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Difference between revisions of "2003 PMWC Problems"

(New page: == Problem I1 == Solution == Problem I2 == Solution == Problem I3 == Solution ==...)
 
 
(One intermediate revision by one other user not shown)
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== Problem I2 ==
 
== Problem I2 ==
−
 
+
<asy>
 +
size(7cm);
 +
pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100);
 +
draw((M--N)^^(origin--a)^^(origin--b));
 +
draw(anglemark(N,origin,a,10)^^anglemark(a,origin,b)^^anglemark(b,origin,M,5));
 +
label("$a$",origin,10*unit(unit(a)+unit(N)));
 +
label("$b$",origin,8*unit(unit(a)+unit(b)));
 +
label("$c$",origin,5*unit(unit(b)+unit(M)));
 +
label("$N$",N,E);
 +
label("$M$",M,W);
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem I2|Solution]]
 
[[2003 PMWC Problems/Problem I2|Solution]]
  
 
== Problem I3 ==
 
== Problem I3 ==
−
 
+
<asy>
 +
size(4cm);
 +
void blackbox(pair p)
 +
{
 +
pair a = (0,1/2.0);
 +
pair b = (1/2.0,0);
 +
filldraw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black);
 +
}
 +
void whitebox(pair p)
 +
{
 +
pair a = (0,1/2.0);
 +
pair b = (1/2.0,0);
 +
fill((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,white);
 +
draw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black);
 +
}
 +
real len = 13/2;
 +
draw((-len,-len)--(len,len),linetype("3 3"));
 +
draw((len,-len)--(-len,len),linetype("3 3"));
 +
path b = (-len,-len)--(-len,len)--(len,len)--(len,-len)--cycle;
 +
draw(b^^(scale(9/13)*b),linetype("3 3"));
 +
for(int i = -2; i <= 2; ++i)
 +
{
 +
for(int j = -2; j <= 2; ++j)
 +
{
 +
if(abs(i)==abs(j))
 +
{
 +
blackbox((j,i));
 +
}
 +
else
 +
{
 +
whitebox((j,i));
 +
}
 +
}
 +
}
 +
for(int i = -1; i <= 1; i=i+2)
 +
{
 +
for(int j = -1; j <= 1; j=j+2)
 +
{
 +
pair c=scale(5)*(i,j);
 +
blackbox(scale(4)*(i,j));
 +
blackbox(c);
 +
blackbox(scale(6)*(i,j));
 +
whitebox(c+(1,0));
 +
whitebox(c+(0,1));
 +
whitebox(c-(1,0));
 +
whitebox(c-(0,1));
 +
if(i==j)
 +
{
 +
whitebox(c-(1,0)+(0,1));
 +
whitebox(c-(0,1)+(1,0));
 +
}
 +
else
 +
{
 +
whitebox(c+(1,0)+(0,1));
 +
whitebox(c-(1,0)-(0,1));
 +
}
 +
}
 +
}
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem I3|Solution]]
 
[[2003 PMWC Problems/Problem I3|Solution]]
  
Line 20: Line 88:
  
 
== Problem I6 ==
 
== Problem I6 ==
−
 
+
<asy>
 +
size(5cm);
 +
pair C=origin, B=(0,1), A= (1,1),D=(4/3,0);
 +
pair E=intersectionpoint(A--C,B--D);
 +
draw(B--C--A--B--D--C);
 +
draw(A--D);
 +
label("$A$",A,NE);
 +
label("$B$",B,NW);
 +
label("$C$",C,SW);
 +
label("$D$",D,SE);
 +
label("$E$",E,S*2);
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem I6|Solution]]
 
[[2003 PMWC Problems/Problem I6|Solution]]
  
Line 36: Line 115:
  
