AIME Cheat Sheet 2026

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

1. If P(x) = x^3 + 2x^2 - 5x - 6 and P(-1) = 0, fully factor P(x). → (x+1)(x-2)(x+3)
2. How many positive integer solutions are there to the equation x+2y=10? → 5
3. Find all values of x where f(x) = x^3 - 3x is increasing. → x 1
4. What is the sum of the infinite geometric series 4 + 2 + 1 + 1/2 + …? → 8
5. What is the sum of interior angles of a hexagon? → 720°
6. How many integer solutions does x^2 < 16 have? → 7
7. Compute (2+3i)(1-i). → 5+i
8. For positive reals a and b satisfying a + b = 1, what is the minimum value of a³ + b³? → 1/4
9. For real numbers x and y satisfying x + y = 6, what is the minimum value of x² + y²? → 18
10. Compute the conjugate of z = 7 - 2i. → 7 + 2i
11. In how many ways can you choose 2 different cards from a standard deck of 52 cards? → 1326
12. Among all triangles with perimeter 12, what is the maximum area? → 4√3
13. In a triangle, the lengths of the sides are in the ratio 3:4:5. If the perimeter of the triangle is 36, what is the area of the triangle? → 24
14. What is the area of an equilateral triangle with side length 4? → 4√3
15. For positive reals x and y with xy = 1, what is the minimum value of (x + 1/x)² + (y + 1/y)²? → 8
16. What is the maximum product of positive integers that sum to 10? → 36
17. What is the period of f(x) = sin(3x)? → 2π/3
18. Find all rational roots of f(x) = x^3 - 6x^2 + 11x - 6. → {1, 2, 3}
19. For what value of c does cx^2 + 8x + 4 = 0 have a double root? → 4
20. For positive reals x, y, z with x + y + z = 9, what is the maximum value of xy + yz + xz? → 27
21. For real numbers, if (x-2)^2 + (y+3)^2 = 0, find x + y. → -1
22. A sphere has radius 3. Find its volume in terms of π. → 36π
23. What is the argument (angle) of the complex number z = -1 + i? → 135°
24. In triangle ABC with sides a=7, b=24, c=25, what type of triangle is it? → Right
25. The diagonals of a rhombus are 10 and 24. Find its perimeter. → 52
26. Express sin(2θ) using a double-angle formula. → 2sin(θ)cos(θ)
27. Chord AB and chord CD intersect inside a circle. If AX=3, XB=8, CX=4, find XD. → 6
28. What is the largest integer n such that n^2 + 4n < 45? → 5
29. Solve |2x - 3| = 7. What is the sum of all solutions? → 3
30. Factor completely: 2x^3 - 8x. → 2x(x-2)(x+2)