Y by Adventure10, Mango247
Here are the results with a question-by-question analysis.
NOTE: If you have not taken the test and would like to, please see the other thread for the questions as this thread has the questions ALONG with the answers.
TOTAL NUMBER OF CONTESTANTS: 15
MOCK AMC I-
CREATED BY: Lucky707 and Singthesorrow
DATE ADMINISTERED: January 30, 2005
Rules:
(Standard AMC Rules)
1. You have 75 minutes to complete the test.
2. Only non-programmable calculators are allowed.
3. 6 points for a correct answer, 2.5 points for a question left unanswered and 0 points for an incorrect answer so it is in your best interest NOT to guess.
4. All other standard AMC Rules apply... We've had plenty of Mock AMCs so you should know the rules by now.
5. At 5:15, you must submit the test. If you do not, it will not count. I will allow a few extra minutes for slow modems, etc. but please submit the test at 5:15.
6. My AOL is hezgotzbeenz if anyone has any questions during the test.
1. The I.Q. of a Martian varies directly with the square of the number of eyes it has. If a Martian with 5 eyes has an I.Q. of 75, find the IQ of a Martian with 8 eyes.
(A) 120
(B) 128
(C) 144
(D) 192
(E) 200
Generally well done, those who missed this question did so because they didn't read it and multiplied 8 by 15 to get 120.
ANSWER: D
AVERAGE SCORE: 5.2 (13 correct/15 attempted)
2. The number of cubic feet in the volume of a cube is the same as the number of square inches of its surface area. What is the length of an edge of it expressed in feet? (NOTE: 12 inches = 1 foot)
(A) 6
(B) 36
(C) 72
(D) 108
(E) 864
Also fairly well done. Those who missed this one did not convert at some point down the line.
ANSWER: E
AVERAGE SCORE: 4.8 (12/15)
3. The first two terms of a geometric progression are
and the fourth root of three. What is the fifth term?
(A)
(B)
(C) fourth root of 3
(D) fourth root of
(E) tenth root of 3
This was somewhat of a trick question. Those who were tricked by it chose E but if we look at the common ratio and work it out as usual, we get the correct answer of A.
CORRECT ANSWER: A
AVERAGE SCORE: 4.8 (12/15)
4. A leaky faucet drips at a steady rate. If we measure from the instant the first drop hits the bottom, to the instant the last drop falls, it takes 36 seconds for 9 drops to fall. How many seconds would it take for 12 drops to fall?
(A) 42 seconds
(B) 44 seconds
(C) 48 seconds
(D) 49.5 seconds
(E) 54 seconds
This question was nasty. So many places where one could make an error. Here is the solution:
We measure from the instant the first drop falls to the instant the last drop falls. 36 seconds for 9 drops to fall means 36/8 seconds between drops hitting the ground. That's the first place one can make a mistake. The other place is multiplying by 12 instead of 11. The answer is 49.5
ANSWER: D
AVERAGE SCORE: 3.0 (7/14)
5. A triangle has sides of length 15, 20 and 25 units. It is inscribed in a circle. What is the circle's radius?
(A)
(B)
(C)
(D)
(E)
This question was very well done. After realizing that the triangle is a right triangle all one needs to do is divide the hypotenuse by 2 to get the answer.
ANSWER: D
AVERAGE SCORE: 6.0 (15/15)
6. How many zeroes are at the end of the product of the first fifty positive integers?
(A) 5
(B) 12
(C) 17
(D) 35
(E) 47
I don't think the average would be this high in real life for this type of question but I was sure that AOPSers could steamroll through this question... and steamroll they did!
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
7. If
, then what is the largest value of
such that
divides
evenly?
(A) 50
(B) 97
(C) 98
(D) 99
(E) 100
See comment above. 6 and 7 were quite similar actually. Nobody made an addition error so everyone got this right.
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
8. I have a red shirt, a blue shirt and a green shirt. I also have blue jeans, black jeans and khakis. How many possible combinations can I make if I absolutely hate wearing the blue shirt with the blue jeans?
(A) 5
(B) 8
(C) 14
(D) 16
(E) 17
This one was a gimme.
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
9. Let
be the smallest perfect square that has 140 as a divisor. Then
(a)
(b)
(c)
(d)
(e)
Wow... there were a lot of prime factorization questions early on. Realizing that 140 = 7*5*2^2.... we find that 7^2*5^2*2^2 will be a perfect square. And that value is 4900, making the answer E.
