Just for kicks, test your math IQ

[GSTamral]

Ok folks, here's a quiz I took today at work (this is a sample ASHME secondary exam from 1999 for the US mathematics Olympiad. Scoring as follows: 10 points for a correct answer, 0 points for a blank, -3 points for an incorrect answer. 20 minutes to complete the test. Scoring multipliers:
Less than 5 minutes: 1.4
5-8 minutes: 1.25
8-12 minutes: 1.1
12-16 minutes: 1.0
16-20 minutes: .9

No textbooks, and only a non-programmable calculator is allowed. I scored a 109 on this thing (9 correct, 1 incorrect, 6m 32s). I'll post the solutions with work sometime later.

The average score for this exam is 79. A score of 100 or greater is needed to qualify to advance (typically it takes 105 or better to move on) However, it is important to note that only people who score above 100 on the original ASHME exam as well as a minimum of 7 correct answers on the secondary exam will take this test (4% of the original test takers, who are supposedly the in the top 10%). In parenthesis by each is the expected percentage of people to get the question correct.

1) What is the maximum value of change you can have in your pocket (quarters nickels dimes and pennies only) and still not be able to make exact change for a dollar?
(92%)

2) If you flip a coin 15 times in succession, what is the probablity that you end up with exactly 6 heads and 9 tails?
(81%) (Lo and behold due to a multiplication error, this is the one I got wrong)

3) A perfectly spherical balloon is being filled with air at the rate of 2 cubic feet/minute. At what rate is the radius of the sphere increasing at time t=6 seconds?
(97%)

4) You are playing a two player game called "50". In this game, a player starts by calling out a number from 1 to 6. Then player B calls out a number from 1-6 and they continue. These numbers are added in succession. A player wins the game by calling out a number that makes the total equal to 50.
(44%)

A sample game as follows:
Total
Player A: 5 5
Player B: 5 10
Player A: 6 16
Player B: 3 19
Player A: 4 23
Player B: 1 24
Player A: 4 28
Player B: 6 34
Player A: 6 40
Player B: 4 44
Player A: 6 50 Player A wins.

You are player A: In order to garauntee success, picking first, what is the correct number to select?
You are player B: Player A has started the game and selected the number 3. What number should you choose to garauntee success?

5) Two cars start out at the same point. At time t=0, car A starts driving south, accelerating at 10 meters / second squared. Car B starts driving west, accelerating at 24 meters per second squared. When the two cars are 52 meters apart, at what rate (velocity) are they increasing the distance between themselves?
(95%)

6) You have 100 feet of barbed wire. What is the maximum amount of space you can enclose?
(94%)

7) What is the first non-prime number that is not divisible by any of the numbers between 2 and 20?
(29%)

8) Given that the sum of a series f(n+1)(x) = r * f(x), where r is between 0 and 1, where f(0)(x) = 1 is 1/r, what is the sum of the following infinite series?
1, -1/2, 1/4, -1/8, 1/16, -1/32………
(82%)

9) You are rolling two dice. What is the probability that 6 consecutive rolls of the dice will add up to be 6 or less?
(96%)

10) (this was really question number 4, but I saved it for last because only 9% got it right, and it is quite the doozy for a 20 minute test):
Maximize the following:

6A + 3B + 5C - 2D

Using the following constraints:
A, B, C, D all greater than or equal to 0
2A - 3D + C <= 0 (less than or equal to)
B + 3C <= 12
A + B + C + D <= 6
A + C <=3

What is the maximum value? What are the values of A,B,C and D?

[GSTamral]

sorry, must clarify something here, in question 8, the sum is not 1/r but rather (1/(1-r)), typo there

[Jorddyn]

Eleventeen.

What do I win?

Jorddyn

[GSTamral]

Also, here are the answers without the work, for those of you wishing to try and check your answers.

1) 1.19 (3quarters, 4 dimes, 4 pennies)
2) 1/3003
3) 1/(6 * (pi^1/3))
4) a)1, b)5
5) 52 meters/second
6) 2500/pi square feet
7) 529
8) 2/3
9) (5/12)^6
10) A=0,B=3,C=3,D=1, function = 22

[GSTamral]

&*()&*)_()_(+_ dammit. I misreported the answer to question number 2, once again showing my amazing ability to do math while on a conference call.

