Difference between revisions of "2021 AIME II Problems/Problem 3"

Revision as of 19:27, 9 September 2021

Problem

Find the number of permutations $x_1, x_2, x_3, x_4, x_5$ of numbers $1, 2, 3, 4, 5$ such that the sum of five products $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$ is divisible by $3$ .

Solution 1

Since $3$ is one of the numbers, a product with a $3$ in it is automatically divisible by $3$ , so WLOG $x_3=3$ , we will multiply by $5$ afterward since any of $x_1, x_2, ..., x_5$ would be $3$ , after some cancelation we see that now all we need to find is the number of ways that $x_5x_1(x_4+x_2)$ is divisible by $3$ , since $x_5x_1$ is never divisible by $3$ , now we just need to find the number of ways $x_4+x_2$ is divisible by $3$ . $x_2$ and $x_4$ can be $(1, 2); (2, 1); (1, 5); (5, 1); (2, 4); (4, 2); (4, 5); (5, 4)$ . We have $2$ ways to designate $x_1$ and $x_5$ for a total of $8 \cdot 2 = 16$ . So the desired answer is $16 \times 5=\boxed{080}$ .

~ math31415926535

~MathFun1000 (Rephrasing for clarity)

Solution 2 (Cyclic Symmetry and Casework)

The expression $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2$ has cyclic symmetry. Without the loss of generality, let $x_1=3.$ It follows that $\{x_2,x_3,x_4,x_5\}=\{1,2,4,5\}.$ We have:

$x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\equiv x_2x_3x_4 + x_3x_4x_5\pmod{3}.$

$x_2,x_3,x_4,x_5$ are congruent to $1,2,1,2\pmod{3}$ in some order.

We construct the following table for the case $x_1=3,$ with all values in modulo $3:$ $\begin{array}{c||c|c|c|c|c||c} & & & & & & \\ [-2.5ex] \textbf{Row} & \boldsymbol{x_2} & \boldsymbol{x_3} & \boldsymbol{x_4} & \boldsymbol{x_5} & \boldsymbol{x_2x_3x_4 + x_3x_4x_5} & \textbf{Valid?} \\ [0.5ex] \hline & & & & & & \\ [-2ex] 1 & 1 & 1 & 2 & 2 & 0 & \checkmark \\ 2 & 1 & 2 & 1 & 2 & 0 & \checkmark \\ 3 & 1 & 2 & 2 & 1 & 2 & \\ 4 & 2 & 1 & 1 & 2 & 1 & \\ 5 & 2 & 1 & 2 & 1 & 0 & \checkmark \\ 6 & 2 & 2 & 1 & 1 & 0 & \checkmark \end{array}$ For Row 1, $(x_2,x_3)$ can be either $(1,4)$ or $(4,1),$ and $(x_4,x_5)$ can be either $(2,5)$ or $(5,2).$ By the Multiplication Principle, Row 1 produces $2\cdot2=4$ permutations. Similarly, Rows 2, 5, and 6 each produce $4$ permutations.

Together, we get $4\cdot4=16$ permutations for the case $x_1=3.$ By the cyclic symmetry, the cases $x_2=3, x_3=3, x_4=3,$ and $x_5=3$ all have the same count. Therefore, the total number of permutations $x_1, x_2, x_3, x_4, x_5$ is $16\cdot5=\boxed{080}.$

~MRENTHUSIASM

Solution 3

WLOG, let $x_{3} = 3$ So, the terms $x_{1}x_{2}x_{3}, x_{2}x_{3}x_{4},x_{3}x_{4}x_{5}$ are divisible by $3$ .

We are left with $x_{4}x_{5}x_{1}$ and $x_{5}x_{1}x_{2}$ . We need $x_{4}x_{5}x_{1} + x_{5}x_{1}x_{2} \equiv 0 (mod 3)$ The only way is when They are $(+1,-1)$ or $(-1, +1)$ (mod 3)

The numbers left with us are $1,2,4,5$ which are $+1,-1,+1,-1$ (mod $3$ ) respectively.

