Download NO CALCULATORS! You have 40 minutes to complete 30 questions

NO CALCULATORS! You have 40 minutes to complete 30 questions . Start no later
than 7:10. You must have submitted your answers to me by PM by 7:50. Stay tuned for
the targets afterwards!
1. Solve: x x =x−x
2. If you sell fine vegetables for $1.50 per pound, you will sell 200 pounds a day. Every
dime the price is increased, you sell 8 pounds less. If you want to maximize your
earnings, how much should you sell your vegetables for? Assume a unlimited supply.
3. How many integral triples (a,b,c) have the property that a<b<c, a=bc, b=ac, and c=ab?
4. Use the pattern below to express 250 2 in the form a 2b 2−c 2 :
122=8292−12
222=18214 2−6 2
322=282192−112
5. The digits 1 through 12 are arranged, one per circle, in the triangle shown such that the
sum of the numbers on each side is the same. What is the range of the possible sums?
6. My solar system contains 7 planets and 1 sun. The planets take 3,5,7,11,15,35,and 37
months to make a complete revolution. If on New Year's Day the planets and the sun
are in a straight line, how many years will it be before they are in a line again on New
Year's? Assume 13 months per year.
7. Find angle A:
8. If there are 53 districts in a competition, and girls have a probability of 2/5 of winning
a district, but only a 1/10 chance of winning the whole country, what is the probability
that a boy will win the country?
1
1
3
9. Find the sum: 160 16 4 16 2 16 4 ...163
10.Find the quotient of 2 numbers whose sum is 10 and whose product is as large as
possible.
11.When I am 35 years older than I am now, I will be the square of my age divided by 4.
How old am I?
12.Find the sum of the x-coordinates of all possible positive integral solutions to
1 1 1
 =
x y 7
13.What is the average of the mean and median of the answers to the previous 12
questions?
14.How many positive 3-digit integers are increased by exactly 99 when the digits are
reversed?
15.A school has x switches and x students, all numbered. Each student goes down the hall
with the switches and changes every switch that their number divides. For example,
student 1 opens all of the lockers, student 2 closes all of the even-numbered switches,
and so on. If after all of the students have gone down the hall, there are 18 switches
open, what is the least number that x could be?
16.A frustrum is a cone with a smaller cone removed from the top. A water container is in
the form of a frustrum, with a top having a radius ½ the radius of the bottom. Halfway
up, the barrel is marked 200 gallons. What is the volume of the barrel?
17.What is the range of all values of x such that   x x ? Express rounded to the
nearest whole number.
18.Express in the form the number 2810  3 .
19.Assume Mr. R starts AoPS and he is the only member. He convinces 1 person to join
the first day, 2 the next day, 4 the next, and so on. Recently the 8192nd member joined.
How many days has it been?
20.What is the sum of the rational coefficients of the number  x35 ?
21. V has a favorite number, between 200 and 400, divisible by 5 and 3, which has a
remainder of 4 when divided by 7, and a remainder of 30 when divided by 120. What
is V's favorite number?
22.A man is running down the train track in a high canyon just wide enough for the train.
He hears a train approaching from behind at 100 miles an hour when he is 2/5 through.
Whether he runs forward or back, he will reach the end of the canyon at the same time
as the train. How long is the train?
23. Solve: x x =2
24.What is the fewest number of square tiles, 1x1, that are required to completely cover
this figure if when you cut a tile, you cannot use the part cut off?
25.Levi is enjoying the elevators at Disney. He decides that he will go up a floor if he
flips a head and down a floor if he flips a tail. He starts at floor 8. What is the
probability that he will be at floor 7 at the end of 5 flips?
26.Another person has taken Mickey hostage in a elevator, and has decided that he will
do the same coin-flipping as Levi. After 2 flips the police have arrived, and are
waiting at floor 8. What is the probability that Mickey will not be freed by the police at
the end of 7 flips? Mickey was taken hostage at floor 6.
27.There are about 3,000,000 words in the Igiliu language. If David knows 500,000 of
them and it is his only language, what is the probability that a word David knows is of
the Igiliu language?
28.What is the sum of all common prime factors of all four-digit number of the form
A00A?
29.Mr. Buzynezzmaan has a wall being built for him. Shown are the top rows. As you can
see, the top row has 2 tiles and each lower row has 3 more tiles than the one above. If
the structure is 15 tiles high, and a tile costs $2, how much did Mr. Buzynezzmaan pay
for the wall?
xx
x
xx
x...
30. What is the value of 5252−4752 ?
Wow! You got through this whole test! Was it easy? Hard? Stupid? Tell me!