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Math Team
Skills for
March Rounds
Round 1 – Alg 2: Simultaneous Equations and
Determinants
The determinant of a 2x2 matrix is determined using a formula
a b 
The determinant of matrix 

c
d


a b
Denoted:
 ad  bc
c d
Alternatively, the negatively sloped diagonal product
minus the positively sloped diagonal product
a b
c d
bc
ad
To find the determinant of a 3x3 or larger square matrix…
The sum of the negatively sloped diagonal products
minus the sum of the positively sloped diagonal products.
Negatively sloped
diagonal products
a b
d e
g h
c
f
i
bfg cdh aei
a b
d e
g h
Positively sloped
diagonal products
a b
d e
g h
ceg afh
c
f
i
bdi
c
f  ( aei  bfg  cdh )  (afh  bdi  ceg )
i
Solving Simultaneous Equations –
Reduce the number of variables then backfill.
Example: (2 equations, 2 unknowns)
2x  y  5
x  3y  6
Multiply both sides
of the top equation
by 3.
Add the two equations
and the y-terms will
cancel.
Solve for x.
3(2 x  y )  (5)3
6 x  3 y  15
x  3y  6
Substitute the known value
into either of the first
equation to find the other
unknown.
7 x  21
x3
2x  y  5
2(3)  y  5
6 y 5
y  1
Example: (3 equations, 3 unknowns)
x  2y  z  8
2x  y  z  7
x  2 y  z  6
1( x  2 y  z )  (8)( 1)
2x  y  z  7
 x  2 y  z  8
2x  y  z  7
x  y  1
2x  y  z  7
x  2 y  z  6
3x  y  1
x  y  1
3x  y  1
x  y  1
3x  y  1
2x  2
x 1
Backfill into
previous
equations to find
y and z.
Round 2 – Alg 1: Exponents and Radicals
Rules of Exponents
x m x n  x mn
xm
m n

x
xn
x n 
n
1
xn
m
y
x

y m x n
x0  1
x 
m n
 x mn
xy  x y
n

x
y
n
n n
x
a
b
xn
 n
y
If n is a positive integer and x is a nonnegative real number, then
x1/ n  n x.
Therefore, rules for radicals are the same as the rules for exponents.
Power
Root
Ex.) Simplify
a)
3
b)
5  52  5x 2  5(12 x 2)  5x 5
c)
3x 2 y 6 z 0 3x5 y 6 y 2  1 x 3 y 8 1 3 8


 x y
5 2
2
9x y
9x
3
3
d)

x 2y
 4
3
2
  27 
2 
3
 32 xy


1
3
27
2
3


2
 4
3

1
 27 
2
3
1
1 10
 8  2  1 
3
9 9
Matching Bases
Solving equations involving exponents often requires
matching the bases of the terms. If the bases are
matched, the exponents must be equal.
Ex.)
Solve
27 x 1  3  9 x  2
3

 3  3

3 
 3  3

3 x 1
3 x 1
1
1
2 x 2
2 x 2
33 x 3  31  32 x 4
33 x 3  32 x 5
3x  3  2 x  5
x 8
All the terms can be
expressed using base 3.
Power to a power
- Multiply (distribute)
Product of powers
- Add (comb. Like terms)
Bases are matched,
exponents are equal
Simplifying and Manipulating Radicals
To simplify a radical, we basically take the perfect squares
out. (Or perfect cubes for cube roots)
Ex)
18 x 9  9  2  x 8  x
 9  x 8  2  x  3x 4 2 x
But remember, if it is convenient, we can put numbers or
variables back into the radical.
Ex) Solve for a and b.
8 3  6  a 24  b
4  2  3  6  a 24  b
4  4  3  6  a 12  b
4  12  6  a 12  b
a  4, b  6
Round 3 – Trigonometry: Anything
• See Trigonometry Rounds and info from
previous months
Round 4 – Alg 1: Anything
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Solving equations
Proportions
Percent and mixture problems
The coordinate plane
Factoring
Solving rational equations
… just to name a few topics
Round 5 – Plane Geometry: Anything
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Regular polygons and their properties
Triangles
Transversals
Interior and exterior angle theorems
Circles
… just to name a few topics
Round 6 – Probability and the Binomial
Theorem
Using the Binomial Theorem for a binomial expansion.
To expand ( x  y )n , we use the nth row of Pascal’s
triangle to find the coefficients. The powers of x start
with n in the first term and decrease by one for each
successive term. The powers of y start with zero and
increase by one for each successive term.
Ex) ( x  y )
2
Row 0
Row 1
 1x 2  2 xy  1 y 2
Row 2
 x 2  2 xy  y 2
Row 3
Row 4
Row 5
Ex) Expand ( x  y )
Row 0
4
Row 1
1x  4 x y  6 x y  4 xy  1 y
4
3
2
2
3
Row 2
4
Row 3
x 4  4 x 3 y  6 x 2 y 2  4 xy 3  y 4
Row 4
Row 5
 2x  y 
1 2 x   3  2 x    y   3  2 x  (  y )
8 x  3  4 x   y   3  2 x  y  y
2 3
Ex) Expand
3
2
3
2
2
2
4
8 x  12 x y  6 xy  y
3
2
2 2
2
4
6
6
 1  y

2 3
Probability
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Independent events
Dependent events
Compound probability
Combinatorics
• Counting principles
• Combinations
• Permutations