# You Gotta Know These Types of Computation Problems

- Pythagorean triples: Sets of small integers that satisfy the equation of the Pythagorean theorem,
*a*^{2}+*b*^{2}=*c*^{2}, and could therefore be the side lengths of a right triangle. The simplest ones are {3, 4, 5}, {5, 12, 13}, {7, 24, 25}, and {8, 15, 17}. Note that any multiple of a Pythagorean triple is also a Pythagorean triple, so (for instance) {6, 8, 10}, {15, 20, 25}, and {300, 400, 500} are also ones by virtue of {3, 4, 5} being one. - Matrices: Every team should be able to add, subtract, multiply, take the determinant of, transpose, and invert matrices, particularly 2×2 ones.
- Vectors: Every team should be able to find the magnitude (length) of a vector, and add, subtract, find the angle between, find the dot product of, and find the cross product of two vectors.
- Solids: Teams should be able to calculate the volume and surface area of simple geometric figures including the sphere, cone, cylinder, pyramid, hemisphere, prism, and parallelepiped.
- Plane figures: Teams should be able to calculate the areas of triangles, trapezoids, parallelograms, rhombi, regular polygons, and circles using several methods.
- Similar figures: The areas of similar figures are related by the square of any corresponding length, and the volumes are related by the cube of any corresponding length. For instance, if a square has a diagonal that is 30% longer than another square, it has an area that is 1.30 × 1.30 = 1.69 times as great (69% greater). Similar reasoning applies to perimeters, side lengths, diameters, and so forth.
- Combinatorics: Teams should be able to compute the number of permutations and combinations of
*n*objects taken*m*at a time. They should also have memorized the first six (or so) values of the factorial function to make this easier. - Logarithms: Teams should be familiar with basic operations of logarithmic math: simplifying the logarithm of a product, difference, or power, and converting from one base to another.
- Complex numbers: Teams should be familiar with the symbol
*i*representing an imaginary square root of –1, basic operations on complex numbers, graphing complex numbers, and converting complex numbers to polar form. - Divisibility rules: Teams should be able to quickly apply the divisibility rules for small integers (2 through 11) to large integers.
- Polynomials: Teams should be able to quickly add, subtract, multiply, divide, factor, and find the roots of low-degree polynomials, and understand how the degree behaves under the first four operations.
- Calculus: Teams should be able to find the derivative, integral, slope at a point, local extrema, points of inflection, and critical points of polynomial, trigonometric, and other common functions.