Difference between revisions of "2021 AIME II Problems/Problem 7"
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==Solution== | ==Solution== | ||
− | From the fourth equation we get < | + | From the fourth equation we get <math> d=\frac{30}{abc}. </math> substitute this into the third equation and you get <math>abc + \frac{30(ab + bc + ca)}{abc} = abc - \frac{120}{abc} = 14</math>. Hence <math>(abc)^2 - 14(abc)-120 = 0</math>. Solving we get <math>abc = -6</math> or <math>abc = 20</math>. From the first and second equation we get <math>ab + bc + ca = ab-3c = -4 \Longrightarrow ab = 3c-4</math>, if <math>abc=-6</math>, substituting we get <math>c(3c-4)=-6</math>. If you try solving this you see that this does not have real solutions in <math>c</math>, so <math>abc</math> must be <math>20</math>. So <math>d=\frac{3}{2}</math>. Since <math>c(3c-4)=20</math>, <math>c=-2</math> or <math>c=\frac{10}{3}</math>. If <math>c=\frac{10}{3}</math>, then the system <math>a+b=-3</math> and <math>ab = 6</math> does not give you real solutions. So <math>c=-2</math>. From here you already know <math>d=\frac{3}{2}</math> and <math>c=-2</math>, so you can solve for <math>a</math> and <math>b</math> pretty easily and see that <math>a^{2}+b^{2}+c^{2}+d^{2}=\frac{141}{4}</math>. So the answer is <math>\boxed{145}</math>. |
==See also== | ==See also== | ||
{{AIME box|year=2021|n=II|num-b=6|num-a=8}} | {{AIME box|year=2021|n=II|num-b=6|num-a=8}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 14:49, 22 March 2021
Problem
Let and be real numbers that satisfy the system of equations There exist relatively prime positive integers and such that Find .
Solution
From the fourth equation we get substitute this into the third equation and you get . Hence . Solving we get or . From the first and second equation we get , if , substituting we get . If you try solving this you see that this does not have real solutions in , so must be . So . Since , or . If , then the system and does not give you real solutions. So . From here you already know and , so you can solve for and pretty easily and see that . So the answer is .
See also
2021 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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