1999 PMWC Problems

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Problem I6

$[asy] /* modified Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(3.62cm); real labelscalefactor = 1.3; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -22.84, xmax = 51.79, ymin = -28.06, ymax = 28.9; /* image dimensions */ draw((0,0)--(7.8,0)--(3.9,6.75)--cycle); /* draw figures */ draw((0,0)--(7.8,0)); draw((7.8,0)--(3.9,6.75)); draw((3.9,6.75)--(0,0)); draw((7.8,0)--(1.95,3.38)); draw((3.9,0)--(1.95,3.38)); draw((1.95,0)--(1.95,3.38)); /* dots and labels */ label("B", (0,0), NW * labelscalefactor); label("C", (7.8,0), NE * labelscalefactor); label("A", (3.9,6.75), NE * labelscalefactor); label("D", (1.95,3.38), NW * labelscalefactor); label("E", (3.9,0), NE * labelscalefactor); label("F", (1.95,0), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ //Credit to dasobson for the diagram[/asy]$ Solution

Problem I10

$[asy] pair P,Q,R,S; S = origin; P = (0,3); R = (5,0); Q = P+R; draw(P--Q--R--S--cycle); label("S",S,SW); label("R",R,SE); label("Q",Q,NE); label("P",P,NW); label("5",(P+Q)/2,N); label("3",(Q+R)/2,E); //Credit to dasobson for the diagram[/asy]$ Solution

Problem I14

$[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(4.62cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -22.84, xmax = 51.79, ymin = -28.06, ymax = 28.9; /* image dimensions */ /* draw figures */ draw(circle((0,0), 1.09)); draw(circle((9.29,0), 1.09)); draw(circle((9.29,6.48), 1.09)); draw(circle((0,6.48), 1.09)); draw(circle((4.65,3.24), 1.09)); draw((1.09,6.48)--(8.2,6.48)); draw((1.09,0)--(8.2,0)); draw((9.29,1.09)--(9.29,5.39)); draw((0,1.09)--(0,5.39)); draw((0.9,0.62)--(3.75,2.62)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ //Credit to dasobson for the diagram[/asy]$ Solution

Problem T2

$[asy] void box(pair p) { real c = .3; pair hvect=(c,0); pair vvect=(0,c); draw((p-hvect-vvect)--(p-hvect+vvect)--(p+hvect+vvect)--(p+hvect-vvect)--cycle,linewidth(0.9)); } size(10cm); real horiscale=1.9; defaultpen(fontsize(10)); usepackage("amssymb"); string[][] array = {{"A","B","C","D","E","F","G"},{"\Box","\Box","\Box","\Box","\Box","\Box","\Box"},{"1","2","3","4","5","6","7"},{"13","12","11","10","9","8",""},{"","14","15","16","17","18","19"}, {"","..","..","..","..","20",""},{"","..","..","..","..","..",".."},{"..","..","..","..","..",".."}}; for(int i = 0; i < array.length; ++i) { for(int j = 0; j < array[i].length;++j) { array[i][j]="\textbf{" + array[i][j] + "}"; } } for(int i = 0; i < array.length; ++i) { for(int j = 0; j < array[i].length; ++j) { if(i==1) { box((horiscale*j,-i)); } else{ label(array[i][j],(horiscale*j,-i)); } } } //Credit to dasobson for the diagram[/asy]$ Solution

Problem T5

$[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(4.86cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.38, xmax = 9.48, ymin = -5.32, ymax = 6.79; /* image dimensions */ /* draw figures */ draw(circle((-2.71,3.64), 1)); draw(circle((-0.71,3.64), 1)); draw(circle((1.29,3.64), 1)); draw(circle((-1.71,2.64), 1)); draw(circle((0.29,2.64), 1)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ //Credit to dasobson for the diagram[/asy]$ Solution

Problem T6

$[asy] size(4cm); pair A,B,C; A=(-5,0); B=(0-A.x,A.y); C=(A+B)/2+(0,10); draw(A--B--C--cycle); pair W,X,Y; W=intersectionpoint(B--(-5,5),A--C); Y = (-W.x, W.y); X = intersectionpoint(A--Y,B--W); draw(A--Y); draw(B--W); label("X",(X+C+W+Y)/4); label("5",(X+A+W)/3); label("8",(X+B+Y)/3); label("10",(X+A+B)/3); //Credit to dasobson for the diagram[/asy]$ Solution