$+1$ (of $x_{4}x_{5}x_{1}$ or $x_{5}x_{1}x_{2}$ ) $= +1 \cdot +1 \cdot +1$ $\;\;\; OR \;\;\;+1$ (of $x_{4}x_{5}x_{1}$ or $x_{5}x_{1}x_{2}$ ) = $-1 \cdot -1 \cdot +1$ .

$-1$ (of $x_{4}x_{5}x_{1}$ or $x_{5}x_{1}x_{2}$ ) $= -1 \cdot -1 \cdot -1$ $\;\;\; OR \;\;\;-1$ (of $x_{4}x_{5}x_{1}$ or $x_{5}x_{1}x_{2}$ ) = $-1 \cdot +1 \cdot +1$

But, as we have just two $+1's$ and two $-1's$ , Hence, We will have to take $+1 = +1 \cdot -1 \cdot -1$ and $-1 = -1 \cdot +1 \cdot +1$ Among these two, we have a $+1$ and $-1$ in common, i.e. $(x_{5}, x_{1}) = (+1, -1) or (-1, +1)$ (because $x_{1}$ and $x_{5}$ are common in $x_{4}x_{5}x_{1}$ and $x_{5}x_{1}x_{2}$ )

So, $(x_{5}, x_{1}) \in {(1,2), (1,5), (4,2), (4,5), (2,1), (5,1), (2,4), (5,4)}$ i.e. $8$ values.

For each value of $(x_{5}, x_{1})$ we get $2$ values for $(x_{2}, x_{4})$ Hence, in total, we have $8 \times 2 = 16$ ways.

But any of the $x_{i} 's$ can be $3$ So, $16 \times 5 = \boxed{080}$

-Arnav Nigam

Video Solution

https://www.youtube.com/watch?v=HikWWhQlkVw

@@ Line 4: / Line 4: @@
 ==Solution 1==
-Since <math>3</math> is one of the numbers, a product with a <math>3</math> in it is automatically divisible by <math>3</math>, so WLOG <math>x_3=3</math>, we will multiply by <math>5</math> afterward since any of <math>x_1, x_2, ..., x_5</math> would be <math>3</math>, after some cancelation we see that now all we need to find is the number of ways that <math>x_5x_1(x_4+x_2)</math> is divisible by <math>3</math>, since <math>x_5x_1</math> is never divisible by <math>3</math>, now we just need to find the number of ways <math>x_4+x_2</math> is divisible by <math>3</math>, after some calculation you will see that there are <math>16</math> ways to choose <math>x_1, x_2, x_4,</math> and <math>x_5</math> in this way. So the desired answer is <math>16 \times 5=\boxed{080}</math>.
+Since <math>3</math> is one of the numbers, a product with a <math>3</math> in it is automatically divisible by <math>3</math>, so WLOG <math>x_3=3</math>, we will multiply by <math>5</math> afterward since any of <math>x_1, x_2, ..., x_5</math> would be <math>3</math>, after some cancelation we see that now all we need to find is the number of ways that <math>x_5x_1(x_4+x_2)</math> is divisible by <math>3</math>, since <math>x_5x_1</math> is never divisible by <math>3</math>, now we just need to find the number of ways <math>x_4+x_2</math> is divisible by <math>3</math>. <math>x_2</math> and <math>x_4</math> can be <math>(1, 2); (2, 1); (1, 5); (5, 1); (2, 4); (4, 2); (4, 5); (5, 4)</math>. We have <math>2</math> ways to designate <math>x_1</math> and <math>x_5</math> for a total of <math>8 \cdot 2 = 16</math>. So the desired answer is <math>16 \times 5=\boxed{080}</math>.
 ~ math31415926535
+~MathFun1000 (Rephrasing for clarity)
 ==Solution 2 (Cyclic Symmetry and Casework)==

2021 AIME II (Problems • Answer Key • Resources)
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