MathDB
CUBES LOOK cOoL

Source: 2024 AMC 8 Problem 20

January 26, 2024
AMC 8geometry3D geometry2024 AMC 8

Problem Statement

Any three vertices of the cube PQRSTUVW,PQRSTUVW, shown in the figure below, can be connected to form a triangle. ((For example, vertices P,Q,P, Q, and RR can be connected to form △PQR.)\triangle{PQR}.) How many of these triangles are equilateral and contain PP as a vertex?
<spanclass=′latex−bold′>(A)</span>0<spanclass=′latex−bold′>(B)</span>1<spanclass=′latex−bold′>(C)</span>2<spanclass=′latex−bold′>(D)</span>3<spanclass=′latex−bold′>(E)</span>6<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>3\qquad<span class='latex-bold'>(E) </span>6 [asy] unitsize(4); pair P,Q,R,S,T,U,V,W; P=(0,30); Q=(30,30); R=(40,40); S=(10,40); T=(10,10); U=(40,10); V=(30,0); W=(0,0); draw(W--V); draw(V--Q); draw(Q--P); draw(P--W); draw(T--U); draw(U--R); draw(R--S); draw(S--T); draw(W--T); draw(P--S); draw(V--U); draw(Q--R); dot(P); dot(Q); dot(R); dot(S); dot(T); dot(U); dot(V); dot(W); label("PP",P,NW); label("QQ",Q,NW); label("RR",R,NE); label("SS",S,N); label("TT",T,NE); label("UU",U,NE); label("VV",V,SE); label("WW",W,SW); [/asy]
Back to ProblemsView on AoPS