2 = 7√2.
Answer: 7√2.
Find the sum 1 + 2 + 3 + … + 100.
1. Use the formula for the sum of the first n integers: n(n + 1)/2.
2. = 100 × 101 / 2 = 10100 / 2.
Answer: 5050.
How many positive divisors does 36 have?
1. Prime factorise: 36 = 2² × 3².
2. The number of divisors is (2 + 1)(2 + 1) = 3 × 3.
Answer: 9.
Get unlimited competition problems
Choose your contest and area, and get a fresh AMC or AIME-style challenge each time with full solutions.
Open Maths Daily Helper →How to train for AMC and AIME
Competition maths is a skill built through volume and reflection. Solve a handful of problems daily, and when one defeats you, study the solution until the key idea is clear, then look for that idea again in later problems. Timed practice matters too: the AMC gives you 75 minutes for 25 questions, so pace is part of the challenge. Rotate across algebra, number theory, combinatorics and geometry so no area lags behind.