AIME 2006 II · 第 13 题

Solution 1

Let the first odd integer be $2n+1$, $n\geq 0$. Then the final odd integer is $2n+1 + 2(j-1) = 2(n+j) - 1$. The odd integers form an arithmetic sequence with sum $N = j\left(\frac{(2n+1) + (2(n+j)-1)}{2}\right) = j(2n+j)$. Thus, $j$ is a factor of $N$.

Since $n\geq 0$, it follows that $2n+j \geq j$ and $j\leq \sqrt{N}$.

Since there are exactly $5$ values of $j$ that satisfy the equation, there must be either $9$ or $10$ factors of $N$. This means $N=p_1^2p_2^2$ or $N=p_1p_2^4$. Unfortunately, we cannot simply observe prime factorizations of $N$ because the factor $(2n+j)$ does not cover all integers for any given value of $j$.

Instead we do some casework:

• If $N$ is odd, then $j$ must also be odd. For every odd value of $j$, $2n+j$ is also odd, making this case valid for all odd $j$. Looking at the forms above and the bound of $1000$, $N$ must be

$$(3^2\cdot5^2),\ (3^2\cdot7^2),\ (3^4\cdot5),\ (3^4\cdot7),\ (3^4\cdot 11)$$ Those give $5$ possibilities for odd $N$.

• If $N$ is even, then $j$ must also be even. Substituting $j=2k$, we get

$$N = 4k(n+k) \Longrightarrow \frac{N}{4} = k(n+k)$$ Now we can just look at all the prime factorizations since $(n+k)$ cover the integers for any $k$. Note that our upper bound is now $250$:

$$\frac{N}{4} = (2^2\cdot3^2),(2^2\cdot5^2),(2^2\cdot7^2), (3^2\cdot5^2), (2^4\cdot3), (2^4\cdot5), (2^4\cdot7), (2^4\cdot11), (2^4\cdot13), (3^4\cdot2)$$ Those give $10$ possibilities for even $N$.

The total number of integers $N$ is $5 + 10 = \boxed{15}$.

Solution 2

Let the largest odd number below the sequence be the $q$th positive odd number, and the largest odd number in the sequence be the $p$th positive odd number. Therefore, the sum is $p^2-q^2=(p+q)(p-q)$ by sum of consecutive odd numbers. Note that $p+q$ and $p-q$ have the same parity, and $q$ can equal $0$. We then perform casework based on the parity of $p-q$.

If $p-q$ is odd, then $p^2-q^2$ must always be odd. Therefore, to have 5 pairs of odd factors, we must have either $9$ (in which case the number is a perfect square) or $10$ factors. Considering the upper bound, the only way this can happen is $p_1^4\cdot{p_2}$ or $p_1^2\cdot{p_2^2}$. N must then be one of

$$(3^2\cdot5^2),\ (3^2\cdot7^2),\ (3^4\cdot5),\ (3^4\cdot7),\ (3^4\cdot 11)$$ So, there are $5$ solutions when $(p+q)(p-q)$ is odd.

If $p-q$ is even, then $(p+q)(p-q)$ must have at least two factors of $2$, so we can rewrite the expression as $4(k)(k-q)$ where $k=\frac{p+q}{2}$. We can disregard the $4$ by dividing by $4$ and restricting our upper bound to $250$. Since $k$ and $q$ don't have to be the same parity, we can include all cases less than 250 that have 9 or 10 factors. We then have

$$(2^2\cdot3^2),(2^2\cdot5^2),(2^2\cdot7^2), (3^2\cdot5^2), (2^4\cdot3), (2^4\cdot5), (2^4\cdot7), (2^4\cdot11), (2^4\cdot13), (3^4\cdot2)$$ as the possibilities.

Therefore, there are $10+5=\boxed{015}$ possibilities for $p^2-q^2=N$ ~sigma