2004 AIME I Problems/Problem 2

Problem

Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$

Simple Solution

Look at the problem... consecutive integers. Now, since set $A$ has the properties of $m$ integers that sum to $2m$, it's obvious that the middle integer is just $2$ (the average that "balances out" everything), and the largest is $2 + \frac{m-1}{2}$. That's because there are $m-1$ to go after taking $2$, and they are "evenly balanced" on either side.

From there, we see that set $B$'s average is $0.5$. How can that happen? Only if the middle TWO values are $0$ and $1$. Since set $B$ has $2m$ integers, its maximum must be $99$ larger than the maximum of set $A$ unless $m=1$, which is impossible.

From there, the largest element of set $B$ is $1 + (m-1) = m$.

Solving, we get $$m - 2 - \frac{m-1}{2} = 99$$ $$m-\frac{m}{2}+\frac{1}{2}=101$$ $$\frac{m}{2}=100\frac{1}{2}.$$ There we go- $m$ is equal to none other than $\boxed{201}$.

Solution 1

Let us give the elements of our sets names: $A = \{x, x + 1, x + 2, \ldots, x + m - 1\}$ and $B = \{y, y + 1, \ldots, y + 2m - 1\}$. So we are given that $$2m = x + (x + 1) + \ldots + (x + m - 1) = mx + (1 + 2 + \ldots + (m - 1)) = mx + \frac{m(m -1)}2,$$ so $2 = x + \frac{m - 1}2$ and $x + (m - 1) = \frac{m + 3}2$ (this is because $x = 2 - \frac{m-1}{2}$ so plugging this into $x+(m-1)$ yields $\frac{m+3}{2}$). Also, $$m = y + (y + 1) + \ldots + (y + 2m - 1) = 2my + \frac{2m(2m - 1)}2,$$ so $1 = 2y + (2m - 1)$ so $2m = 2(y + 2m - 1)$ and $m = y + 2m - 1$.

Then by the given, $99 = |(x + m - 1) - (y + 2m - 1)| = \left|\frac{m + 3}2 - m\right| = \left|\frac{m - 3}2\right|$. $m$ is a positive integer so we must have $99 = \frac{m - 3}2$ and so $m = \boxed{201}$.

Solution 2

The thing about this problem is, you have some "choices" that you can make freely when you get to a certain point, and these choices won't affect the accuracy of the solution, but will make things a lot easier for us.

First, we note that for set $A$

$$\frac{m(f + l)}{2} = 2m$$

Where $f$ and $l$ represent the first and last terms of $A$. This comes from the sum of an arithmetic sequence.

Solving for $f+l$, we find the sum of the two terms is $4$.

Doing the same for set B, and setting up the equation with $b$ and $e$ being the first and last terms of set $B$,

$$m(b+e) = m$$

and so $b+e = 1$.

Now we know, assume that both sequences are increasing sequences, for the sake of simplicity. Based on the fact that set $A$ has half the number of elements as set $B$, and the difference between the greatest terms of the two two sequences is $99$ (forget about absolute value, it's insignificant here since we can just assume both sets end with positive last terms), you can set up an equation where $x$ is the last term of set A:

$$2(x-(-x+4)+1) = 1+(x+99)-(-x-99+1)$$

Note how i basically just counted the number of terms in each sequence here. It's made a lot simpler because we just assumed that the first term is negative and last is positive for each set, it has absolutely no effect on the end result! This is a great strategy that can help significantly simplify problems. Also note how exactly i used the fact that the first and last terms of each sequence sum to $4$ and $1$ respectively (add $x$ and $(-x+4)$ to see what i mean).

Solving this equation we find $x = 102$. We know the first and last terms have to sum to $4$ so we find the first term of the sequence is $-98$. Now, the solution is in clear sight, we just find the number of integers between $-98$ and $102$, inclusive, and it is $m = \boxed{201}$.

Note how this method is not very algebra heavy. It seems like a lot by the amount of text but really the first two steps are quite simple.

Solution 3 (Sketchy solution to use when you don't have enough time)

First, calculate the average of set $A$ and set $B$. It's obvious that they are $2$ and $1/2$ respectively. Let's look at both sets. Obviously, there is an odd number of integers in the set with $2$ being in the middle, which means that $m$ is an odd number and that the number of consecutive integers on each side of $2$ are equal. In set $B$, it is clear that it contains an even number of integers, but since the number in the middle is $1/2$, we know that the range of the consecutive numbers on both sides will be $(x$ to $0)$ and $(1$ to $-y)$.

Nothing seems useful right now, but let's try plugging an odd number, $3$, for $m$ in set $B$. We see that there are $6$ consecutive integers and $3$ on both sides of $1/2$. After plugging this into set $A$, we find that the set equals $${1,2,3}$$. From there, we find the absolute value of the difference of both of the greatest values, and get 0.

Let's try plugging in another odd number, $55$. We see that the resulting set of numbers is $(-54$ to $0)$, and $(1$ to $55)$. We then plug this into set $A$, and find that the set of numbers is $(-25$ to $-29)$ which indeed results in the average being $2$. We then find the difference of the greatest values to be 26.

From here, we see a pattern that can be proven by more trial and error. When we make $m$ equal to $3$, then the difference is $0$ whearas when we make it $55$, then the difference is $26$. $55-3$ equals to $52$ and $26-0$ is just $0$. We then see that $m$ increases twice as fast as the difference. So when the difference is $99$, it increased $99$ from when it was $0$, which means that $m$ increased by $99*2$ which is $198$. We then add this to our initial $m$ of $3$, and get $\boxed{201}$ as our answer.