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

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==Solution==
 
==Solution==
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We can't have a solution without a problem.
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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>.
  
 
==See also==
 
==See also==
 
{{AIME box|year=2021|n=II|num-b=2|num-a=4}}
 
{{AIME box|year=2021|n=II|num-b=2|num-a=4}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 15:14, 22 March 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\[x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2\]

Solution

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 $3$, after some calculation you will see that there are $16$ ways to choose $x_1, x_2, x_4,$ and $x_5$ in this way. So the desired answer is $16 \times 5=\boxed{080}$.

See also

2021 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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