Skills for December Rounds Math Team Skills for December Rounds Round 1 Round 2 Round 3 Round 4 Round 5 Round 6

Round 1 – Trig: Right Angle Problems Law of Sines and Cosines For right triangles: Pythagorean Theorem

For oblique triangles: (ASA, AAS, or SSA)

For oblique triangles: (SAS or SSS)

Round 2 – Arithmetic/Number Theory Percent of Change = Positive percent of change  percent increase Negative percent of change  percent decrease

Number of Factors Example: 90 has 12 positive integer factors To find the number of positive integer factors of a number: Find the number’s prime factorization (using exponents to describe repeating factors) Add 1 to each of the exponents Find the product of the numbers generated by adding 1 to each exponent Example: 90 has 12 positive integer factors

Base 10 Numbers We use base 10 numbers all of the time. We have memorized the places of base 10 numbers since we were young children. For Example, the number has a 2 in the one’s place, a 9 in the ten’s place and a 6 in the hundred’s place. (The subscript of 10 simply means that it is a base 10 number. When there is no subscript, we always assume that the number is base 10) To understand other bases, we need to know where the names of the place values come from. So the number 69210 is really: Six 100’s = 600 + Nine 10’s = 90 + Two 1’s = 2 = 692 Base 10 place values: Evaluate to get the names:

Base 2 Numbers Base 2 numbers’ place values work the same as base 10: Base 2 place values: In base 2, each place value is worth… When we are in base 10, remember that we can only use the numbers from 0 to 9. In base 2, we can only use numbers from 0 to 1. So to evaluate 101102 : Evaluating this base 2 number: Using these place values: One 16 + Zero 8’s + One 4 + One 2 + Zero 1’s Yields: 16 + 0 + 4 + 2 + 0 =22 Which is: 101102 = 2210 Therefore:

Base 3 Numbers If you can figure out the base 10 equivalent to 12013 then you’ve got it. Solution: 27 + 18 + 0 + 1 = 46

Round 3 – Coordinate Geometry of lines and circles Equation of a Circle: {with center (h,k) and radius r} If given this form Divide both sides by ac, then complete each square to change back to general form.

Ex) Circle with center (2, -1) and radius 4

Equations of lines: slope y-intercept Given Point (x1 , y1) Typically A > 0 and A, B, C are Integers Given Point (x1 , y1)

Center of a circle is the midpoint of a diameter Slope of a line is constant. If given slope, use to find additional points

Round 4 – Log and Exponential Functions Logarithmic Form vs. Exponential Form exponent base base exponent Remember: Log is exponent

Properties of Logs Properties of Exponents Ex) Ex) Ex) Ex) Ex) Ex)

Inverse Properties: Ex) Ex) Special logs: Special Values:

Round 5 – Alg 1: Ratio, Proportion or Variation If the ratio of x to y is 3 : 4, then Cross multiply to solve proportions Ratios can be reduced the same as fractions: 25 is to 100 as 1 is to 4.

Direct Variation Inverse Variation Joint Variation “y varies directly as x” Direct variation is a line that intersects the origin (0, 0) and has slope (or constant of variation), k. Inverse Variation “y varies inversely as x” To solve variation problems, use the initial values of x and y to find the constant of variation, k. Then substitute k back into the equation. Joint Variation “z varies jointly with x and y”

Round 6 – Plane Geometry: Polygons (no areas) Vocabulary: Midpoint, segment bisector, segment trisector, angle bisector, perpendicular, altitude, etc. Sum of interior angles of a polygon with n sides: For a regular polygon with n sides, each Interior angle of the polygon would be: Know your shapes and their properties: square, rhombus, rectangle, triangle, quadrilateral, etc.

Length of the sides of a Tangential (inscribed) polygon (a polygon in which each side is tangent to a circle) For a tangential polygon with an even number of sides, if you number the sides consecutively, the sum of the even sides is always equal to the sum of the odd sides. a In this example, a + c + e = b + d + f f b e c d