Here are comments and solutions to problems 11-15 on the 2011 Math Prize for Girls contest that took place at MIT on September 17, 2011.
This problem is about a linear recurrence relation. The theory of linear recurrence relations is well understood. The better your facility with this topic, the easier you’ll find this problem. I’ll start by giving a solution that takes advantage of a sound understanding of the theory.
Using that theory, I know that the given sequence can be written
where , , and are constants and , , and are roots of the cubic equation associated with the recurrence relation: . Because 1 is a root of this cubic, we can factor completely and find that the three roots are:
, , and .
These three roots are sixth roots of unity. This tells us that the sequence is periodic with a period of 6 (or possibly something that divides evenly into 6).
We’re given the values of , , and , which, knowing about the periodicity, is the same as knowing that , , and . We’re asked to compute which is equal to since 1331 is congruent to -1 modulo 6.
Thus, the answer is .
Now, let’s pretend we know nothing about the theory of linear recurrence relations and we find ourselves confronted with this problem. What one might try to do is suppose that the sequence begins a, b, c, and then compute the next few values looking for some pattern (and since this is a contest math problem, there is very likely some simplifying pattern). So we just compute:
a, b, c, 2c – 2b + a, 2c – 3b + 2a, c – 2b + 2a, a (!), b (!), c (!)…
and we see we come right back to the starting 3 values. So then we can work modulo 6 and find a = 100, b = 10, and c = 1 and that we need to find , which is c – 2b + 2a = 181. (I put the exclamation marks in the last three computed values of the sequence above because if I knew nothing about the theory of recurrence relations, I’d be thrilled during the contest to discover that a, b, and c return so wonderfully because that means that the entire sequence is just 6 numbers that repeat over and over.)
If you had trouble, study linear recurrence relations. They sprout up all over the place: Fibonacci numbers, continued fractions, combinatorics… They’re also a good way to learn more about matrices, characteristic equations, and determinants.
In this problem, we seek values of where . The function rule that defines looks complicated with greatest integer functions and logarithms all over the place so it would be very unpleasant if we had to use that function rule and directly compute . Fortunately, this is a math contest, so you can bet that there is a less obscure description of the function. Just keep your wits about you and start unraveling the function rule.
Since all logarithms are base 10, we know that is the largest power of 10 less than or equal to . Therefore, is the leftmost digit of . That means that the expression in parentheses in the function rule yields the number you get if you erase the first (leftmost) digit of . Combining these observations, we can see that the net effect of the function rule is to move the leftmost digit of over so that it becomes its rightmost digit. For example, S(7438) = 4387.
Applying S twice to a number moves its leftmost two digits over to the right…unless the digit second in from the left is zero, because in that case, after applying S once, all zeros that come directly after the (original) leftmost digit of n are erased.
Now that we have a clearer description of S, we can set about finding those for which . If the second or third digit (from the left) of n is zero, then won’t even have the same number of digits as n, so we can ignore all numbers whose second or third digit (from the left) is zero. Suppose is such a decimal number whose digits are the . (The underscore reminds us that this expression does not stand for the product of the .) Then
For this to equal , we need , , . . ., , , and . We get two types of solutions depending on whether is even or odd. If is even, then all these equations tell us that all digits are equal. If is odd, then all the odd digits are equal and all the even digits area equal. So let’s divide the solutions into sets according to how many digits the numbers have and start counting:
For 1 digit numbers, all work, so that gives 9 solutions.
For 2 digit numbers, all work provided that the units digit isn’t zero, so that gives 81 solutions.
For 3 digit numbers (i.e. ), we have 9 solutions (namely, 111, 222, 333, etc.)
For 4 digit numbers, we can specify the first two digits freely subject to the condition that we don’t use zero, and then the rest of the number is determined. But also, we need numbers less than or equal to 2011, so that gives us the following solutions: 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, and 1919, for a total of 9 more solutions.