 
== Problem I10 ==
 
== Problem I10 ==
−
 
+
<asy>
 +
size(8cm);
 +
pair A=(2,0),B=(2,1),C=(0,1),D=origin,E=(1,1),F=(B+A)/2,H=(1.0/2,0);
 +
draw(A--B--C--D--cycle);
 +
filldraw(D--E--B--F--H--cycle,gray(.8));
 +
label("$A$",A,SE);label("$B$",B,NE);label("$C$",C,NW);label("$D$",D,SW);label("$E$",E,N);label("$F$",F,E);label("$H$",H,S);
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem I10|Solution]]
 
[[2003 PMWC Problems/Problem I10|Solution]]
  
Line 52: Line 137:
  
 
== Problem I14 ==
 
== Problem I14 ==
−
 
+
<asy>
 +
size(14cm);
 +
path[] a = arc((0,0),1,0,180)^^((-1,0)--(1,0));
 +
draw(a);
 +
draw(shift((5,0))*scale(2)*a);
 +
draw(shift((12,0))*scale(3)*a);
 +
label("I",origin,S*2.5);
 +
label("II",(5,0),S*2.5);
 +
label("III",(12,0),S*2.5);
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem I14|Solution]]
 
[[2003 PMWC Problems/Problem I14|Solution]]
  
Line 72: Line 166:
  
 
== Problem T4 ==  
 
== Problem T4 ==  
−
 
+
<asy>
 +
for(int i = -4; i < 5; ++i){
 +
for(int j = -4; j < 5; ++j){
 +
if(abs(i)+abs(j)==4) label("S",(i,j));
 +
else if(abs(i)+abs(j)==3) label("H",(i,j));
 +
else if(abs(i)+abs(j)==2) label("T",(i,j));
 +
else if(abs(i)+abs(j)==1) label("A",(i,j));}}
 +
label("M",(0,0));
 +
//Credit to chezbgone2 for the diagram</asy>
 
[[2003 PMWC Problems/Problem T4|Solution]]
 
[[2003 PMWC Problems/Problem T4|Solution]]
  
Line 96: Line 198:
  
 
== Problem T10 ==
 
== Problem T10 ==
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(1cm);
 +
draw((-1.38,0)--(1.38,0)--(0,2.39)--cycle);
 +
draw((-1.38,0)--(1.38,0)--(0,-2.39)--cycle);
 +
/* draw figures */
 +
draw((-1.38,0)--(1.38,0));
 +
draw((1.38,0)--(0,2.39));
 +
draw((0,2.39)--(-1.38,0));
 +
//Credit to dasobson for the diagram</asy>
 +
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(1.5cm);
 +
draw((-1.38,0)--(1.38,0)--(0,2.39)--cycle);
 +
draw((2.76,2.39)--(1.38,0)--(0,2.39)--cycle);
 +
draw((1.38,0)--(4.14,0)--(2.76,2.39)--cycle);
 +
/* draw figures */
 +
draw((-1.38,0)--(1.38,0));
 +
draw((1.38,0)--(0,2.39));
 +
draw((0,2.39)--(-1.38,0));
 +
//Credit to dasobson for the diagram</asy>
 +
 +
<asy>
 +
/* File unicodetex not found. */
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(5cm);
 +
draw((7,0)--(9,0)--(8,1.73)--cycle);
 +
draw((9,0)--(11,0)--(10,1.73)--cycle);
 +
draw((8,1.73)--(10,1.73)--(9,3.46)--cycle);
 +
draw((12,1.73)--(14,1.73)--(13,3.46)--cycle);
 +
draw((12,1.73)--(14,1.73)--(13,3.46)--cycle);
 +
draw((15,3.46)--(14,1.73)--(13,3.46)--cycle);
 +
draw((12,1.73)--(14,1.73)--(13,3.46)--cycle);
 +
draw((14,1.73)--(16,1.73)--(15,3.46)--cycle);
 +
draw((16,1.73)--(15,0)--(14,1.73)--cycle);
 +
draw((1,1.73)--(3,1.73)--(2,0)--cycle);
 +
draw((1,1.73)--(3,1.73)--(2,0)--cycle);
 +
draw((4,0)--(3,1.73)--(2,0)--cycle);
 +
draw((3,1.73)--(5,1.73)--(4,0)--cycle);
 +
draw((6,0)--(5,1.73)--(4,0)--cycle);
 +
//Credit to dasobson for the diagram</asy>
  
