8 Proven AMC 10 Problems to Strengthen Problem-Solving Skills

It’s a weekday evening. Your child stares at an AMC 10 question, exhales, and says, “I don’t get where to start.” That moment shows what many AMC 10 learners truly need: not tougher problems, but clarity through strategy.

If you’ve ever seen that mix of frustration and self-doubt and want to help but aren’t sure how, this guide is for you.

8 Must-Try Problems with Step-by-Step Solutions

Each problem in this set targets specific AMC 10 skill gaps: strategy development, pattern recognition, and concise reasoning.

They represent core competition topics: algebraic manipulation, geometry insight, number theory tricks, and counting strategies.

Our selection criteria focus on problems that:

• Reward thinking over computation
• Demonstrate multiple solution paths
• Build confidence through achievable complexity levels

Let’s go through them.

1. Algebra: Smart Equation Rearrangement

Problem: If x + 1/x = 5, find x² + 1/x²

Solution Steps:

• Square both sides: (x + 1/x)² = 25
• Expand: x² + 2(x)(1/x) + 1/x² = 25
• Simplify middle term: x² + 2 + 1/x² = 25
• Isolate: x² + 1/x² = 23

Try asking your child: “What happens when we square the entire equation?” “Can you simplify x times 1/x?”

Success indicator: Solves in ≤6 minutes with ≤1 hint.

2. Geometry: Angle Bisector Insight

Problem: In triangle ABC, ∠A = 40°, ∠B = 60°. AD bisects ∠A. Find ∠ADB.

Solution Steps:

• AD bisects ∠A → angles BAD = CAD = 20°
• In triangle ABD: ∠ADB = 180° − 60° − 20° = 100°

Try asking your child: “If a line bisects an angle, what does that tell us about its two parts?”

Success indicator: Applies the same idea in a new triangle unaided.

3. Number Theory: Counting Factors

Problem: How many positive integers less than 100 have exactly 6 factors?

Solution Steps:

• Numbers with 6 factors are of the form p⁵ or p²q (p, q are primes)
• p⁵: 2⁵ = 32 (3⁵ = 243 > 100)
• p²q: 2²×3=12, 2²×5=20, … up to 3²×11=99 → 15 numbers
• Total = 16

Try asking your child: “Can you find smaller examples before listing them all?”

Success indicator: Completes a similar task in < 8 minutes with ≥ 80% accuracy.

4. Arithmetic: Reverse Average

Problem: The Average of 5 numbers is 12. One number is removed, the average becomes 11. Find the removed number.

Solution Steps:

• Total sum = 5×12 = 60
• New sum = 4×11 = 44
• Removed number = 60 − 44 = 16

Try asking your child: “How do the totals before and after removal compare?”

Success indicator: Solves correctly within 5 minutes.

5. Combinatorics: Handshake Count

Problem: In a room of 10 people, each shakes hands once with everyone else. How many handshakes?

Solution Steps:

• Use formula n(n−1)/2 = 10×9/2 = 45

Try asking your child: “Why divide by 2 at the end?”

Success indicator: Confidently applies the idea to 12 people.

6. Sequences: Pattern Recognition

Problem: Find the next number: 2, 4, 8, 16, …

Solution Steps:

• Pattern: multiply by 2
• Next number = 32

Try asking your child: “What operation links each term to the next?”

Success indicator: Extends correctly for 5 more terms.

7. Coordinate Geometry: Distance and Midpoint

Problem: Find the midpoint of points (2,3) and (6,7).

Solution Steps:

• Midpoint = ((2+6)/2, (3+7)/2) = (4,5)

Try asking your child: “How do we find a point halfway between two?”

Success indicator: Solves new example in ≤2 minutes.

8. Probability: Simple Fraction

Problem: A bag contains 3 red and 2 blue balls. One ball is drawn at random. What is the probability that it is red?

Solution Steps:

• Total balls = 5
• Probability = 3/5

Try asking your child: “How many total outcomes do we have?”

Success indicator: Answers variant problems unaided.

Together, these eight problems create a balanced foundation: building speed, strategy, and confidence through steady, structured thinking rather than rote memorization.

Turn AMC Practice into a Confidence-Boosting Journey

You don’t need to solve every AMC 10 problem alongside your child. Your real role is to guide their thinking, not their calculations. Start small: pick one problem from this list tonight, set a 20-minute timer, and celebrate each step they figure out on their own. Every small win builds confidence, focus, and a love for math that lasts far beyond the test.

When your child is ready for structured, strategy-driven learning, book a free demo class with Bhanzu. Our expert mentors turn AMC preparation into an engaging journey of curiosity, speed, and creativity, helping your child see math not as a challenge to fear, but a skill to master with joy.