Download Gryphon Gladiatorial Mathematics

Gryphon Gladiatorial
Mathematics
practice problems
Rules
1. Teams of 4 students will compete against one another
to answer mathematical problems and puzzles rapidly
and accurately.
2. One player from each team will work each problem on a
white board or chalk board at the same time with points
for the problem accruing to the team that produces a
correct answer first.
3. Answers must be written on the board and circled before
the referee can judge them.
4. Team members may help verbally as long as the next
step or exact answer is not given. Violation of this rule
is a foul that awards the points for the problem to the
other team.
5. Each team member must come to the board once before
any team member returns to the board.
6. Ranking of teams is by total points earned.
7. Problems come in categories with stated point values
and the teams chose categories in alternation with a
coin flip determining first choice.
8. The referee may always require a contestant to explain
their reasoning or show their work. A correct answer
only counts if it is well supported in the opinion of the
referee. Guessing answers may be cool but it is not
mathematics.
For the teacher.
The following sheets contain two problems per
sheet. They need to be cut, possibly ten sheets at
a time with a good paper cutter. The questions
are arranged on a table not easily visible from the
perspective of the students.
You do not need to use all the categories in a
given practice session - choose several categories
appropriate to your students and arrange them by
topic and point value.
The actual questions at the Olympics will be different and some of the categories will be different.
These categories are intended for practice for both
junior and senior students.
√
Calculators are not used: an answer like 8 is
left in that form and most of the answers will work
out to whole numbers or other values that can
be computed without calculators. Many of the
questions require that students recognize patterns
rather than compute numerical values.
Trigonometry 1
Find the tangent of θ if:
6
!
11
6
Answer: 11
Trigonometry 2
Find H if:
H
Answer:
!
5
12
52 + 122 =
√
169 = 23
Trigonometry 3
Suppose that sin(θ) = 0.4, what is cos(θ)?
"
!
cos2(θ)
Answer:
= 1−
√
√
1 − 0.16 = 0.84 .
sin2(θ)
"
so
1 − 0.42 =
Trigonometry 4 Suppose that sin(θ) =
0.337, what is cos(φ)?
"
!
Answer: cos(φ) = sin(90 − φ) = sin(θ) = 0.337
Trigonometry 1
What are the angles of an equilateral triangle?
!
!
!
π, π
Answer: 60,60,60 (degrees) or π
3 3 3
(radians).
Trigonometry 2
suppose that X = 8 what are Y and Z?
Z
Y
45
o
X
√
Answer Y = 8, Z = 8 2 .
Trigonometry 3
Give an example of three numbers that could be
the side lengths of a triangle that is neither and
equilateral or a right triangle.
Answer: any three positive numbers a ≤ b < c, not
all equal, with a2 + b2 $= c and a + b ≥ c.
Trigonometry 4
√
suppose that X = 5 3 what are Y and
Z?
Z
Y
30 o
X
Answer Y = 5, Z = 10 .
Trigonometry 1
If A = 67◦ and C = 22◦ then what is B?
A
B
C
Answer: 180 − 67 − 22 = 91◦
Trigonometry 2 If A > B > C then which of
a, b, or c has the smallest cosine?
a
C
B
b
A
Answer: a .
c
Trigonometry 3 Demonstrate the following
is or is not a right triangle:
(4,4)
(2,2)
(7,2)
Answer: It is not . The slopes of the sides are 0,
and so no slope is a negative reciprocal
1, and −2
5
of the other. This means none of the three angles
are right angles.
Trigonometry 4
Can you have a right equilateral triangle
in the plane? If not, say why, if yes, say
where.
Answer: No 3 × 90 = 270 > 180 and every triangle in the plane has angles summing to 180 degrees.
Patterns 1
What comes next?
1, 8, 15, 22, 29, 36, ?
Answer: 36+7= 43 .
Patterns 2
What comes next?
1, 3, 7, 15, 31, ?
Answer: (double and add one) 63 .
Patterns 3
What comes next?
2, 4, 6, 10, 16, 26, ?