The correct answer for number 2 is
5005/35736, not 1/3003.

[Latrinsorm]

I only got like 4 right. The wording is pretty atrocious. For instance, in number 7, the world "inclusive" would be nice. I still don't know what number 8 is trying to say. The only thing I can think of is it's trying to talk about derivatives or something with those two seperate arguments.

The rest I got wrong because I'm bad with probability and games that don't involve a baseball.

[GSTamral]

Sorry for the wording. I was paraphrasing some of the wording from what I remembered, because I don't have the test in front of me anymore.

[Latrinsorm]

Originally posted by GSTamral
Sorry for the wording. I was paraphrasing some of the wording from what I remembered, because I don't have the test in front of me anymore. No problem. :)

[GSTamral]

ok, now that I am home from work and have some time, here's the work for each problem.

1) 3 quarters, 4 dimes, 4 pennies, no other work required.

2) Probability Law

(15!/(6! * 9!)) * .5^15

3) V = 4/3 * pi * r^3
DV/DT = 2 = 4 *pi* r^2 * dr/dt
after 6 seconds, V = 2* 6 = 12
Solve for R, using V =12, then solve for dr/dt using the obtained V = 12, yielding r = (9/pi) ^ 1/3, plug that r into
2 = 4 * pi * r^2 dr/dt and solve for dr/dt

4) rule of 7. If you can make your opponent call a number when the total =43, you can always make 50. Thus, if he calls 6 you call 1, at the stated pattern, 43,36,29,22,15,8,1

By starting with 1, you can always make the total equal to 8 with the next number, and so forth.

b) same reasoning, make the total =8

5) Right triangle law.
At t =2, a =20, da/dt = 20
b=48, db/dt =48
c = 52

ada/dt + bdb/dt = cdc/dt

dc/dt = (20*20 + 48*48)/52

6) you make a circle with circumference 100/pi
radius = 50/pi
area = pi * r *r = 2500/pi

7) first prime number after 20 is 23. 23 squared is 529.

8) this reduces to the following:

1/2 + 1/8 + 1/32, or

1/2 * (1 + 1/4 +1/16...)

or 1/2 as the front constant and r = 1/4

1/2 * 1/(3/4) = 2/3

9) probability in one roll of rolling 6 or below = 15/36 = 5/12
6 in a row = (5/12) ^6

10) this one is nasty.

Using the ratio A = 6/5C, the first equation eliminates A, set A to 0
Solve equation 1 using the parameter A + C < 3, set C to 3. D then equals 1.
Solve paramter A + B + C + D < 6. Set B to 3. Ensure it falls under the constraint of B and C. Solve.

Alternatively, you can solve this as a second order Markov Matrix, or using the Big M method, however, there is no way to do that in under 20 minutes, unless you can write 3 pages worth of work and calculations that quick.

[Mistomeer]

Originally posted by Jorddyn
Eleventeen.

What do I win?

Jorddyn

Eleventy billion dollars.

[imported_Kranar]

<< 1) 3 quarters, 4 dimes, 4 pennies, no other work required. >>

3 quarters, 9 dimes, 4 pennies is the maximal set. That gives you $1.69 and the set can not be decomposed to a value of 100.

<< 7) first prime number after 20 is 23. 23 squared is 529. >>

1 is the first number that is indivisible by any number in the set 2 <= x <= 20.

1 is not a prime number, although many people confuse it for one.

[Faent]

"1" was prime for Goldbach.

-Scott

[GSTamral]

<<<
<< 1) 3 quarters, 4 dimes, 4 pennies, no other work required. >>

3 quarters, 9 dimes, 4 pennies is the maximal set. That gives you $1.69 and the set can not be decomposed to a value of 100.

<< 7) first prime number after 20 is 23. 23 squared is 529. >>

1 is the first number that is indivisible by any number in the set 2 <= x <= 20.

1 is not a prime number, although many people confuse it for one.
>>>

1) uhhhh, 2 quarters and 5 dimes = 1 dollar.

Number 7 actually should have also included >20, yes. I thought it was implied in the number would have been greater than 20. None of these questions are "trick" questions.