AIME 2015 I · 第 1 题

Solution 1

We have

$$|A-B|=|1+3(4-2)+5(6-4)+ \cdots + 37(38-36)-39(1-38)|$$ $$\implies |2(1+3+5+7+ \cdots +37)-1-39(37)|$$ $$\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\implies \boxed{722}$$

Solution 2

We see that

$A=(1\times 2)+(3\times 4)+(5\times 6)+\cdots +(35\times 36)+(37\times 38)+39$

and

$B=1+(2\times 3)+(4\times 5)+(6\times 7)+\cdots +(36\times 37)+(38\times 39)$.

Therefore,

$B-A=-38+(2\times 2)+(2\times 4)+(2\times 6)+\cdots +(2\times 36)+(2\times 38)$ $=-38+4\times (1+2+3+\cdots+19)$ $=-38+4\times\frac{20\cdot 19}{2}=-38+760=\boxed{722}.$

Solution 3 (slower solution)

For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations.

We write down the pairs of numbers after multiplication and solve each layer:

$$2, 12, 30, 56, 90...(39)$$ $$10, 18, 26, 34...$$ $$8, 8, 8...$$ and

$$(1) 6, 20, 42, 72...$$ $$14, 22, 30...$$ $$8, 8, 8...$$ Then we use Newton's Little Formula for the sum of $n$ terms in a sequence.

Notice that there are $19$ terms in each sequence, plus the tails of $39$ and $1$ on the first and second equations, respectively.

So,

$$2\binom{19}{1}+10\binom{19}{2}+8\binom{19}{3}+1$$ $$6\binom{19}{1}+14\binom{19}{2}+8\binom{19}{3}+39$$ Subtracting $A$ from $B$ gives:

$$4\binom{19}{1}+4\binom{19}{2}-38$$ Which unsurprisingly gives us $\boxed{722}.$

-jackshi2006

Solution 4 (Cheezy Peezy!)

We can stockpile A onto B, like this, so an expression like $(1 \times 2) + (3 \times 4) + (5 \times 6) + ... + (37\times 38) + 39 - [(2 \times 3) + (4 \times 5) + ... + (38 \times 39) +1]$ is formed. This can be achieved by rearranging the one in this expression to be under the 39, so that there are neat even multpiles.

Next, using paper, one can stock up the expressions like the image shown below. We notice that we have stacked up a lot of even multiples, such as multpiles of 2, 4, 6, 8, et cetera, going on to 38.

Now, notice in the first "column", of $(1 \times 2) - (2 \times 3)$, the bottom number multiplying 2 is always two greater than the top number multiplying 2. This goes for all of the even multpiles, up to 38. This means that when the bottom expression is subtracted from the top expression, we are left with a lot of $-2$s being multiplied by even numbers. It cascades all the way to 38, meaning that the value of the chunk (excluding the 39-1) adds up to $-2\times 2 + -2 \times 4 + -2 \times 6 + ... + -2 \times 38$.

By the distributive property, the number is also equivalent to $-2(2+4+6+...+38)$. A two can also be factored out of this expression from the parentheses, giving $-2(2(1+2+...+19))$. Finally, because $-2 \times 2 = 4$, as well as the fact that the sum of the first $n$ positive integers is $\frac{(n)(n+1}{2}$, we get the expression $-4(\frac{(19)(20)}{2}) = -4 \times 380 \div 2 = -4 \times 190 = -760.$

However, there is also another thing at the end; the $39 - 1$, using the 1 that was moved earlier! This means we need to add 38 to the number. Because we are in the negatives, adding 38 gives a lower absolute value. Using arithmetic, $-760 + 38 = -722$, but because the answer is the absolute value, the answer is $\boxed{722}$.

~AlgowheelAZ1