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Answers
G2.1) In a triangle, the interior angles add to 180 degrees. What do the interior
angles in a hexagon (6 sides) add up to?
Not 121
N3.1) The regular decimal (base ten) number 69 expressed in base 7 is 126Base seven , or
1×72+2×71+6×70=49+14+6. What is the decimal number 199 converted to base seven?
In other words,
199Base ten = ________ Base seven
Ar4.1) Take a number, raise it to the third power, add five, double it, and then take the sixth
root. You get 2. What is that original number plus one?
Al5.1) What is |-5| – 3 · |-1| - (-1) ?
G2.2) A certain polygon has three times as many diagonals as sides. A diagonal in a polygon
is any line segment that connects two vertices and is not a side. What is the number of
diagonals in this polygon?
N3.2) In base two, also known as “binary”, you can use the digits 0 and 1. Using three digits,
the number 111Base two is the same as 7Base ten (22+21+20=4+2+1=7). What is the biggest
number you can express in binary with six 1 digits and one 0 digit? Give your answer in base
ten.
Ar4.2) What is 5-2 · 103 ÷ 22 ?
Hint: 10=2·5
1
Al5.2) (Hard) With a line graph for G like this:
If |X-5| < 2 draw the line graph of possible values of X:
2
3
then we know 1 ≤ G < 3 .
G2.3) Priyanha takes her robotic dog Sam for a walk. They go East 8 meters and then North
N meters. Sam is now 17 meters from the starting point. What is N?
5 × 106 ⋅ 3 × 10 −8
N3.3) What is
? Give your answer in scientific notation.
1 × 10 − 4
Ar4.3) What is the average of the integers that are greater than the cube root of 2007 but less
than the square root of 207?
Al5.3) If you know that V is an integer, then for how many different values of V can
9
also be an integer?
V +1
A
G2.4) In this figure, lines that look straight are straight,
and angles that look like right angles are right angles.
The drawing may not be to scale, however. If segment
FB is 15, and segment EF is 17, and segment AD is 10,
what is the length of segment ED? Express your answer
as a mixed number in simplest terms.
E
F
D
B
C
N3.4) The three problems below are all written in the same base. What is the
value of the base?
23 + 42 = 65
53 − 35 = 15
23 × 4 = 125
Ar4.4) Evaluate the expression below. Express your answer as a mixed number
in simplest form.
5
 3  9 
−2
  +   + 8  − 5 ⋅ (10)
9
 2   16 
−1
Al5.4) If x =
2
0
12
and k is an integer (integers can be positive or negative), what
k
values of k make x > 3 ? List all the values in increasing order.
Genius Questions (try all previous questions first.)
6.1) Alicia has a red shirt and a red hat, Brittany has a blue shirt and blue hat, and Caroline
has a pink shirt and pink hat. They agree to shuffle hats so that just one girl has a hat that is
the same color as her shirt. How many ways can they arrange the hats to do this?
6.2) How would you write the year 2007 as a base 5 number?
That is, 2007Base ten = ______Base five
.
6.3) The supermarket sells cookies in packages of 7 or 3 cookies per bag. You can’t buy just
one cookie, but if you wanted 10 cookies, you could buy one bag of 7, and one bag of 3.
How many positive integer numbers of cookies are there that you could not buy from this
store by combining zero or more of each kind of package?
6.4) (Very hard) You have 7 circles arranged in a hexagon as shown. Each
circle could be colored white or black. How many different arrangements of
colored circles are there? For example, all white; the center black; all
black. Rotations and/or reflections don’t count as different, so having
black just in the upper left but white everywhere else is the same as having
black just in the lower right.
6.5) If a hen and a half lays an egg and a half in a day and a half, how many eggs could 4 hens
lay in 3 days?
6.6) If 5x = 36, what is the square root of 5x+2 ?
Answers 01/2007 Try-out:
G2.1) 720° Four triangles of 180°
N3.1) 4037. 4x49+3=199
Ar4.1) 4
Start with 3: 33+5=32. 32x2=64.
Al5.1) 3
|-5| - 3·|-1| - (-1) = 5 – 3·1 +1 = 3
G2.2)
N3.2)
Ar4.2)
Al5.2)
6
64 = 2
27
126
10
3<x<7
G2.3)
N3.3)
Ar4.3)
Al5.3)
S(S-3)/2 = 3S, S(S-3) = 6S, (S-3) = 6, S=9. Diagonals=9(6)/2=27.
111 11102 = 64+32+16 + 8+4+2 (or 128-2)
103 / (52 · 22) = 103 / 102 = 10
(Hard) First, assume X-5ù0 or 5≤X (so we can ignore absolute value).
Now you get X-5 < 2, or X < 7, so we have solutions: 5≤X<7
Second, assume X-5<0 or X<5 so we can negate it and drop absolute value:
-(X-5) < 1, or –X+5 < 2, or –X < -3, or X>3, or 3<X<5
Now combine these two: 3<X<7 Easier: Assume for a moment |X-5| = 2.
Solve, giving X=3 or 7. Then try values
<3, between 3 and 7, and >7
15
152 + 82 = 172 = 289
1.5 x 103
13.5 123=1728, 133=2197, 142=196, 152=225. So, only 13 and 14 are in that range.
6
(Hard) Denominator must be 1, 2, or 4. V+7 can be -1, -2, -4, or +1,+2,+4.
G2.4)
N3.4)
Ar4.4)
Al5.4)
5
all triangles are similar to a 8-15-17 triangle, so sides have same proportions
7
Use 2nd problem
5
3,-2,-1, 1,2,3
Genius
6.1) 3
Each girl takes a turn being the only one with same color.
6.2) 310125.
Divide 2007 by 5: Quotient = 401, Remainder = 6. 6 is the low digit
of the answer. Divide 401 by 5: Q=80, R=1. 1 is the next digit of the answer. 80/5=16 R 0,
so the next digit is a zero, and 16/5=3R1, so the next two digits are 1 and then 3: 31012.
6.3) 6
The highest you can’t buy is 7x3-7-3=11. The numbers you can’t buy:
1,2,4,5,8,11, but everything else is OK. Thus there are 6. Once you get three in a row such as
12,13,14, then all numbers after that are possible, by adding 3’s.
6.4)
26
6.5)
8 eggs.
6.6)
30
0 black: 1 arrangement.
1 black: 2 arrangement (center, edge)
2 blacks: 4 arrangements: 1c1e, 2e together, 2e sep by 1, 2e sep by 2
3 blacks: 6: 1c2e*3, 3e:111,1011,10101. 4-7: 6 4 2 1
Total = 1+2+4+6 + 6+4+2+1
Each hen can lay one egg in 1.5 days
So in 3 days, one hen could lay 2 eggs.
Four hens could lay 8.
5x+2 = 5x·52 = 36 · 52 = 62 · 52 = (6·5) 2, take sq root: 30