ANSWER: E
AVERAGE SCORE: 6.0 (15/15)
10. What is the sum of the first twenty powers of 2 (2^1 + 2^2 + 2^3 ... + 2^20)?
(A)
(B)
(C)
(D)
(E) None of the above
This question can easily be done by induction... or if you want to stay honest (which is not a good idea during the AMC)... geometric series.
ANSWER: D
AVERAGE SCORE: 5.2 (13/15)
11.
and
represent the same line.
and
can both be expressed as fractions in lowest terms. What is the numerator of
plus the denominator of
equal to?
(A) 212
(B) 144
(C) 72
(D) 48
(E) 24
Generally those who attempted this question got it right. One person got it wrong and I'm assuming it's because he forgot to reduce.
ANSWER: E
AVERAGE SCORE: 5.1 (12/13)
12. A square of side length 8 has its corners cut off to make a regular octagon. Find the area of the octagon to the nearest whole number.
(A) 32
(B) 48
(C) 53
(D) 55
(E) 59
Anyone who attempted this question got it right. The trick was not to be confused and make the cut along the midpoints or else it would be a hexagon. Luckily, nobody fell for that trick.
ANSWER: C
AVERAGE SCORE: 5.5 (13/13)
13. A piece of string is cut in two at a random point. What is the probability that the larger piece is at least
times as large as the shorter piece? (where
)
(A)
(B)
(C)
(D)
(E) None of the Above
This is the type of question thta is far easier when it is a multiple choice question than when it is a written question. After a little bit of trial and error, one eventually arrives at the correct answer.
CORRECT ANSWER: C
AVERAGE SCORE: 4.5 (10/12)
14. Let
denote the sum of all distinct four-digit numbers that contain only 1,2,3,4 or 5, each at most once. Then
(A)
(B)
(C)
(D)
(E)
A disguised combinatorics problem here. The answer is C but the number itself is so very close to D that it is easy to make a mistake if using the wrong method. Luckily, nobody made THAT mistake (Even though few got it wrong).
ANSWER: C
AVERAGE SCORE: 3.9 (8/11)
15. How many digits are in
if
and
~ 
(A) 277
(B) 278
(C) 282
(D) 283
(E) 284
Once again, a question AOPS knows how to do but I can't say the same about the rest of the mathematical world. If you know how to do this, which almost everyone did, then you will get the correct answer (unless you forget to take the floor, which nobody did!)
ANSWER: A
AVERAGE SCORE: 5.5 (13/13)
16. If
, what is the value of
?
(A)
(B)
(C)
(D)
(E) Undefined
This was a nice question. Anyone who attempted it got it right but it wasn't exactly easy.
ANSWER: A
AVERAGE SCORE: 4.8 (10/10)
17. Let the parabola
meet the
-axis at two distinct points,
and
. Let the vertex of said parabola be
. The area of triangle
can be expressed as
, where
and
are relatively prime.
and
are both...
(A) Prime (but not in the form
)
(B) Perfect Squares
(C) Perfect Cubes
(D) In the form
(E) None of the Above
A question dealing with co-ordinates. After finding the vertex of the parabola, it becomes pretty straightforward. The answer is 27/8, which corresponds to C.
ANSWER: C
AVERAGE SCORE: 5.0 (12/14)
18. Let
be the number of positive integral factors of
. The sum of the digits of
equals
(A) 7
(B) 10
(C) 13
(D) 16
(E) None of the Above
ANSWER: B (I believe it was 1900)
AVERAGE SCORE: 5.3 (12/12)
19. Bette visits her friend Keith and the nreturns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then the total distance she walks, in km, is
(A) 9
(B) 12
(C) 18
(D) 22
(E) 36
I like this question. One must realize that she will walk the same distance uphill and downhill. If one finds her average speed uphill and downhill, it will be 3 km/h... the same as on level ground! Thus, in 6 hours, she walks 18 km.
ANSWER: C
AVERAGE SCORE: 3.7 (8/12)
20. If
is a regular pentagon with side length
and diagonal length
, what is the value of
?
(A)
(B) 1
(C)
(D) 2
(E) Depends on
One could approach this using Ptolemy's Theorem, trigonometry or just having seen it before. E was intended to throw some people off but wasn't that effective.
ANSWER: B
AVERAGE SCORE: 4.7 (10/11)
21. Let
be a point outside circle
. Draw two tangents from
such that they intersect
at
and
. If
and the larger of the two arcs
is twice as large as the smaller of the two arcs, find the length of
.
(A)
(B)
(C)
(D)
(E) None of the Above
I came up with this geometry question myself after looking at a random theorem.
ANSWER: D
AVERAGE SCORE: 4.3 (9/11)
22. The system of equations


