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Omni-MATH / Real numbers x,y,z satisfy x+xy+xyz=1, y+yz+xyz=2, z+xz+xyz=4 The largest possible value of xyz is (a+b√(c))/(d), where a,b,c,d are integers, d is positive, c is square-free, and …

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problem

Real numbers x,y,zx, y, z satisfy x+xy+xyz=1,y+yz+xyz=2,z+xz+xyz=4x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4 The largest possible value of xyzx y z is a+bcd\frac{a+b \sqrt{c}}{d}, where a,b,c,da, b, c, d are integers, dd is positive, cc is square-free, and gcd(a,b,d)=1\operatorname{gcd}(a, b, d)=1. Find 1000a+100b+10c+d1000 a+100 b+10 c+d.
Plain-text mathematical notation (without MathML)
Real numbers x,y,z satisfy x+xy+xyz=1, y+yz+xyz=2, z+xz+xyz=4 The largest possible value of xyz is (a+b√(c))/(d), where a,b,c,d are integers, d is positive, c is square-free, and gcd(a,b,d)=1. Find 1000a+100b+10c+d.
Original LaTeX notation
Real numbers $x, y, z$ satisfy $$x+x y+x y z=1, \quad y+y z+x y z=2, \quad z+x z+x y z=4$$ The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.

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Source and history

task_id
25298ba8-7fab-5c2d-b4fb-b567c22c6729
task_key
test--25298ba8-7fab-5c2d-b4fb-b567c22c6729
task_revision_id
1
config / split
/ test

Official source

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