Answer: 16+26= 42 .
Patterns 4
What comes next?
2, 3, 5, 9, 17, 33,
Answer:(double and subtract one) 65 .
Patterns 1
What comes next?
1, 2, 3, 4, 5, 6, 7, ?
Answer: 8 .
Patterns 2
What comes next?
1, 4, 9, 16, 25, 36, ?
Answer: (sequence of squares) 49 .
Patterns 3
What comes next?
1, 2, 4, 5, 7, 8, 10, 11, ?
Answer: add 1, add 2, repeat,
11+2= 13 .
Patterns 4
What comes next?
1, 3, 6, 18, 36, 108, 216, ?
Answer:
648 .
(triple then double,
repeat)
Patterns 1
What comes next?
1, 3, 2, 4, 5, 7, 6, 8, ?
Answer: two odds, two evens and repeat,
always increasing = 9 .
Patterns 2
What comes next?
2, 6, 18, 54, 162,?
Answer: triple the previous number
3*162= 486 .
Patterns 3
2, 3, 4, 8, 9, 16, 27, ?
Answer: powers of 2 and 3 32 .
Patterns 4
Where does 8 go?
1 → 4, 2 → 1, 3 → 10, 4 → 2,
5 → 15, 6 → 3, 7 → 22, 8 →?,
Answer:
The hailstone sequence.
8 → 4 Even numbers are divided by two,
odd numbers you triple and add one.
Patterns 1
Fill in the square so that there is an A, B,
C , or D in each square but never the same
thing twice in the same row or column.
A B D C
B C
Answer, e.g.
A D
D
A C B
C
D B A
Patterns 2
Where does the missing star go?
* . . . . . .
. . . . * . .
. . * . . . .
. * . . . . .
. . . . . . .
. . . . . . *
. . . . . * .
Answer: Row 5, Column 4
Patterns 3
Each symbol may appear at most once in
a row or column. Fill in the blanks!
A
B
C
B
D
A
C
D
D
B
D
C
A
B
C
D
B
D
A
C
C
A
D
B
D
C
B
A
A
Answer:
Patterns 4
Which does not belong?
Up, Down, Left and Right.
1.
2.
3.
4.
5.
URDRUULL
RULURRDD
UUURDDLU
UULLDDLL
UURDDRUU
Answer: Number 4 , it does not cover a
3x3 grid.
Arithmetic 1
Compute: 8-7+6-5+4-3+2-1=?
Answer: 4 .
Arithmetic 2
26 × 35
=?
3
2
4 ×9
Answer: Cancel powers of 2 and 3 to get
3.
Arithmetic 3
Simplify the fraction
6×7
.
2
1
2 +2 +1
Answer: 6
1 or 6 .
Arithmetic 4
Find a positive number so that the reciprocal of the number plus two is equal to
the number.
√ ∼
Answer: 1 + 2 = 2.4142136 If 2 + 1
x =
x then x2 − 2x − 1 = 0 and use the
quadratic formula.
Arithmetic 1
Compute: 1+2+4+8+16=?
Answer: 31 .
Arithmetic 2
45 × 37
=?
10
2
2 ×9
Answer: Cancel to 33 = 27 .
Arithmetic 3
Simplify the fraction
1+2+3+4+5+6+7+8
.
4×5×6
3 .
Answer: 10
Arithmetic 4
((1 + 2) × 3)4 =?
Answer: 6561
Arithmetic 1
1 + 2 + 2 + 3 + 3 + 3 + 4 + 4 + 4 + 4 =?
Answer: 1+4+9+16= 30 .
Arithmetic 2
Compute: If we multiply 2 by itself four
times and subtract one what is the largest
prime divisor of the resulting number?
Answer: 24 − 1 = 15 = 3 × 5 so 5 .
Arithmetic 3
Simplify
24 − 22
=?
25 − 23
Answer:
cancel)
1/2 .
(factor out a 2 and
Arithmetic 4
Find the largest prime divisor of 63 − 1
Answer: 63 − 1 = 215 so 5 .