has how many solutions where
,
and
are all integers?
(A)
(B)
(C)
(D)
(E) None of the Above
I presented this question for my math class and wow was it hard. This is actually a spinoff of an old Cayley question but I like my version better. The only solution is (2,3,7)
ANSWER: B
AVERAGE SCORE: 3.0 (5/9)
23. How many arrangements of the letters of the word "INDEPENDENT" have no adjacent E's?
(A)
(B)
(C)
(D)
(E) None of the Above
I swear I made up options B, C and D off the top of my head. Question was well done after people figured out that I was randomly putting factorials wherever I wanted.
ANSWER: E
AVERAGE SCORE: 4.4 (9/10)
24. If
for all
and
, and
never equals
, then
equals...
(A)
(B)
(C)
(D) :pm: 1
(E) None of the above
This one was a killer. The only person to get this correct was towersfreak2006. Most people determined f(x)^2=1 correctly but then mistakenly thought f(x) = :pm: 1
But f(x) is a function. How can f(x) = :pm: 1
The correct method involves setting y = x/2 and working from there. Try it yourself. The correct answer is simply 1.
ANSWER: E
AVERAGE SCORE: 1.2 (1/10)
25. Pieces of paper numbered from
to
are each placed in one of three hats such that there is at least one piece of paper in each hat. In how many ways can this be done such that if two hats are selected and a card is taken from each, then the knowledge of their sum alone is always sufficient to identify the third hat?
(A)
(B)
(C)
(D)
(E) None of the Above
Nobody attempted this question. I put it on here to simulate an extremely difficult question 25 and to prevent any perfects. I felt that if you can get perfect on AMC10, you don't really need practice for it.
SOURCE OF THIS QUESTION: IMO 2000 B1 ... yeah, don't feel upset if you didn't solve it... it was only supposed to be a simulation. Nobody got anywhere near perfect anyway.
And for whitehorseking88, here's what the question is asking: Let's say you had 3 hats and you knew the sum of the numbers in all 3. Let's say the sums were A, B and C respectively. Now if you randomly pick two pieces of paper out of any two hats and are told their sum, you are able to identify which hat has sum A, which hat has sum B and which hat has sum C. Now how many possible sums A, B and C exist for this to work?
(HINT: Try sums like 1 and 2...)
ANSWER: I won't tell ya... IMO 2000 if you really need to know. kalva.demon.co.uk
AVERAGE SCORE: 2.5 (0 correct/ 0 attempted)
Results up soon.
NOTE: If you have not taken the test and would like to, please see the other thread for the questions as this thread has the questions ALONG with the answers.
TOTAL NUMBER OF CONTESTANTS: 15
MOCK AMC I-
CREATED BY: Lucky707 and Singthesorrow
DATE ADMINISTERED: January 30, 2005
Rules:
(Standard AMC Rules)
1. You have 75 minutes to complete the test.
2. Only non-programmable calculators are allowed.
3. 6 points for a correct answer, 2.5 points for a question left unanswered and 0 points for an incorrect answer so it is in your best interest NOT to guess.
4. All other standard AMC Rules apply... We've had plenty of Mock AMCs so you should know the rules by now.
5. At 5:15, you must submit the test. If you do not, it will not count. I will allow a few extra minutes for slow modems, etc. but please submit the test at 5:15.
6. My AOL is hezgotzbeenz if anyone has any questions during the test.
1. The I.Q. of a Martian varies directly with the square of the number of eyes it has. If a Martian with 5 eyes has an I.Q. of 75, find the IQ of a Martian with 8 eyes.
(A) 120
(B) 128
(C) 144
(D) 192
(E) 200
Generally well done, those who missed this question did so because they didn't read it and multiplied 8 by 15 to get 120.
ANSWER: D
AVERAGE SCORE: 5.2 (13 correct/15 attempted)
2. The number of cubic feet in the volume of a cube is the same as the number of square inches of its surface area. What is the length of an edge of it expressed in feet? (NOTE: 12 inches = 1 foot)
(A) 6
(B) 36
(C) 72
(D) 108
(E) 864
Also fairly well done. Those who missed this one did not convert at some point down the line.
ANSWER: E
AVERAGE SCORE: 4.8 (12/15)
3. The first two terms of a geometric progression are

