2019 AMC 10A Problems/Problem 19

Problem

What is the least possible value of $(x+1)(x+2)(x+3)(x+4)+2019$ where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Solution 1

Grouping the first and last terms and two middle terms gives $(x^2+5x+4)(x^2+5x+6)+2019$ , which can be simplified to $(x^2+5x+5)^2-1+2019$ . Noting that squares are nonnegative, and verifying that $x^2+5x+5=0$ for some real $x$ , the answer is $\boxed{\textbf{(B) } 2018}$ .

Solution 2

Let $a=x+\tfrac{5}{2}$ . Then the expression $(x+1)(x+2)(x+3)(x+4)$ becomes $\left(a-\tfrac{3}{2}\right)\left(a-\tfrac{1}{2}\right)\left(a+\tfrac{1}{2}\right)\left(a+\tfrac{3}{2}\right)$ .

We can now use the difference of two squares to get $\left(a^2-\tfrac{9}{4}\right)\left(a^2-\tfrac{1}{4}\right)$ , and expand this to get $a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}$ .

Refactor this by completing the square to get $\left(a^2-\tfrac{5}{4}\right)^2-1$ , which has a minimum value of $-1$ . The answer is thus $2019-1=\boxed{\textbf{(B) }2018}$ .

Solution 3 (calculus)

Similar to Solution 1, grouping the first and last terms and the middle terms, we get $(x^2+5x+4)(x^2+5x+6)+2019$ .

Letting $y=x^2+5x$ , we get the expression $(y+4)(y+6)+2019$ . Now, we can find the critical points of $(y+4)(y+6)$ to minimize the function:

$\frac{d}{dx}(y^2+10y+24)=0$

$2y+10=0$

$2y(y+5)=0$

$y=-5,0$

To minimize the result, we use $y=-5$ . Hence, the minimum is $(-5+4)(-5+6)=-1$ , so $-1+2019 = \boxed{\textbf{(B) }2018}$ .

Note: We could also have used the result that minimum/maximum point of a parabola $y = ax^2 + bx + c$ occurs at $x=-\frac{b}{2a}$ .

Solution 4(guess with answer choices)

The expression is negative when an odd number of the factors are negative. This happens when $-2 < x < -1$ or $-4 < x < -3$ . Plugging in $x = -\frac32$ or $x = -\frac72$ yields $-\frac{15}{16}$ , which is very close to $-1$ . Thus the answer is $-1 + 2019 = \boxed{\textbf{(B) }2018}$ .

Solution 5 (using the answer choices)

Answer choices $C$ , $D$ , and $E$ are impossible, since $(x+1)(x+2)(x+3)(x+4)$ can be negative (as seen when e.g. $x = -\frac{3}{2}$ ). Plug in $x = -\frac{3}{2}$ to see that it becomes $2019 - \frac{15}{16}$ , so round this to $\boxed{\textbf{(B) }2018}$ .

We can also see that the limit of the function is at least -1 since at the minimum, two of the numbers are less than 1, but two are between 1 and 2.

Solution 6 (no words)

$\left(x+1\right)\left(x+2\right)\left(x+3\right)\left(x+4\right)+2019$ $=\left(x+2\right)\left(x+3\right)\left(x+1\right)\left(x+4\right)+2019$ $=\left(x^{2}+5x+6\right)\left(x^{2}+5x+4\right)+2019$ $=\left(\left(x^{2}+5x+5\right)+1\right)\left(\left(x^{2}+5x+5\right)-1\right)+2019$ $=\left(x^{2}+5x+5\right)^{2}-1^{2}+2019$ .

$x^{2}+5x+5=0 \Rightarrow x=\frac{-5\pm\sqrt{25-20}}{2}=\frac{-5\pm\sqrt{5}}{2}$ .

$\left(x^{2}+5x+5\right)^{2}=0$

$0-1^{2}+2019=2018 \Rightarrow \boxed{B}$ .

Solution 7

Arcticturn

Video Solutions

https://www.youtube.com/watch?v=Lis8yKT9WXc (less than 2 minutes)

https://youtu.be/NRa3VnjNVbw - Education, the Study of Everything
https://www.youtube.com/watch?v=Mfa7j2BoNjI
https://youtu.be/tIzJtgJbHGc - savannahsolver
https://youtu.be/3dfbWzOfJAI?t=3319 - pi_is_3.14
https://youtu.be/GmUWIXXf_uk?t=1134 ~ pi_is_3.14

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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