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1)Donatello starts with a marble cube of side length $10.$ He then slices a pyramid off each corner, so that in the resulting polyhedron, all the edges have the same side length $s.$ Find $s.$

 

[asy] import three; size(7cm); unitsize(1 cm); currentprojection=perspective(6,3,2); triple A, B, C, D, E, F, G, H; A = (1,1,0); B = (1,0,0); C = (0,0,0); D = (0,1,0); E = (1,1,1); F = (1,0,1); G = (0,0,1); H = (0,1,1); draw(A--B--F--G--H--D--cycle); draw(E--A); draw(E--F); draw(E--H); draw((2*F + B)/3--(2*F + E)/3--(2*F + G)/3); draw((2*E + A)/3--(2*E + F)/3--(2*E + H)/3--cycle); draw((2*A + B)/3--(2*A + E)/3--(2*A + D)/3); draw((2*D + A)/3--(2*D + H)/3); draw((2*B + A)/3--(2*B + F)/3); draw((2*H + D)/3--(2*H + E)/3--(2*H + G)/3); draw((2*G + F)/3--(2*G + H)/3); //dot("$C$", E, SW); //dot("$X$", (2*E + H)/3, SE); //dot("$Y$", (2*E + F)/3, SW); //dot("$Z$", (2*E + A)/3, SW); [/asy]

 

2)Let $ABCDEFGH$ be a rectangular prism, where $EF = 4$, $EH = 5$, and $EA = 6$. Find the volume of pyramid $CFAH$.

 

[asy] unitsize(1 cm); pair A, B, C, D, E, F, G, H; A = (0,0); B = (0,2); C = (0,2) + 2*dir(-30); D = 2*dir(-30); E = A + 3*dir(20); F = B + 3*dir(20); G = C + 3*dir(20); H = D + 3*dir(20); fill(A--H--F--cycle,gray(0.8)); draw(A--B--F--G--H--D--cycle); draw(C--B); draw(C--G); draw(C--D); draw(E--A,dashed); draw(E--F,dashed); draw(E--H,dashed); draw(C--A); draw(C--F); draw(C--H); draw(A--H--F--cycle,dashed); label("$A$", A, SW); label("$B$", B, NW); label("$C$", C, N); label("$D$", D, S); label("$E$", E, S); label("$F$", F, N); label("$G$", G, NE); label("$H$", H, SE); [/asy]

 

3)Let $A_1 A_2 A_3 A_4$ be a regular tetrahedron. Let $P_1$ be the center of face $A_2 A_3 A_4,$ and define vertices $P_2,$ $P_3,$ and $P_4$ the same way. Find the ratio of the volume of tetrahedron $A_1 A_2 A_3 A_4$ to the volume of tetrahedron $P_1 P_2 P_3 P_4.$

 

[asy] import three; size(7cm); unitsize(1 cm); currentprojection=perspective(6,3,2); triple A, B, C, D, E, F, G, H; real ang = 40; A = (Cos(ang), Sin(ang), 0); B = (Cos(ang + 120), Sin(ang + 120), 0); C = (Cos(ang + 240), Sin(ang + 240), 0); D = (0,0,sqrt(2)); E = (B + C + D)/3; F = (A + C + D)/3; G = (A + B + D)/3; H = (A + B + C)/3; draw(B--A--C--D--cycle); draw(A--D); draw(B--C,dashed); draw(E--F--G--H--cycle,dashed); draw(E--G,dashed); draw(F--H,dashed); label("$A_1$", A, S); label("$A_2$", B, dir(0)); label("$A_3$", C, W); label("$A_4$", D, N); dot("$P_1$", E, N); dot("$P_2$", F, W); dot("$P_3$", G, dir(0)); dot("$P_4$", H, S); [/asy]

 

Note: A tetrahedron is called regular if all the edges lengths are equal, so all the faces are equilateral triangles.

 Feb 20, 2020
 #1
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Stop attempting to cheat on your AoPS homework

you didnt even bother to remove the asymptote

 Mar 12, 2020

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