# 2002 PMWC Problems

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## Problem I13

$[asy] size(8cm); int[][] triangle; int nrows=5; int count = 1; for(int i = 0; i < nrows; ++i) { int[] temp; for(int j = 0; j <= i; ++j) { temp.push(count); count+=2; } triangle.push(temp); } for(int i = 0; i < nrows; ++i) { for(int j = 0; j <= i; ++j) { label( string(triangle[i][j]), (-i/2+j,-i*sqrt(3)/2) ); } } real correction=.175; string[] ord = {"st", "nd", "rd"}; while(ord.length < nrows) { ord.push("th"); } for(int i = 1; i <= nrows; ++i) { label("" + string(i) + "^\textrm{" + ord[i-1] + "} Row:", (-5,(-i+1)*sqrt(3)/2+correction)); } int orows=3; real cradius=.07; for(int i = nrows; i < orows + nrows; ++i) { for(int j = 0; j <= i; ++j) { fill( circle((-i/2+j,-i*sqrt(3)/2), cradius) ); } } //Credit to dasobson for the diagram[/asy]$

$[asy] size(2cm); defaultpen(fontsize(15)); pen dts=linetype("5 5"); real correction=2/5; label("\textbf{A}",(-1,0)); label("\textbf{B}",(1,0)); label("\textbf{C}",(0,-sqrt(3))); draw((-1 + correction,0)--(1 - correction,0),dts); draw( (1-cos(pi/3)*correction,-sin(pi/3)*correction)--(0+cos(pi/3)*correction,-sqrt(3)+sin(pi/3)*correction), dts ); draw( (-1+cos(pi/3)*correction,-sin(pi/3)*correction)--(0-cos(pi/3)*correction,-sqrt(3)+sin(pi/3)*correction), dts); //Credit to dasobson for the diagram[/asy]$

## Problem T3

$[asy] size(8cm); import patterns; pair A,B,C,D,E,F,G; A=(0,4); B=origin; C=(4,0); D=(4,4); E=(7,0); F=(7,3); G=(4,3); draw(A--B--E--F--G--C--D--cycle); add("crosshatch",crosshatch(.6mm)); fill(A--G--E--cycle,pattern("crosshatch")); label("A",A,NW); label("B",B,SW); label("C",C,S); label("D",D,NE); label("E",E,SE); label("F",F,NE); label("G",G,NE); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem T7

$[asy] size(7cm); real eta=pi/2; pair[] hexvertices; for(int i = 0; i < 6; ++i) { hexvertices[i] = expi(i*4*eta/6+eta); } filldraw(hexvertices[0]--hexvertices[1]--hexvertices[2]--hexvertices[3]--hexvertices[4]--hexvertices[5]--cycle,gray); pair[] overtices; for(pair h : hexvertices) { overtices.push(rotate(30)*h+(sqrt(3)-1)*unit(rotate(30)*h)); } draw(overtices[0]--overtices[1]--overtices[2]--overtices[3]--overtices[4]--overtices[5]--cycle); string[] labels = {"F","A","B","C","D","E"}; for(int i = 0; i < overtices.length; ++i) { draw(overtices[i]--overtices[(i+2)%overtices.length]); label(labels[i],overtices[i],unit(overtices[i])); } //Credit to dasobson for the diagram[/asy]$ Solution

## Problem T9

$[asy] size(6cm); pair R = (25,0),U=(0,20); draw(origin--R--(R+U)--U--cycle); real radius=2; pair ccenter=U+(radius,-radius); draw(circle(ccenter,radius)); real lift=1; real space =3; real shift = 1; draw(shift((shift,0))*circle(ccenter,radius),linetype("4 4")); draw(shift((2shift,0))*circle(ccenter,radius),linetype("4 4")); draw(shift(0,lift)*((U+R/2+(space,0))--(R+U)),EndArrow(HookHead,8)); draw(shift(0,lift)*((U+R/2-(space,0))--(U)),EndArrow(HookHead,8)); label("25\textrm{cm}",U+R/2,N); draw(shift(lift,0)*((R+U/2+(0,space))--(R+U)),EndArrow(HookHead,8)); draw(shift(lift,0)*((R+U/2-(0,space))--(R)),EndArrow(HookHead,8)); label(rotate(270)*Label("20\textrm{cm}",U/2+R,E)); draw((ccenter+(radius,0))--(ccenter+R-(2radius,0)),EndArrow(HookHead,8)); draw((ccenter+R-(2radius,0))--(R-(radius,0)+(0,radius)),EndArrow(HookHead,8)); draw((R-(radius,0)+(0,radius))--(ccenter.x,radius),EndArrow(HookHead,8)); draw((ccenter.x,radius)--(ccenter.x,U.y-2radius),EndArrow(HookHead,8)); //Credit to dasobson for the diagram[/asy]$ Solution

## Problem T10

$[asy] size(8cm); void hexagon(pair p) { draw((p.x,p.y-1)--(p.x-sqrt(3)/2,p.y-.5)--(p.x-sqrt(3)/2,p.y+.5)--(p.x,p.y+1)--(p.x+sqrt(3)/2,p.y+.5)--(p.x+sqrt(3)/2,p.y-.5)--cycle); return; } for(int i = 0; i < 10; ++i) { for(int j = 0; j <= i; ++j) { hexagon((-sqrt(3)*i/2+sqrt(3)*j,0-3/2*i)); } } //Credit to dasobson for the diagram[/asy]$ Solution