State Tournament

1A Prealgebra Topics:

• Fractions to add and express as the quotient of two relatively prime integers

• Complex fractions and continued fractions

• Decimals, repeating decimals

• Percentage, interest, and discount

• Least common multiple, greatest common divisor

• Number bases; change of base
  
  

2A Linear Equations in One Unknown:

• Solving numeric equations (perhaps involving a second degree term which drops out)

• Solving literal equations

• Story problems leading to linear equations in one variable

• Linear inequalities

• Absolute value equations and inequalities
  
  

3A Systems of Linear Equations in Two (or on occasion three) Variables:

• Numeric and literal systems

• Relation to graphical procedures

• Word problems leading to such systems

• Systems of inequalities used to define a region in the plane

• Determinants   or   Determinants

• Area of Triangles using Determinants

• Solving a System of Equations using Determinants
  
  

4A Algebraic Manipulation:

• Factoring (including x3 + y3, x3 – y3)

• Sums, products, quotients of rational expressions

• Solving equations (including radical equations) involving these skills, but ultimately solvable by factoring or the quadratic formula (but no complex roots)

• Rational exponents

• Simplifying radical expressions

• Function notation and variational dependencies
  
  

5A Puzzle Problems:

• Word problems, one or more variables

• Max-min problems not requiring calculus

• Problems found in "brain-teaser" type books

• Logic puzzles, including the use of Venn Diagrams

1B Angles and Special Triangles:

• The Theorem of Pythagoras; familiar Pythagorean triples

• Complementary, supplementary, and vertical angles

• Interior and exterior angles for triangles and polygons

• Angles formed by transversals cutting parallel lines
  
  

2B Triangular Figures and Solids:

• Medians, angle bisectors, and altitudes

• Ceva’s and Stewart’s Theorems (Stewart's Theorem Practice Problems)

• Area of a triangle (including Hero’s Formula)

• Triangular prisms & pyramids (including volume and surface area)
  
  

3B Polygonal Figures and Solids:

• Special quadrilaterals and regular polygons (including area formulas)

• Intersecting diagonals

• Ptolemy's Theorem

• Polygonal prisms & pyramids (including volume and surface area)
  
  

4B Circular Figures and Solids:

• Central, inscribed, tangential, and exterior angles

• Power of a point (chords, secants, tangents)

• Interior and exterior tangents of two circles

• Intercepted arcs

• Area of circles, sectors, circular segments

• Cylinders, cones, & spheres (including volume and surface area)
  
  

5B Congruence and Similarity:

• Ratio and proportion

• Segments intercepted by parallel lines

• Identification of similar/congruent figures

• Ratios of areas and volumes

• Elementary trigonometric ratios

1C Elementary Trigonometry:

• Definitions and solution of right triangles

• Elementary identities

• Radian measure and graphs of elementary functions

• Trigonometric functions of multiples of  π/6, π/4, π/3, π/2.
  
  

2C Trigonometry:

• Functions of sums of angles and sums of functions of angles

• Half and double angle formulas

• Reduction formulas

• (here's a link for the necessary identities and more) 

• (Not required: formulas for sin A + sin B, etc.)
  
  

3C Trigonometry:

• Law of sines, law of cosines

• Inverse functions and their graphs

• Solving trigonometric equations

• De Moivre's Theorem and the roots of unity

• Method to remember common values on the unit circle
  
  

4C Miscellaneous Topics:

• Sequences: patterns and recursion formulas, arithmetic and geometric sequences

• Series: partial sums, formulas for 1+2+…+n, 12 + 22 + … + n2, 13 + 23 + … + n3

• Function notation; factorial notation and Binomial Theorem
  
  

5C Counting and Probability:

• Permutations, with and without replacement

• Combinations, with and without replacement

• Using the principle of inclusion, exclusion

• Using the binomial and multinomial expansions

• Nonnegative integer solutions to x1+x2+...+xn = b.

• Definition, simple applications of probability (when to multiply, when to add)

1D Roots of Quadratic and Polynomial Equations:

• Solution of quadratic equations by factoring, by completing the square, by formula

• Complex roots of quadratic equations; the discriminant and the character of the roots

• Relations between roots and coefficients

• Synthetic Division, Synthetic Division example, Remainder Theorem

• Function notation
  
  

2D Analytic Geometry of Straight Lines and Circles:

• Slope, families of parallel, perpendicular, or coincident (lines that cross) lines

• Point-slope, slope-intercept, intercept (scroll down to the intercept-intercept section), normal ("normal" means perpendicular) forms of the straight line

• Intersections (solution of simultaneous systems)
  
  

3D Exponents and Logarithms:

• Use of fractional, negative exponents

• Simplifying expressions involving radicals (including denesting radical expressions... this is tough!)

• Solving equations involving radicals

• Use of logarithms; identities involving logarithms

• Solving logarithmic equations

• Relationships between logarithms to different bases
  
  

4D Analytic Geometry of the Conic Sections:

• Using the standard forms of equations of the conic sections

• Graphs, including the location of foci, directrices (plural of directrix), and asymptotes

• Use of properties of conics to solve applied problems, including max-min for parabolas
  
  

5D Variations of Problems Appearing on the Previous Year's AMC 12 (Contest A and B):

• AMC 2023 Contest A

  • Instructional Videos: #1-10 #11-15

• AMC 2023 Contest B

  • Instructional Videos: #1-10 #11-15 A better response to 12B #11