1986 AIME Problem 4

Attempt Problem 4 of the 1986 AIME below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1986 AIME solutions, or check the answer key.

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4.

Determine 3x4+2x53x_4+2x_5 if x1,x_1, x2,x_2, x3,x_3, x4,x_4, and x5x_5 satisfy the system 2x1+x2+x3+x4+x5=6,x1+2x2+x3+x4+x5=12,x1+x2+2x3+x4+x5=24,x1+x2+x3+2x4+x5=48,x1+x2+x3+x4+2x5=96. \begin{aligned} 2x_1+x_2+x_3+x_4+x_5&=6,\\ x_1+2x_2+x_3+x_4+x_5&=12,\\ x_1+x_2+2x_3+x_4+x_5&=24,\\ x_1+x_2+x_3+2x_4+x_5&=48,\\ x_1+x_2+x_3+x_4+2x_5&=96. \end{aligned}

Answer: 181
Concepts:algebraic manipulationsystem of equations
Difficulty rating: 1760
Small Hint:

Let S=x1+x2+x3+x4+x5S=x_1+x_2+x_3+x_4+x_5

Big Hint:

Each equation has the form S+xi=S+x_i= a constant

Solution:

Put S=x1+x2+x3+x4+x5.S=x_1+x_2+x_3+x_4+x_5. The five equations say that S+xiS+x_i equals 6,6, 12,12, 24,24, 48,48, 96,96, respectively. Adding them gives 6S=6+12+24+48+96=186, \begin{aligned} 6S&=6+12+24+48+96\\ &=186, \end{aligned} so S=31.S=31. Hence x4=4831=17x_4=48-31=17 and x5=9631=65.x_5=96-31=65. The requested value is 3(17)+2(65)=181.3(17)+2(65)=181.

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