AMC12 Cheat Sheet 2026

The 30 highest-yield AMC12 facts, distilled from real exam questions. Print it, save it as a PDF, or study it here — free, no sign-up.

1. How many distinct diagonals does a convex octagon have? → 20
2. A bag has 3 red and 5 blue balls. One ball is drawn at random. What is P(red)? → 3/8
3. If n=2^3*3*5^2, how many ordered pairs (a,b) satisfy lcm(a,b)=n? → 105
4. Which expression equals cos^2(x) - sin^2(x)? → cos(2x)
5. A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? → 5/16
6. Solve |2x-3|=7. What is the sum of all solutions? → 3
7. What is the sum from k=1 to 100 of (1/k - 1/(k+1))? → 100/101
8. What is the probability of rolling a sum of 7 with two standard dice? → 1/6
9. What is ∑(k=1 to 20) (2k − 1)? → 400
10. If tan(alpha)=2 and tan(beta)=3, what is tan(alpha+beta)? → -1
11. What is the sum to infinity of 1 + 1/2 + 1/4 + ...? → 2
12. How many ways can 4 books be arranged on a shelf? → 24
13. What is the sum of the first 100 positive integers? → 5050
14. What is the range of y = arcsin(x)? → [-pi/2,pi/2]
15. How many ways can 5 identical balls be distributed into 3 distinct boxes? → 21
16. How many diagonals does a convex octagon have? → 20
17. What is sin(pi/4) * cos(pi/4)? → 1/2
18. If sin(θ) = 3/5 and θ is in quadrant I, what is cos(θ)? → 4/5
19. For how many integers n is (n-1)/(n+1) also an integer? → 4
20. A fair coin is flipped 4 times. What is the probability of exactly 2 heads? → 3/8
21. Using De Moivre's theorem, what is (cos(π/4) + i·sin(π/4))^8? → 1
22. How many ordered selections of 2 items from 7 are possible (order matters)? → 42
23. What is the units digit of 7^53? → 7
24. How many solutions does x^2 = 1 (mod 8) have among {0,1,2,...,7}? → 4
25. What is the sum of k^2 for k=1 to 5? → 55
26. An arithmetic sequence has first term 7 and sum of first 10 terms S₁₀ = 160. Find the common difference. → 2
27. What is the amplitude of y = -3sin(2x)? → 3
28. Let z₁ and z₂ be complex numbers with |z₁| = 2 and |z₂| = 3. What is the maximum possible value of |z₁ + z₂|? → 5
29. If P(A)=0.4 and P(B)=0.5 and A and B are independent, what is P(A and B)? → 0.2
30. How many bit strings of length 8 contain exactly three 1s? → 56