Difference between revisions of "2018 AIME I Problems/Problem 1"

Revision as of 17:39, 15 October 2018

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 non-negative

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 (less complicated)

Notice that for $x^2+ax+b$ to be true, for every $a$ , $b$ will always be the product of the possibilities of how to add two integers to $a$ . For example, if $a=3$ , $b$ will be the product of $(3,0)$ and $(2,1)$ , as those two sets are the only possibilities of adding two integers to $a$ . Note that order does not matter. If we just do some simple casework, we find out that:

if $a$ is odd, there will always be $\left\lceil\frac{a}{2}\right\rceil$ $\left(\text{which is also }\frac{a+1}{2}\right)$ possibilities of adding two integers to $a$ .

if $a$ is even, there will always be $\frac{a}{2}+1$ possibilities of adding two integers to $a$ .

Using the casework, we have $1+2+2+3+3+...50+50+51$ possibilities. This will mean that the answer is $\frac{(1+51)\cdot100}{2}\Rightarrow52\cdot50=2600$ possibilities.

Thus, our solution is $2600\bmod {1000}\equiv\boxed{600}$ .

Solution by IronicNinja~

Solution 3

Let's write the linear factors as (x+n)(x+m).

Then we can write them as: a=n+m, b=nm.

n or m has to be a positive integer as a cannot be 0.

n+m has to be between 1 and 100, as a cannot be over 100.

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 number of b-values in relation to the a-values: 1, 2, 2, 3, 3, 4, 4…

The following link is the URL to the graph I drew showing the relationship between a-values and b-values http://artofproblemsolving.com/wiki/index.php?title=File:Screen_Shot_2018-04-30_at_8.15.00_PM.png#file

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.

Solution provided by- Yonglao

@@ Line 8: / Line 8: @@
 Then, obviously <math>a=c+d</math> and <math>b=cd</math>.
-We know that <math>1\le a\le 100</math> and <math>b\ge 0</math>, so <math>c</math> and <math>d</math> both have to be positive.
+We know that <math>1\le a\le 100</math> and <math>b\ge 0</math>, so <math>c</math> and <math>d</math> both have to be non-negative
 However, <math>a</math> cannot be <math>0</math>, so at least one of <math>c</math> and <math>d</math> must be greater than <math>0</math>, ie positive.

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