2018 AIME I Problems/Problem 1

Problem 1

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$ .

Solution

You let the linear factors be as $(x+c)(x+d)$ .

Then, obviously $a=c+d$ and $b=cd$ .

We know that $1\le a\le 100$ and $b\ge 0$ , so $c$ and $d$ both have to be positive.

However, $a$ cannot be $0$ , so at least one of $c$ and $d$ must be greater than $0$ , ie positive.

Also, $a$ cannot be greater than $100$ , so $c+d$ must be less than or equal to $100$ .

Essentially, if we plot the solutions, we get a triangle on the coordinate plane with vertices $(0,0), (0, 100),$ and $(100,0)$ . Remember that $(0,0)$ does not work, so there is a square with top right corner $(1,1)$ .

Note that $c$ and $d$ are interchangeable, since they end up as $a$ and $b$ in the end anyways. Thus, we simply draw a line from $(1,1)$ to $(50,50)$ , designating one of the halves as our solution (since the other side is simply the coordinates flipped).

We note that the pattern from $(1,1)$ to $(50,50)$ is $2+3+4+\dots+51$ solutions and from $(51, 49)$ to $(100,0)$ is $50+49+48+\dots+1$ solutions, since we can decrease the $y$ -value by $1$ until $0$ for each coordinate.

Adding up gives $\dfrac{2+51}{2}\cdot 50+\dfrac{50+1}{2}\cdot 50.$ This gives us $2600$ , and $2600\equiv 600 \bmod{1000}.$

Thus, the answer is: $\boxed{600}.$

Solution 2

a-value b-value(s) amount of b-value(s)

a=1	b=0	1

a=2 b=1, 0 2 a=3 b=2, 0 2 a=4 b=4, 3, 0 3 a=5 b=6, 4, 0 3 a=6 b=9, 8, 5, 0 4 a=7 b=12, 10, 6, 0 4 a=8 b=16, 15, 12, 7, 0 5 a=9 b=20, 18, 14, 8, 0 5 a=10 b=25, 24, 21, 16, 9, 0 6 a=11 b=30, 28, 24, 18, 10, 0 6 Excluding a=1, we can see there is always a pair of 2 a-values for a certain amount of b-values.

For instance, a=2 and a=3 both have 2 b-values. a=4 and a=5 both have 3 b-values.

We notice the pattern of the amount of b-values in relation to the a values: 1, 2, 2, 3, 3, 4, 4…

The pattern continues until a=100, and in total, there are 49 pairs of a-value with the same amount of b-values. The two lone a-values without a pair are, the (a=1, amount of b-values=1) in the beginning, and (a=100, amount of b-values=51) in the end.

Then, we add numbers from the opposite ends of the spectrum, and quickly notice that there are 50 pairs each with a sum of 52. 52x50 gives 2600 ordered pairs:

1+51, 2+50, 2+50, 3+49, 3+49, 4+48, 4+48…

When divided by 1000, it gives the remainder 600, the answer.

2018 AIME I (Problems • Answer Key • Resources)
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2018 AIME I Problems/Problem 1

Problem 1

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