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Difference between revisions of "1999 PMWC Problems"

(New page: == Problem I1 == Solution == Problem I2 == Solution == Problem I3 == Solution ==...)
 
(→‎Problem T2)
 
(11 intermediate revisions by 2 users not shown)
Line 4: Line 4:
  
 
== Problem I2 ==
 
== Problem I2 ==
 +
  
 
[[1999 PMWC Problems/Problem I2|Solution]]
 
[[1999 PMWC Problems/Problem I2|Solution]]
Line 21: Line 22:
 
== Problem I6 ==
 
== 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>
 
[[1999 PMWC Problems/Problem I6|Solution]]
 
[[1999 PMWC Problems/Problem I6|Solution]]
  
Line 37: Line 63:
 
== Problem I10 ==
 
== 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>
 
[[1999 PMWC Problems/Problem I10|Solution]]
 
[[1999 PMWC Problems/Problem I10|Solution]]
  
Line 53: Line 93:
 
== Problem I14 ==
 
== 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>
 
[[1999 PMWC Problems/Problem I14|Solution]]
 
[[1999 PMWC Problems/Problem I14|Solution]]
  
Line 65: Line 128:
 
== Problem T2 ==  
 
== 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>
 
[[1999 PMWC Problems/Problem T2|Solution]]
 
[[1999 PMWC Problems/Problem T2|Solution]]
  
Line 77: Line 174:
 
== Problem T5 ==
 
== 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>
 
[[1999 PMWC Problems/Problem T5|Solution]]
 
[[1999 PMWC Problems/Problem T5|Solution]]
  
 
== Problem T6 ==
 
== 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>
 
[[1999 PMWC Problems/Problem T6|Solution]]
 
[[1999 PMWC Problems/Problem T6|Solution]]
  

Latest 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

[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 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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