AIME

Question 1

(2012 AIME I #10)

Let S be the set of all perfect squares whose rightmost three digits in base 10 are 256. Let T be the set of all numbers of the form {x-256}/{1000}, where x is in S. In other words, T is the set of numbers that result when the last three digits of each number in S are truncated. Find the remainder when the tenth smallest element of T is divided by 1000.

What are the first steps?

Answer 1

1. Replace x with n^2
2. Convert to equation n^2-256=1000k (n, k are integers)
3. Diophantine, so factor (n-16)(n+16)=2^35^3k

Question 2

Given (n-16)(n+16)=2^35^3k, what do you do next?

Answer 2

Since the 2 factors on the left by 32, exactly one of them must be a multiple of 125, and both of them must be multiples of 4. Therefore, one of them is a multiple of 500. Therefore, the solutions for n are n=16, 500-16, 500+16, 1000-16, 1000+16,… Therefore the answer to 2012 AIME I #10 is [(2500-16)^2-256]/1000=6170 –> 170

Question 3

When something is changing in a complex manner, what do you do?

Answer 3

Look for something that doesn’t change (an invariant), e.g., 2012 AIME I #11 x+y varies by 3m, x-y changes by 5n

Question 4

What are the (complex) solutions to z^n=1?

Answer 4

cos(2kpi/n)+isin(2kpi/n), k=1,2,3, … n-1

Question 5

If you have angle trisectors, what do you also have?

Answer 5

Angle bisectors, which you can use the angle bisector theorem with

Question 6

When looking for the sin, cosine, and tangent of an angle (especially in a triangle), what should you do?

Answer 6

Look for law of sines/cosines, drop altitudes, and look for right triangles with the values you need in them.

Question 7

A point on a figure, when a transformation applies, needs to map onto another figure. What do you do?

Answer 7

Apply the transformation to the first figure. Note that not all points will be able to satisfy this.

Question 8

You need to count the numbers that are not multiples of j or k between m and n. What do you do?

Answer 8

Count how many are not between 1 and gcd(j, k), count how many multiples of gcd(j, k) are between m and n, multiply those numbers, and add the extra.

Question 9

What are Grant Sanderson’s 9 tips for problem solving?

Answer 9

1. Use the defining features of the setup.
2. Give things meaningful names.
3. Leverage symmetry
4. Describe 1 object 2 ways
5. Draw a picture (use numbers as coordinates)
6. Ask simpler versions of the problem
7. Read a lot, and think about problems a lot
8. Always sanity check
9. Learn a bit of programming

Question 10

What is 1/log2(100!)+1/log3(100!)+…+1/log100(100!)?

Answer 10

Two methods:

1. Using logb(a) = 1/loga(b), we get log100!(2)+log100!(3)+…log100!(100) = log100!(23…*100) = 1
2. Using change of base logb(a)=log(b)/log(a), we get log(2)/log(100!)+log(3)/log(100!)+…+log(100)/log(100!) = log(23…*100)/log(100!) = 1

Question 11

What do you do if you see a strange restriction?

Answer 11

Forget about it if you can - incorporate it later.

Question 12

Suppose you have a solution that satisfies some, but not all of the constraints, what can you do?

Answer 12

Try to establish a relationship between some of the properties of your solution. If they hold, you may be able to find the answer without the actual solution.

For example: You need to find the length of a hypotenuse, and you know one of the legs. If you find a solution that isn’t perfect, but can scale appropriately, and you know the ratio between the leg and hypotenuse, you can find the hypotenuse.

Question 13

You have a+b+c = 0. When you have a value of a+b, what should you call it?

Answer 13

-c. It will simplify things.