Easy AIME?
What do you think?
Two problems from the regular administration of this year’s American Invitational Mathematics Examination look, to me, unusually easy. However, I have not carefully followed this exam ever year. For those of you familiar, what do you think? I’m also vaguely curious about variations on solutions (for #2, not really #1)
These two problems are copyright 2008, Mathematics Association of America
1. Of the students attending a school party, 60% of the students are girls, and 40% of the students like to dance. After these students are joined by 20 more boy students, all of whom like to dance, the party is now 58% girls. How many students now at the party like to dance?
2. Square AIME has sides of length 10 units. Isosceles triangle GEM has base EM, and the area common to triangle GEM and square AIME is 80 square units. Find the length of the altitude to EM in triangle GEM.
JD2718 
wow, those are … pretty straightforward.
The altitude will be the same as for a right triangle with the same overlap, so I just did it that way (not that it saves much, you can just as easily do the same calculation on half the isoceles triangle and half the square). I computed the unknown side length of the triangle for the part of the square not in common then used similar triangles to compute the altitude.
Not familiar with the exam.
2: Let the intersection of GEM with AI be y. Then 80 = 1/2 . 10 (y+10), since it’s a trapezium (or do you guys call them trapezoids? I’m never sure), yielding y=6. Now, the triangles at the edge “cut off” from the square by the big triangle are similar to half the big triangle (alternate angles), so 10:altitude = 4:10 and the altitude’s 25.
Efrique’s solution is prettier.
I wonder if it has anything to do with calculator’s not being allowed on the AMC starting this year?
Thanks Efrique, Doc.
Jackie, this is the AIME which has never allowed calculators. I can’t figure out why these questions were (comparatively) so easy. Maybe there is a twist I’ve missed? Or previous years just weren’t so hard?
I received the score report from the MAA. The average number of questions correct on the AIME this year was 5. I recall that in previous years it was closer to 2. So, I need to go read Jackie’s link!
My way was messier:
I’ll call the points where GM and GE intersect the segment AI, P & Q respectively.
Since the shared area is 80 and the whole area of AIME is 100, each of the cut-off triangles on the sides (APM, IQE) has area 10, and thus base 2. That makes the base of GPQ 6. Then I used the fact that GPQ is similar to GME, with ratio .6 to find the area of GME as follows (using the name of the triangle to denote its area)
GPQ = .6^2 GEM
= .36(GPQ + 80)
GPQ = 45
thus GEM = 80 + 45 = 125
in GEM 125 = .5 x 10 x height thus the height is 25, which is the desired altitude.
I agree that these both seem pretty easy for the AIME.
I know calculators have never been allowed on the AIME. I was just wondering if there was a connection. I would think calc/no calc wouldn’t matter for those students who do qualify via AMC, just wondering. I know some of our students were a bit put off by the change this year. Sadly the number of our students who took the AMC was down – but the calculator policy was only one factor.