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Difference between revisions of "1999 PMWC Problems"

(→‎Problem T6)
(→‎Problem I2)
Line 5: Line 5:
 
== Problem I2 ==
 
== Problem I2 ==
  
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<asy>
+
 
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void box(pair p)
 
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{
 
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real c = .3;
 
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pair hvect=(c,0);
 
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pair vvect=(0,c);
 
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draw((p-hvect-vvect)--(p-hvect+vvect)--(p+hvect+vvect)--(p+hvect-vvect)--cycle,linewidth(0.9));
 
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}
 
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size(10cm);
 
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real horiscale=1.9;
 
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defaultpen(fontsize(10));
 
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usepackage("amssymb");
 
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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",""},{"","..","..","..","..","..",".."},{"..","..","..","..","..",".."}};
 
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for(int i = 0; i < array.length; ++i)
 
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{
 
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for(int j = 0; j < array[i].length;++j)
 
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{
 
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array[i][j]="\textbf{" + array[i][j] + "}";
 
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}
 
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}
 
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for(int i = 0; i < array.length; ++i)
 
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{
 
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for(int j = 0; j < array[i].length; ++j)
 
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{
 
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if(i==1)
 
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{
 
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box((horiscale*j,-i));
 
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}
 
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else{
 
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label(array[i][j],(horiscale*j,-i));
 
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}
 
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}
 
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}
 
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//Credit to dasobson for the diagram</asy>
 
 
[[1999 PMWC Problems/Problem I2|Solution]]
 
[[1999 PMWC Problems/Problem I2|Solution]]
  

Revision as of 21:21, 21 April 2014

Problem I1

Solution

Problem I2

Solution

Problem I3

Solution

Problem I4

Solution

Problem I5

Solution

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 I7

Solution

Problem I8

Solution

Problem I9

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 I11

Solution

Problem I12

Solution

Problem I13

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 I15

Solution

Problem T1

Solution

Problem T2

Solution

Problem T3

Solution

Problem T4

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

Problem T7

Solution

Problem T8

Solution

Problem T9

Solution

Problem T10

Solution

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