Download McCall 05

Answers
G2.1) In a triangle, the interior angles add to 180 degrees. What do the interior
angles in a pentagon add up to? Hint: You can divide a pentagon into triangles.
N3.1) In base 3, you count like this: 1, 2, 10, 11, 12, 20, 21, 22, 100. That’s 9 numbers to
count from 1 to 100. How many numbers would it take to count to 100 in base 5? That is,
count up to 1005 (not decimal 100). Hint: Start with 1,2,3,4,10 and end with 44,100.
Ar4.1) What number, if you square it and then take the cube root, gives you 4?
Al5.1) What is |-6| + 3 · |7| ?
G2.2) [From Jan’05] A certain polygon has twice as many diagonals as sides. How many sides
are there in this polygon? Note: A diagonal in a polygon is any line segment that connects
two vertices and is not a side.
Hint: The number of diagonals in and S-sided polygon is S(S-3)/2. Make an equation and solve.
N3.2) (Hard) In base two, using 3 binary digits, the number 1112 is the same as 7 in base ten
(22+21+20=4+2+1=7). What is the biggest number you can express in a byte, where a byte is
8 binary digits? Give your answer in base ten.
Ar4.2) What is 3-2 · 63 ÷ 22 ?
Hint: 6=2·3. Answer is not 96.
1
Al5.2) (Hard) With a line graph for G like this:
If |X-3| < 1 draw the line graph of possible values of X:
2
3
then we know 1 ≤ G < 3
Hint: X=3.1 works.
My Name: __________________
G2.3) Priyanha walks her robotic dog Aibo East 20 feet and then North N feet. Aibo is now
29 feet from the starting point. What is N? Hint: N is an integer. Remember Pythagoras’ theorem.
N3.3) What is 3 x 104 · 4 x 10-5 ÷ (1 x 10-2)? Give your answer in scientific notation.
Hint: x and · both mean “times” here, and are completely equivalent.
Ar4.3) Bob Brainiac is so smart that if you took his IQ and divided it by two, then added 49,
took the square root, and then reversed the digits, and then added one, and then took the fifth
root, you would get two. What is Bob’s IQ?
Al5.3) If you know that V is an integer, then for how many different values of V can
4
also be an integer? (Hint: Don’t list the values, just say how many there are.)
V +7
Better hint: Imagine D=|V+7|. How many values of D work?
A
G2.4) (Very Hard) 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 8, and segment EF is 10, and segment
AD is 4, what is the length of segment BC? Express your answer
as a mixed number in simplest terms.
F
D
E
B
N3.4) Solve this base 8 equation for c. Write your answer in base 8. All values are in base 8.
2(C+12) +4 = 232
Hint: Base 8 32-4 is not 28
Ar4.4) How many whole numbers are there between the cube root of 100 and the cube root of
2006? (Hint: You don’t need to know what those cube roots are; just count perfect cubes.)
Al5.4) Evaluate:
|-6| + 3·|7| – 4·|-2| + 5·|0|
C
Genius Questions
6.1) Alicia has a red shirt and a red hat, Brittney has a blue shirt and blue hat, and Caroline
has a pink shirt and pink hat. They agree to shuffle hats so that at least one girl has a hat that
is a different color from her shirt. How many ways can they arrange the hats to do this?
6.2) How would you write the year 2006 as an octal (base 8) number?
That is, 200610 = ______8
.
6.3) The supermarket sells cookies in packages of 3 or 5 cookies per bag. You can’t buy just
one cookie, but if you wanted 11 cookies, you could buy one bag of 5, and two bags 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) You have 4 coins arranged in a square as shown. Each coin shows either
Heads or Tails. How many different arrangements of the coins are there if all
you do is flip coins over but leave them in the square? A rotation doesn’t count
as different, so having a Head in the upper left but tails everywhere else is the
same as having a Head in the lower right and tails everywhere else.
H
H
T
T
6.5) If a hen and a half lays an egg and a half in a day in a half, how many eggs could 4 hens
lay in 3 days?
6.6) If 11x = 64, what is the cube root of 11x+3 ?
Answers 1/12/2006 Try-outs:
G2.1) 540° [Not: 600, 360, 300, 300, 900]
N3.1) 25
(1 2 3 4 10, 11 12 13 14 20, 21 22 23 24 30, 31 32 33 34 40, 41 42 43 44 100)
Ar4.1) 8
Al5.1) 27
G2.2)
N3.2)
Ar4.2)
Al5.2)
7
255
6
2<x<4
G2.3)
N3.3)
Ar4.3)
Al5.3)
S(S-3)/2 = 2S, S(S-3) = 4S, (S-3) = 4, S=7
(Hard) 128+64+32+16+8+4+2+1 (or 256-1)
63 / (32 · 22) = 63 / 62 = 6
(Hard) First, assume X-3ù0 or 3≤X (so we can ignore absolute value).
Now you get X-3 < 1, or X < 4, so we have solutions: 3≤X<4
Second, assume X-3<0 or X<3 so we can negate it and drop absolute value:
-(X-3) < 1, or –X+3 < 1, or –X < -2, or X>2, or 2<X<3
Now combine these two: 2<X<4 Easier: Assume for a moment |X-3| = 1.
Solve, giving X=2 or 4. Then try values
<2, between 2 and 4, and >4
21
202 + 212 = 292 = 841
1.2 x 102
.Not 12x101, or 1.2x101.
240
start with 2 and reverse: 2 32 31 13 169 120 240
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)
8¼
101
8
19
all triangles are similar to a 3-4-5 triangle, so sides have same proportions
Subtract 4 from each side: 2(C+12) = 226. Divide by 2: C+12=113. Subtr 12.
125=53, 216=63, 73, 83, 93, 103, 113, 123 =1728 (but 133 > 2006)
6+21-8
Genius
6.1) 5
(Hard) Each person can keep the same hat, while the other pair swaps. That’s
3 ways. Or, each person could pass their hat to the left, or pass their hat to the right. 2 more
ways, for a total of 5.
6.2) 37268.
Divide 2006 by 8: Quotient = 250, Remainder = 6. 6 is the low digit of
the answer. Divide 250 by 8: Q=31, R=2. 2 is the next digit of the answer. 31/8=3 R 7, so
the next two digits are 7 and then 3.
6.3)
4
You can’t buy just 1 or 2 cookies, or 4 or 7, but everything else is OK.
Once you get three in a row, such as 8-9-10 (8=5+3, 9=3+3+3,10=5+5)
Then all numbers after that are possible, by adding 3’s to 8, 9 or 10.
6.4)
6
6.5)
8 eggs.
6.6)
44
0 heads: 1 arrangement.
1 head: 1 arrangement.
2 heads: 2 arrangements, with heads side-by-side or diagonal.
3 heads: 1.
4 heads: 1.
Total = 1+1+2+1+1 = 6
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.
11x+3 = 11x·113 = 64 · 113 = 43 · 113 = (4·11) 3, take cube root: 44