be an integer. Find all positive real solutions to the following system of 
equations:
for which 
is a divisor of 
,
"
.
Y by sicilianfan, forthegreatergood, sat113, El_Ectric, Math_Fan, SMOJ, MSTang, Tan, Einstein314, tim9099xxzz, VietaFan, blasterboy, ahaanomegas, JRY, flamefoxx99, Williamgolly, Adventure10, and 7 other users
Obviously there is little time between now and the AMC A, but hopefully people will find this useful anyway. I wrote this for the AAST math team as a practice exam for today, but evidently it proved extremely difficult as we would have had <10 AIME qualifiers based on this test (compared to our usual very high number).
Because this was written specifically for the math team there are a couple of small differences between this and the AMC:
If you want to take this "officially", you can send me answers and I'll score them (and probably post them here unless requested otherwise). The only reason I'm not just posting this with solutions and leaving it at that is because I haven't actually typed up solutions (or an answer key for that matter), so I'll post those here when that actually happens. I'm not asking for signups or anything because this is so late, so don't fill up this thread with a list of people.
On a final note, I only had about a week to write this test due to being away, so I found it easier to "rip off" (and make easier in some cases) some problems for later AMC 12 problems. I believe I only did this 3 times (#21 [which was specifically placed for a couple of select people], 22 [which was made easier], 25 [which is classic] on AMC 12), but obviously you may have seen problems of similar natures before. Furthermore the shortened timeframe means that there may very well be errors in the problems/answers/solutions (note that this is already a "fixed" version from the earlier 5-errata one). The problem ordering is also a little haphazard - I had to make quick guesses about which problems would supposedly be harder than others. Finally, the mock tests are pretty similar (I formed the AMC 10 by copying and pasting the AMC 12, deleting some hard questions, and adding some easy ones).
I'm not sure what a good normalization factor is, so hopefully some submissions here might help me to figure that out. As such I'd prefer you don't send me answers to grade if you didn't take the test under normal conditions (unless you tell me you didn't take the test under normal conditions), so that I can attempt to figure out just how hard this mock test is. The AMC 10 is much harder than a normal AMC 10, while the AMC 12 seems to only be somewhat harder (who knew that trig wasn't supposed to be on an AMC 10 anyway?).
Enjoy the tests!
EDIT: By the way, there may be an ASIA TEAM V2 at some point, though Kelvin the Frog is quickly becoming tired and is threatening to go on strike.
EDIT2: Corrected tests updated in this post. Previous download counts were 121 AMC 12 and 147 AMC 10 if anyone was keeping track for some reason.
Because this was written specifically for the math team there are a couple of small differences between this and the AMC:
- Several problems ask for the usual "the answer can be expressed in the form
, find 
", but this test allows cases such as 
. This is done when, for example, it's not immediately obvious that the answer turns out to be an integer without actually solving the problem, so it makes bssolving a little harder. - Most problems reference the name of a math team member (possibly obfuscated) followed by some animal's name. Missing the reference is not a big deal (though many will recognize the return of Kelvin the Frog anyway
) - These problems are harder than usual. Part of this was intentional to reflect the fact that they should have been paying attention to lectures, so some of the harder problems utilized lecture topics. However, a bigger part of this is that gauging difficulty of easy/mid-level problems is difficult for me to do, so most of the early/middle problems are much more difficult than usual (#8 on the mock 10 caused some excitement for example).
If you want to take this "officially", you can send me answers and I'll score them (and probably post them here unless requested otherwise). The only reason I'm not just posting this with solutions and leaving it at that is because I haven't actually typed up solutions (or an answer key for that matter), so I'll post those here when that actually happens. I'm not asking for signups or anything because this is so late, so don't fill up this thread with a list of people.
On a final note, I only had about a week to write this test due to being away, so I found it easier to "rip off" (and make easier in some cases) some problems for later AMC 12 problems. I believe I only did this 3 times (#21 [which was specifically placed for a couple of select people], 22 [which was made easier], 25 [which is classic] on AMC 12), but obviously you may have seen problems of similar natures before. Furthermore the shortened timeframe means that there may very well be errors in the problems/answers/solutions (note that this is already a "fixed" version from the earlier 5-errata one). The problem ordering is also a little haphazard - I had to make quick guesses about which problems would supposedly be harder than others. Finally, the mock tests are pretty similar (I formed the AMC 10 by copying and pasting the AMC 12, deleting some hard questions, and adding some easy ones).
I'm not sure what a good normalization factor is, so hopefully some submissions here might help me to figure that out. As such I'd prefer you don't send me answers to grade if you didn't take the test under normal conditions (unless you tell me you didn't take the test under normal conditions), so that I can attempt to figure out just how hard this mock test is. The AMC 10 is much harder than a normal AMC 10, while the AMC 12 seems to only be somewhat harder (who knew that trig wasn't supposed to be on an AMC 10 anyway?).
Enjoy the tests!
EDIT: By the way, there may be an ASIA TEAM V2 at some point, though Kelvin the Frog is quickly becoming tired and is threatening to go on strike.
EDIT2: Corrected tests updated in this post. Previous download counts were 121 AMC 12 and 147 AMC 10 if anyone was keeping track for some reason.
This post has been edited 2 times. Last edited by BOGTRO, Feb 3, 2014, 6:38 AM
(rather than 
or whatever it previously asked for). Will be updated at some point.