Hence the answer is 9 + 81 + 9 + 9 = 108.
By the way, this problem is a good example of the kind of math problem that only lives in the math contest world. That’s because good mathematicians would never bother describing the function using such a complex looking, obfuscating formula. When good mathematicians aren’t being whimsical, they seek clarity, not obfuscation, so they’d just describe the function S by describing its effect on the digits of n. Even more, such a function in mathematical research isn’t likely because the function has a special affinity for a specific base, and the choice of 10 as a base is arbitrary. And then there’s that silly, seemingly traditional, math contest invocation of the year of the exam in the problem statement.
If you had trouble, perhaps the complex looking function intimidated you? If you have a basic, sound knowledge of arithmetic, the greatest integer function, and exponents and logarithms, then you have all you need to unravel this problem. Don’t be intimidated. Just think through it step by step. Especially on a math contest, you have good reason to expect that there will be a much simpler description. And after you figure out the nicer description, be careful not to trip up on details such as the effect of zero digits.
This problem is about pattern recognition and Sophie Germain’s identity. We’re told that 104,060,465 has a 5 digit prime factor and are asked to find it. Unfortunately, I’m not sure I can say something that would be very helpful if you had trouble with this one. Here’s what happened when I tried to solve it.
First, I noticed that the number is close to and so I thought to try to factor it by finding and such that
I fiddled with this approach for a little while and didn’t get anywhere, so I thought, “hey, this is a contest math problem…if this approach seems laborious, then it’s probably the wrong approach…there has to be some ‘nice’ way of finding the factor…I must be missing some neat observation.” So I looked at the 9 digit number again and noticed that it amusingly looks a little like the 1, 4, 6, 4, 1 row of Pascal’s triangle…and then I realized how to do the problem because I know that the rows of Pascal’s triangle are the digits of powers of 11 (if you take a sufficiently high base, but here, 14641 is exactly ). That is, . Therefore, , which is exactly in the form of one side of Sophie Germain‘s identity:
Substituting and , we find that
Since we are told that there is a 5 digit prime factor, it must be 10613.
Because this approach works so snugly and factoring directly would be tedious and time-consuming, I’m confident that the contest designers had this solution in mind when they set the problem. If you aren’t able to recognize the given number as , I’m not sure what you could do to solve this problem in a reasonable amount of time. While you’re taking a math competition, be careful not to allot too much time to a tedious approach even if you know it should work in principle. Instead, try to look for something you hadn’t noticed before. In this case, I’m not sure how one would become prepped to notice an application of Sophie Germain’s identity other than to say that if you enjoy math and keep on studying it, you’ll put yourself in an ever better position to recognize such things.
This is an optimization problem. When you fix , the function is linear in , so the value of will be equal to or depending on whether the coefficient of (i.e. the slope) is positive or negative, respectively. The coefficient of is . This is positive or negative according to whether or . (If the slope is zero, it doesn’t matter whether we take or because they’ll be equal!) Thus, if and if . Because the slope of is negative when and positive when , the minimal value of occurs right where .
Because all outcomes are equally likely, we have to count how many ways the desired condition holds and divide by to get the probability of that condition happening. The desired condition is:
This can be rearranged to .
Now let’s exploit the fact that the variables are all powers of 2: let , , , and . The condition becomes:
(And note that .) The only ways that this equation can hold are if (if it helps, think of this equation as an identity involving binary numbers):
So we just have to count these possibilities being careful not to double count any cases. The first case is equivalent to and . The second is equivalent to . Since the first case is more restrictive than the second, we can simply ignore the first case and worry about the second and third cases only.
The third case is equivalent to . That is, how many ways can two pairs of normal dice produce the same sums? If there are n ways to get the sum s, then there are ways for two pairs to get the sum s. So the total number of ways to get this third case is . Of these cases, there are where ; these are the cases that overlap with the second condition above. And the number of cases that fall within the second condition is .
Thus, the total number of possibilities is and the answer is .