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Let ABCD be a regular tetrahedron with edge length s. Points X, Y and Z lie on rays AB, AC and AD respectively (beyond A or between A and the vertex) such that XY = YZ = 7 and XZ = 5. The lengths AX, AY, AZ are all distinct. If the volume of tetrahedron AXYZ equals sqrt(lambda), find the value of lambda.

1. 100
2. 122
3. 144
4. 169

Correct answer: 122

Solution

In a regular tetrahedron, angle between edges from A = 60 deg (cos theta = 1/2). Let AX=p, AY=q, AZ=r. By cosine rule: XY² = p²+q²-pq=49, YZ²=q²+r²-qr=49, XZ²=p²+r²-pr=25. Subtracting eq1 from eq2: (r²-p²)-q(r-p)=0 => (r-p)(r+p-q)=0. Since all three are distinct, p!=r, so q=p+r. Substituting into eq1: p²+pr+r²=49. From eq3: p²+r²-pr=25. Subtracting: 2pr=24, pr=12. Thus p²+r²=37, q²=(p+r)²=37+24=61, q=sqrt(61). The Gram determinant of three unit vectors with pairwise angle 60 deg = 1-3*(1/4)+2*(1/8) = 1/2. V = (pqr/6)*sqrt(1/2) = (12*sqrt(61))/(6*sqrt(2)) = 2*sqrt(61)/sqrt(2) = sqrt(122). So lambda=122.

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