I recently came across an AIME problem that compared what I call “mixed up functions:”
How many real numbers x satisfy the equation:
By mixed up, I mean we have the periodic sine curve equated with the monotonically increasing log function. Often with logs we can replace them with an exponential function, in this case with base 2, but that looks messy.
The approach I often see with these mis-matched functions is to draw a graph and see where the 2 functions intersect. But first, I’m going to multiply both sides by 5 because who likes fractions?
Ah that’s better. Notice that the maximum and minimum values of occur when and , and the maximum and minimum values are 5 and -5. And of course this is a periodic function that is defined for all real numbers.
on the other hand is defined only over positive reals, and since it is always increasing from left to right, at some point the value of the function will exceed 5. Specifically, this will occur for . Likewise, will be less than -5 for .
And it’s apparent that in the interval the log function is going to intersect the sine function as it goes up and down on its periodic path. How many times does it intersect? AIME writers like to catch students on “off by one” errors, so take care to examine what is happening close to and before entering your answer!
The best way to prepare for any test is to locate old versions of the exam and practice with those, especially studying the problems you answered incorrectly. This is true for the AMCs and MathCounts. AoPS also offers online prep classes. For those of you who prefer learning from textbooks, here are few recommendations. These books take problems from old competitions and organize them by topic so you can ramp up your skills.
Competition Math for Middle School by J. Batterson. In particular I enjoyed the chapters on counting and probability for their concise and clear introductions to the subjects. You’ll be up to speed on these topics in no time. Other chapters are algebra, geometry, and number theory.
First Steps for Math Olympians by J. Douglas Faires. This is the next step in difficulty is preparing for the AMC 10/12. This book uses questions from those exams as exercises. Content includes: ratios, polynomials, functions, triangles, circles, polygons, counting, probability, primes, number theory, sequences and series, statistics, trig, 3D geometry, logs, and complex numbers.
A Gentle Introduction to the American Invitational Mathematics Exam by Scott Annin. This title seemed to be speaking to me. I qualified for AIME my senior year of high school, but bombed the test, so I’ve been living in fear of the AIME every since. I’m working through the chapter 1 and I’m pleased to find myself actually solving some problems without assistance. Contents include: algebra, combinatorics, probability, number theory, sequences/series, logs, trig, complex numbers, polynomials, plane geometry, and 3D geometry. Nearly all exercises are former AIME problems.
But if you want to save more time, I’ve actually transcribed that portion of the talk, just because I think it’s so important.
“Right after I started Art of Problem Solving I received an email from someone who attended Princeton right around the time I did:”
“I want you think for a minute what this student’s middle school and high school teachers thought when he went off to Princeton. They thought, “We succeeded. He went off to Princeton; we’re awesome.” They never saw this. I’m sure he didn’t go back to his middle school teachers and say, “Yeah what’s up?!? You didn’t prepare me for this.”
“So they didn’t get this feedback, and this happens a lot. I saw this a lot at Princeton, this happens a lot now. Kids go through school, some very good schools, they get perfect scores on everything, and then they come to place like MIT, a place like Princeton, they walk into that first year math class, and they see something they’ve never seen before: problems they don’t know how to solve. And they completely freak out. And that’s a bad time to have these first experiences. Having to overcome initial failure.”
I was just helping a student with his AoPS homework, when I came across the following related problems:
Eight people, including Fred, are in a club. They decide to form a 3 person committee. How many possible committees can be formed?
So we are choosing 3 people out of 8 or .
How many possible committees include Fred?
Since Fred is taking one committee seat, that means we need to choose 2 more people from the remaining seven, or .
How many possible committees do not include Fred?
Since we can’t choose Fred, we need to choose 3 members from the remaining 7 or .
Since the total number of committees is equal to the number of of committees with Fred plus the number of committees without Fred, then we can say
Generally we call this Pascal’s Rule: .