A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
For finding the triangle with least surface area, base your casework on a.
Case 1) a=1 c<b+1 We know a<b<c⇒b+1≤c×
Case 2) a=2 c<a+b=b+2a<b<c⇒b+1≤c⇒c=b+1⇒(a,b,c)=(2,b,b+1) ABC is acute, so (b+1)2<b2+42b<3b≤1×
Case 3) a=3
Similarly, (a,b,c)=(3,b,b+1) or (3,b,b+2) Case 3.1) (3,b,b+1) (b+1)2<b2+92b<8b<4b≤3× Case 3.2) (3,b,b+2) (b+2)2<b2+94b<5b≤1×
Case 4) a=4 a<b<c⇒b≥5c≥6 (4,5,6) satisfies triangle inequality and acuteness since c<a+b6<4+5=9✓c2<a2+b236<16+25=41✓
Among acute triangles, increasing any side while keeping the others fixed increases the area, so 4,5,6 gives the minimum possible face area.Area1=21⋅4⋅5sinα<21⋅4⋅5sinβ=Area2⟺sinα<sinβ⟺α<β(since 0<α,β<90∘)
s=2a+b+c=24+5+6=215[ABC]=215⋅27⋅25⋅23=41⋅5⋅37=4157least total surface area=4(4157)=157