Geometry 1 Suppose that we draw
four line segments, what is the largest
possible number of intersections?
Answer:6 See above diagram.
Geometry 2
What is the smallest value that the largest
angle in a quadrilateral can take on?
Answer: 90◦ or π
2 radians, as in a square
or rectangle.
Geometry 3 If one angle of a isosceles
triangle is 30◦ when what other values can
its angles take on?
Answer: 120◦ or 75◦ depending if there
are one or two 30◦ angles. Both answers
are required.
Geometry 4 The points (1,2), (3,0),
(5,2), and (3,4) are:
a)
b)
c)
d)
e)
A triangle.
A non-square rectangle.
A non-rectangular quadrilateral.
A square.
None of the above
Answer: d, a square.
Geometry 1 If the grid is made of 1 × 1 cm
squares, find the area of the shape:
8
6
4
2
0
0
2
4
6
8
Answer:Base 6 (vertically), height 4,
1
6
2
× 4= 12cm2
Geometry 2 If the grid is made of 1 × 1 cm
squares, find the area of the shape:
8
6
4
2
0
0
2
4
6
8
Answer: break into 7 2 × 2 rectangle for a total of
28cm2 .
Geometry 3 If the grid is made of 1 × 1 cm
squares, find the perimeter of the shape:
8
6
4
2
0
0
2
4
6
8
Answer: Add it up, 32cm2 .
Geometry 4 If the grid is made of 1 × 1 cm
squares, find the area of the shape:
8
6
4
2
0
0
2
4
6
8
Put the diamond in a 4 × 6 rectangle and subtract
the right triangle corners. Area is 12cm2 .
Geometry 1
If angle a is 44◦ then what is angle b?
b
a
Answer: 180 − 44 = 136◦.
Geometry 2 Give points with wholenumber coordinates (a, b), (c, d), (e, f ) so
that the points are the vertices of a triangle with area 8.
Answer: E.g.: (0,0) (0,4) (4,0), check
the answer directly.
Geometry 3 Suppose L is a line through (1,3)
and (3,9) while M is a line through (1,5) and
(2,6). Demonstrate that L and M are or are not
parallel.
Answer: 9−3
= 3 while 6−5
= 1 so the lines have
3−1
2−1
different slopes and are not parallel.
Geometry 4
Suppose a rectangle can be cut in half to yield
two rectangles exactly the same shape but with
half the area:
What is the ratio of the longer side to the shorter
side?
Answer
√
2.
Algebra 1
Solve for x
3x + 1 = x + 7
Answer: x = 3
Algebra 2
Solve BC + CD − AB = 0 for C.
AB
Answer: C = B+D
Algebra 2 How many real solutions does
x2 − 3x + 5 = 0
have? You may only give an answer once.
Answer: Compute the discriminant
(−3)2 − 4 × 1 × 5 = −11,
so it has no real solutions.
Algebra 4 Find a, b, c all not zero so
that a2 + bx + c = 0 has no solutions.
Answer: Any values so that b2 < 4ac.
Algebra 1
Solve for x:
3x + 5 = 8
Answer: x = 1 .
Algebra 2
Compute and simplify:
(2 − 2x)(4 − x)
Answer: 2x2 − 10x + 8
Algebra 3
Solve for y:
x=
y+1
y−1
Answer: y = x+1
x−1 .
Algebra 4
Compute and simplify:
(x2 + x + 2)2
Answer: x4 + 2x3 + 5x2 + 4x + 4 .
Algebra 1
If ab + ac − bc = 3 solve for a.
Answer: a = 3+bc
b+c
Algebra 2
What is the value of
(x − 1)(x + 1)(x + 2)
if x = 3?
Answer: 2 × 4 × 5 = 40
Algebra 3
Suppose that y = xn + axn−1 + · · ·+
bx + c. If the graph of this function is:
Y
X
then what is the smallest value n could have?
Answer: 4 , this is the shape of a quartic (4thpower) curve.
Algebra 4
In
(x2 − x + 3)2
what is the coefficient of x2?
Answer: 7 .