e. Topics for Events
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
· Challenge topic: the Euclidean algorithm
1B Measuring Angles
· Angle sums for triangles and polygons
· Complementary, supplementary, and vertical angles
· Interior and exterior angles of a triangle
· Angles formed by transversals cutting parallel lines
· Familiarity with isosceles, equilateral and 30°-60°-90° triangles
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.
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
· Challenge Topic: Rational functions and their graphs
· Function notation
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
2B Familiar Geometric Figures, Congruence and Similarity
· Ratio and proportion
· Segments intercepted by parallel lines
· Medians, angle bisectors, and altitudes
· The Theorem of Pythagoras; familiar Pythagorean triples
· Relationships in 30°-60°-90° and 45°-45°-90° triangles
· Equilateral and isosceles triangles, associated terminology
· Challenge Topic: Number theoretic aspects of Pythagorean triples
2C Trigonometry
· Functions of sums of angles and sums of functions of angles
· Half and double angle formulas
· Reduction formulas
· (Not required: formulas for sin A + sin B, etc.)
2D Analytic Geometry of Straight Lines and Circles
· Slope, families of parallel, perpendicular, or coincident lines
· Point-slope, slope-intercept, intercept, normal forms of the straight line
· Intersections (solution of simultaneous systems)
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
3B Quadrilaterals and Polygons
· Rectangles, parallelograms, the rhombus, and the trapezoid
· Intersecting diagonals
· Ptolemy's Theorem
· Regular polygons and inscribed or circumscribed circles
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
3D Exponents and Logarithms
· Use of fractional, negative exponents
· Simplifying expressions involving radicals
· Solving equations involving radicals
· Use of logarithms; identities involving logarithms
· Solving logarithmic equations
· Relationships between logarithms to different bases
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
4B Circles
· Inscribed angles
· Secants, intersecting chords
· Interior and exterior tangents of two circles
· The measurement of angles by intercepted arcs
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
4D Analytic Geometry of the Conic Sections
· Using the standard forms of equations of the conic sections
· Graphs, including the location of foci, directrices, and asymptotes
· Use of properties of conics to solve applied problems, including max-min for
parabolas
5A Puzzle Problems (20 minutes)
· 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
5B Areas, Perimeters, and Volumes
· Triangles - including Heron's formula and ability to use ideas of elementary
trigonometry to find certain lengths
· Trapezoids and parallelograms
· Circles, sectors of circles
· Volumes of familiar 3-dimensional objects
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)
5D Variations of Problems appearing on the previous year’s AMC 12 (contest A)
f. League Comments on Building a Team, Level of Difficulty
League policy requires that no more than 6 members of a school's 8-member team
can be
beyond 10th grade. The general expectation is that 2 members will be 10th
graders, and 6
members will be 11th or 12th graders. Schools may include 9th or even 8th
graders from their
system, but to compete effectively, such students would have to be familiar with
mathematical
topics not usually taught at their grade level. A coach wishing to build for the
future might
encourage a 9th grader to come to meets and compete as an extra, particularly in
Event A.
Event A is generally restricted to topics covered in Algebra One and Two. To
emphasize the
importance of geometry and to guarantee a second event in which a 10th grader
has a good
chance of scoring points, Event B always covers topics in geometry.
It must be understood that the topics listed for a particular event are intended
as an indication of
the primary emphasis, not as a complete list of everything a participant must
know. An effort is
made to draw upon material generally covered prior to the topics listed, but
varying order of
presentation from school to school makes this difficult, and certain topics (the
theorem of
Pythagoras, proportions, solving simple equations) are likely to crop up
everywhere. Event A
of a meet may use all topics of previous Event A's; similarly for events B, C,
and D. By
keeping the same topics for corresponding events from one year to the next, it
is intended that a
file of exams from previous years will help participants anticipate the kind of
questions to be
expected.
Review is, in fact, to be encouraged at every level. Team Events always
emphasize topics
drawn from Individual Events of the meet, sometimes using a question that is
only a slight
variation of one used in an Individual Event earlier in the meet. Also watch in
Team Events for
variations of some of the more difficult questions that were used in meets
earlier in the season.
NOTE that to prepare questions for event 5D, it is necessary for the problem
writer to read
through the previous year's AMC 12 (contest A) very carefully. The influence of
this reading
may be detected throughout the season, and the AMC contest materials can always
be
recommended as a source of sample questions.
Event A of meet 5, focuses on puzzle problems. This is because media coverage,
if we get it,
commonly reports on the final meet. Puzzle problems are the problems most easily
worked into
an article intended for the general public. At the same time, such problems
frequently require
more careful reading and more time to do some experimenting and guessing; and
since we do
want to legitimately claim that our students do solve such problems, the time
limit for this
Individual Event is 20 minutes.
g. Division Coordinator
Each division should, as its last act of business at the conclusion of a season,
appoint a Division
Coordinator for the following season. It is permissible, even advisable, to have
the same person
serve as Coordinator for several successive years.
The Division Coordinator becomes a member of the League's Board of Directors and
represents
the division at the Annual Fall Meeting and at any special meetings of the
Board. A Division
Coordinator who cannot attend a meeting of the Board should appoint another
coach from the
division who then becomes a voting member for that meeting. An annual meeting
will normally
be scheduled in late September. At this meeting, any pending questions about
League rules will
be settled, alignment of teams into divisions will be tentatively settled, and
host schools in each
division should be designated for the coming season. Coordinators should contact
schools in
their division before this meeting and come to the meeting with a list of host
schools to give to
the League Associate Director.