(A)

(B)

(C) fourth root of 3
(D) fourth root of

(E) tenth root of 3
This was somewhat of a trick question. Those who were tricked by it chose E but if we look at the common ratio and work it out as usual, we get the correct answer of A.
CORRECT ANSWER: A
AVERAGE SCORE: 4.8 (12/15)
4. A leaky faucet drips at a steady rate. If we measure from the instant the first drop hits the bottom, to the instant the last drop falls, it takes 36 seconds for 9 drops to fall. How many seconds would it take for 12 drops to fall?
(A) 42 seconds
(B) 44 seconds
(C) 48 seconds
(D) 49.5 seconds
(E) 54 seconds
This question was nasty. So many places where one could make an error. Here is the solution:
We measure from the instant the first drop falls to the instant the last drop falls. 36 seconds for 9 drops to fall means 36/8 seconds between drops hitting the ground. That's the first place one can make a mistake. The other place is multiplying by 12 instead of 11. The answer is 49.5
ANSWER: D
AVERAGE SCORE: 3.0 (7/14)
5. A triangle has sides of length 15, 20 and 25 units. It is inscribed in a circle. What is the circle's radius?
(A)

(B)

(C)

(D)

(E)

This question was very well done. After realizing that the triangle is a right triangle all one needs to do is divide the hypotenuse by 2 to get the answer.
ANSWER: D
AVERAGE SCORE: 6.0 (15/15)
6. How many zeroes are at the end of the product of the first fifty positive integers?
(A) 5
(B) 12
(C) 17
(D) 35
(E) 47
I don't think the average would be this high in real life for this type of question but I was sure that AOPSers could steamroll through this question... and steamroll they did!
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
7. If




(A) 50
(B) 97
(C) 98
(D) 99
(E) 100
See comment above. 6 and 7 were quite similar actually. Nobody made an addition error so everyone got this right.
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
8. I have a red shirt, a blue shirt and a green shirt. I also have blue jeans, black jeans and khakis. How many possible combinations can I make if I absolutely hate wearing the blue shirt with the blue jeans?
(A) 5
(B) 8
(C) 14
(D) 16
(E) 17
This one was a gimme.
ANSWER: B
AVERAGE SCORE: 6.0 (15/15)
9. Let

(a)

(b)

(c)

(d)

(e)

Wow... there were a lot of prime factorization questions early on. Realizing that 140 = 7*5*2^2.... we find that 7^2*5^2*2^2 will be a perfect square. And that value is 4900, making the answer E.
ANSWER: E
AVERAGE SCORE: 6.0 (15/15)
10. What is the sum of the first twenty powers of 2 (2^1 + 2^2 + 2^3 ... + 2^20)?
(A)

(B)

(C)

(D)

(E) None of the above
This question can easily be done by induction... or if you want to stay honest (which is not a good idea during the AMC)... geometric series.
ANSWER: D
AVERAGE SCORE: 5.2 (13/15)
11.






(A) 212
(B) 144
(C) 72
(D) 48
(E) 24
Generally those who attempted this question got it right. One person got it wrong and I'm assuming it's because he forgot to reduce.
ANSWER: E
AVERAGE SCORE: 5.1 (12/13)
12. A square of side length 8 has its corners cut off to make a regular octagon. Find the area of the octagon to the nearest whole number.
(A) 32
(B) 48
(C) 53
(D) 55
(E) 59
Anyone who attempted this question got it right. The trick was not to be confused and make the cut along the midpoints or else it would be a hexagon. Luckily, nobody fell for that trick.
ANSWER: C
AVERAGE SCORE: 5.5 (13/13)
13. A piece of string is cut in two at a random point. What is the probability that the larger piece is at least


(A)

(B)

(C)

(D)

(E) None of the Above
This is the type of question thta is far easier when it is a multiple choice question than when it is a written question. After a little bit of trial and error, one eventually arrives at the correct answer.
CORRECT ANSWER: C
AVERAGE SCORE: 4.5 (10/12)
14. Let

(A)

(B)

(C)

(D)

(E)

A disguised combinatorics problem here. The answer is C but the number itself is so very close to D that it is easy to make a mistake if using the wrong method. Luckily, nobody made THAT mistake (Even though few got it wrong).
ANSWER: C
AVERAGE SCORE: 3.9 (8/11)
15. How many digits are in




