Difference between revisions of "2009 AIME I Problems/Problem 8"

Revision as of 09:24, 23 January 2023

Problem

Let $S = \{2^0,2^1,2^2,\ldots,2^{10}\}$ . Consider all possible positive differences of pairs of elements of $S$ . Let $N$ be the sum of all of these differences. Find the remainder when $N$ is divided by $1000$ .

Solution 1 (Extreme bash)

Find the positive differences in all $55$ pairs and you will get $\boxed{398}$ .

Solution 2

When computing $N$ , the number $2^x$ will be added $x$ times (for terms $2^x-2^0$ , $2^x-2^1$ , ..., $2^x - 2^{x-1}$ ), and subtracted $10-x$ times. Hence $N$ can be computed as $N=10\cdot 2^{10} + 8\cdot 2^9 + 6\cdot 2^8 + \cdots - 8\cdot 2^1 - 10\cdot 2^0$ .

We can now simply evaluate $N\bmod 1000$ . One reasonably simple way: $\begin{align*} N & = 10(2^{10}-1) + 8(2^9 - 2^1) + 6(2^8-2^2) + 4(2^7-2^3) + 2(2^6-2^4) \\ & = 10(1023) + 8(510) + 6(252) + 4(120) + 2(48) \\ & = 10(1000+23) + 8(500+10) + 6(250+2) + 480 + 96 \\ & \equiv (0 + 230) + (0 + 80) + (500 + 12) + 480 + 96 \\ & \equiv \boxed{398} \end{align*}$

Solution 3

This solution is a generalized approach that work when $10$ is replaced by other values.

In similarity to the logic in Solution 2, $N = \sum_{x=0}^{10} (2x-10) 2^x$ .

Let $A = \sum_{x=0}^{10} x2^x$ and let $B=\sum_{x=0}^{10} 2^x$ . Then $N=2A - 10B$ .

$B$ can be calculated easily by geometric series with sum $2^{11}-1 = 2047$ . Hence $B\bmod 1000 = 47$ .

We can compute $A$ using a trick known as the change of summation order.

Imagine writing down a table that has rows with labels 0 to 10. In row $x$ , write the number $2^x$ into the first $x$ columns. You will get a triangular table. Obviously, the row sums of this table are of the form $x2^x$ , and therefore the sum of all the numbers is precisely $A$ .

Now consider the ten columns in this table. Let's label them 1 to 10. In column $x$ , you have the values $2^x$ to $2^{10}$ , each of them once. And this is just a geometric series with the sum $2^{11}-2^x$ . We can now sum these column sums to get $A$ . Hence we have $A = (2^{11}-2^1) + (2^{11}-2^2) + \cdots + (2^{11}-2^{10})$ . This simplifies to $10\cdot 2^{11} - (2^1 + 2^2 + \cdots + 2^{10}) = 10\cdot 2^{11} - 2^{11} + 2$ .

Hence $A = 10\cdot 2048 - 2048 + 2 \equiv 480 - 48 + 2 = 434 \pmod{1000}$ .

Then $N = 2A - 10B \equiv 2\cdot 434 - 10\cdot 47 = 868 - 470 = \boxed{398}$ .

Solution 4

Consider the unique differences $2^{a + n} - 2^a$ . Simple casework yields a sum of $\sum_{n = 1}^{10}(2^n - 1)(2^{11 - n} - 1) = \sum_{n = 1}^{10}2^{11} + 1 - 2^n - 2^{11 - n} = 10\cdot2^{11} + 10 - 2(2 + 2^2 + \cdots + 2^{10})$ $= 10\cdot2^{11} + 10 - 2^2(2^{10} - 1)\equiv480 + 10 - 4\cdot23\equiv\boxed{398}\pmod{1000}$ . This method generalizes nicely as well.

Video Solution

https://youtu.be/JIVs2eexmVQ

@@ Line 28: / Line 28: @@
 This solution is a generalized approach that work when <math>10</math> is replaced by other values.
 In similarity to the logic in Solution 2, <math>N = \sum_{x=0}^{10} (2x-10) 2^x</math>.
-Let <math>A = \sum_{x=0}^{10} x2^x</math> and let <math>B=\sum_{x=0}^{10} 2^x</math>. Then obviously <math>N=2A - 10B</math>.
+Let <math>A = \sum_{x=0}^{10} x2^x</math> and let <math>B=\sum_{x=0}^{10} 2^x</math>. Then <math>N=2A - 10B</math>.
-Computing <math>B</math> is easy, as this is simply a geometric series with sum <math>2^{11}-1 = 2047</math>. Hence <math>B\bmod 1000 = 47</math>.
+<math>B</math> can be calculated easily by geometric series with sum <math>2^{11}-1 = 2047</math>. Hence <math>B\bmod 1000 = 47</math>.
 We can compute <math>A</math> using a trick known as the change of summation order.

2009 AIME I (Problems • Answer Key • Resources)
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