# 2001 PMWC Problems

## Problem I3

$[asy] void wdot(pair p) { fill(circle(p,.025),white); draw(circle(p,.025),black); return; } import patterns; real eta=pi/2; pair A,B,C,D,E,F,G,O; A=origin; B=(1,0); C=(1,1); D=(0,1); O=(A+B+C+D)/4; draw(A--B--C--D--cycle); E=(O.x+cos(-eta/4),O.y+sin(-eta/4)); G = rotate(90,O)*E; F=O+(G-O)+(E-O); draw(O--G--F--E--cycle); add("crosshatch",crosshatch(.6mm)); pair a1, a2; a1 = intersectionpoint(O--G,D--C); a2 = intersectionpoint(B--C,O--E); fill(O--a1--C--a2--cycle,pattern("crosshatch")); wdot(F); wdot(G); wdot(O); wdot(a1); wdot(a2); wdot(A); wdot(B); wdot(C); wdot(D); wdot(E); label("A",A,SW); label("B",B,SW); label("C",C,E); label("D",D,W); label("E",E,S); label("F",F,E); label("G",G,NE); label("O",O,W); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem I11

$[asy] size(8cm); defaultpen(fontsize(11)); pair A,B,C,D; A=origin; B=(0.523713039560712, 0.150288579021811); C=(0.564402523837359, 1.185901582062919); D=(-0.640531236354945, 1.247592090482351); pair Ap,Bp,Cp,Dp; Ap=rotate(180,B)*A; Bp=rotate(180,C)*B; Cp=rotate(180,D)*C; Dp=rotate(180,A)*D; draw(A--Ap--Bp--Cp--Dp--D--C--B); draw(Dp--Ap); draw(Bp--C); draw(Cp--D); label("A",A,SW); label("B",B,S); label("C",C,E); label("D",D,N); label("A'",Ap,unit(Ap)); label("B'",Bp,unit(Bp)); label("C'",Cp,W); label("D'",Dp,unit(Dp)); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem I14

$[asy] size(9cm); real horiscale=11/8; for(int i = 0; i < 4; ++i) { for(int j = 0; j < 3; ++j) { draw((i*horiscale,j)--((i+1)*horiscale,j)--((i+1)*horiscale,j+1)--(i*horiscale,j+1)--cycle); if((i == 0 && j == 1) || (j == 0 && i > 0) || (i == 2 && j == 2)) { label( "\bullet",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 ); } else { label( "\circ",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 ); } } } draw(origin--(-1.3*3,0)--(-1.3*3,3)--(0,3)--cycle); label("Hotel",(-1.3*3/2,5/2)); label("This side up",(-1.3*3/2,3/2)); label("(This is an example of one possibility)", ((-1.3*3+1.3*4)/2,-.4),fontsize(11)); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem T3

$[asy] pair A = (0,4); pair B = (7,0); void drawvert(int xstart, int len) { draw((xstart,A.y)--(xstart,A.y-len)); } void drawhori(pair start, int len) { draw(start--(start.x+len,start.y)); } //draw verticals int[] verts = {4,4,4,2,4,3,4,4}; for(int i = 0; i < verts.length; ++i) { drawvert(i,verts[i]); } //draw horizontals drawhori(A,7); drawhori(A - (0,1),7); drawhori(A - (0,2),4); drawhori((B.x-1,2),1); drawhori((0,1),2); drawhori((4,1),3); drawhori(origin,7); //labels label("A",A,NW); label("B",B,SE); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem T9

$[asy] void wdot(pair p) { fill(circle(p,.025),white); draw(circle(p,.025),black); return; } draw(circle(origin,1)); pair z = expi(3pi/10); pair[] rotations; int max = 7; for(int i = 0; i < max; ++i) { rotations.push(rotate(-360*i/7)*z); } int i = 1; for(pair x : rotations) { wdot(x); label("" + string(i) + "", x , unit(x)); ++i; } //Credit to dasobson for the diagram[/asy]$ Solution