(A) 277
(B) 278
(C) 282
(D) 283
(E) 284
Once again, a question AOPS knows how to do but I can't say the same about the rest of the mathematical world. If you know how to do this, which almost everyone did, then you will get the correct answer (unless you forget to take the floor, which nobody did!)
ANSWER: A
AVERAGE SCORE: 5.5 (13/13)
16. If


(A)

(B)

(C)

(D)

(E) Undefined
This was a nice question. Anyone who attempted it got it right but it wasn't exactly easy.
ANSWER: A
AVERAGE SCORE: 4.8 (10/10)
17. Let the parabola











(A) Prime (but not in the form

(B) Perfect Squares
(C) Perfect Cubes
(D) In the form

(E) None of the Above
A question dealing with co-ordinates. After finding the vertex of the parabola, it becomes pretty straightforward. The answer is 27/8, which corresponds to C.
ANSWER: C
AVERAGE SCORE: 5.0 (12/14)
18. Let



(A) 7
(B) 10
(C) 13
(D) 16
(E) None of the Above
ANSWER: B (I believe it was 1900)
AVERAGE SCORE: 5.3 (12/12)
19. Bette visits her friend Keith and the nreturns home by the same route. She always walks 2 km/h when going uphill, 6 km/h when going downhill, and 3 km/h when on level ground. If her total walking time is 6 hours, then the total distance she walks, in km, is
(A) 9
(B) 12
(C) 18
(D) 22
(E) 36
I like this question. One must realize that she will walk the same distance uphill and downhill. If one finds her average speed uphill and downhill, it will be 3 km/h... the same as on level ground! Thus, in 6 hours, she walks 18 km.
ANSWER: C
AVERAGE SCORE: 3.7 (8/12)
20. If




(A)

(B) 1
(C)

(D) 2
(E) Depends on

One could approach this using Ptolemy's Theorem, trigonometry or just having seen it before. E was intended to throw some people off but wasn't that effective.
ANSWER: B
AVERAGE SCORE: 4.7 (10/11)
21. Let









(A)

(B)

(C)

(D)

(E) None of the Above
I came up with this geometry question myself after looking at a random theorem.
ANSWER: D
AVERAGE SCORE: 4.3 (9/11)
22. The system of equations


has how many solutions where



(A)

(B)

(C)

(D)

(E) None of the Above
I presented this question for my math class and wow was it hard. This is actually a spinoff of an old Cayley question but I like my version better. The only solution is (2,3,7)
ANSWER: B
AVERAGE SCORE: 3.0 (5/9)
23. How many arrangements of the letters of the word "INDEPENDENT" have no adjacent E's?
(A)

(B)

(C)

(D)

(E) None of the Above
I swear I made up options B, C and D off the top of my head. Question was well done after people figured out that I was randomly putting factorials wherever I wanted.
ANSWER: E
AVERAGE SCORE: 4.4 (9/10)
24. If






(A)

(B)

(C)

(D) :pm: 1
(E) None of the above
This one was a killer. The only person to get this correct was towersfreak2006. Most people determined f(x)^2=1 correctly but then mistakenly thought f(x) = :pm: 1
But f(x) is a function. How can f(x) = :pm: 1
The correct method involves setting y = x/2 and working from there. Try it yourself. The correct answer is simply 1.
ANSWER: E
AVERAGE SCORE: 1.2 (1/10)
25. Pieces of paper numbered from


(A)

(B)

(C)

(D)

(E) None of the Above
Nobody attempted this question. I put it on here to simulate an extremely difficult question 25 and to prevent any perfects. I felt that if you can get perfect on AMC10, you don't really need practice for it.
SOURCE OF THIS QUESTION: IMO 2000 B1 ... yeah, don't feel upset if you didn't solve it... it was only supposed to be a simulation. Nobody got anywhere near perfect anyway.
And for whitehorseking88, here's what the question is asking: Let's say you had 3 hats and you knew the sum of the numbers in all 3. Let's say the sums were A, B and C respectively. Now if you randomly pick two pieces of paper out of any two hats and are told their sum, you are able to identify which hat has sum A, which hat has sum B and which hat has sum C. Now how many possible sums A, B and C exist for this to work?
(HINT: Try sums like 1 and 2...)
ANSWER: I won't tell ya... IMO 2000 if you really need to know. kalva.demon.co.uk
AVERAGE SCORE: 2.5 (0 correct/ 0 attempted)
Results up soon.
This post has been edited 1 time. Last edited by Lucky707, Jan 29, 2005, 11:25 PM