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
size(12cm);
 +
label( "$=$" , ((-0.87+-2)/2,0),fontsize(15));
 +
label( "$=$" , ((13)/2,0),fontsize(15));
 +
label( "$=$" , (19/2,0),fontsize(15));
 +
draw((-3,0)--(-2,0)--(-2.5,0.87)--cycle);
 +
draw((-3,0)--(-2,0)--(-2.5,-0.87)--cycle);
 +
draw((0,0.5)--(0,-0.5)--(0.87,0)--cycle);
 +
draw((0,0.5)--(0,-0.5)--(-0.87,0)--cycle);
 +
draw((4,0)--(5,0)--(4.5,0.87)--cycle);
 +
draw((5,0)--(6,0)--(5.5,0.87)--cycle);
 +
draw((5,0)--(5.5,0.87)--(4.5,0.87)--cycle);
 +
draw((6,0)--(5,0)--(5.5,-0.87)--cycle);
 +
draw((7,0)--(8,0)--(7.5,0.87)--cycle);
 +
draw((8,0)--(7,0)--(7.5,-0.87)--cycle);
 +
draw((8,0)--(7.5,-0.87)--(8.5,-0.87)--cycle);
 +
draw((8,0)--(8.5,-0.87)--(9,0)--cycle);
 +
draw(shift(0,0)*((11,-1)--(11,0)--(10.13,-0.5)--cycle));
 +
draw(shift(0/2,0)*((11,0)--(11,-1)--(11.87,-0.5)--cycle));
 +
draw(shift(0/2,0)*((11,0)--(11.87,-0.5)--(11.87,0.5)--cycle));
 +
draw(shift(0/2,0)*((11,0)--(11.87,0.5)--(11,1)--cycle));
 +
//Credit to dasobson for the diagram</asy>
 
[[2003 PMWC Problems/Problem T10|Solution]]
 
[[2003 PMWC Problems/Problem T10|Solution]]

Latest revision as of 16:15, 22 April 2014

Problem I1

Solution

Problem I2

[asy] size(7cm); pair M=(-1,0), N=(1,0),a=4/5*expi(pi/10),b=expi(37pi/100); draw((M--N)^^(origin--a)^^(origin--b)); draw(anglemark(N,origin,a,10)^^anglemark(a,origin,b)^^anglemark(b,origin,M,5)); label("$a$",origin,10*unit(unit(a)+unit(N))); label("$b$",origin,8*unit(unit(a)+unit(b))); label("$c$",origin,5*unit(unit(b)+unit(M))); label("$N$",N,E); label("$M$",M,W); //Credit to dasobson for the diagram[/asy] Solution

Problem I3

[asy] size(4cm); void blackbox(pair p) { pair a = (0,1/2.0); pair b = (1/2.0,0); filldraw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black); } void whitebox(pair p) { pair a = (0,1/2.0); pair b = (1/2.0,0); fill((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,white); draw((p-a-b)--(p+a-b)--(p+a+b)--(p-a+b)--cycle,black); } real len = 13/2; draw((-len,-len)--(len,len),linetype("3 3")); draw((len,-len)--(-len,len),linetype("3 3")); path b = (-len,-len)--(-len,len)--(len,len)--(len,-len)--cycle; draw(b^^(scale(9/13)*b),linetype("3 3")); for(int i = -2; i <= 2; ++i) { for(int j = -2; j <= 2; ++j) { if(abs(i)==abs(j)) { blackbox((j,i)); } else { whitebox((j,i)); } } } for(int i = -1; i <= 1; i=i+2) { for(int j = -1; j <= 1; j=j+2) { pair c=scale(5)*(i,j); blackbox(scale(4)*(i,j)); blackbox(c); blackbox(scale(6)*(i,j)); whitebox(c+(1,0)); whitebox(c+(0,1)); whitebox(c-(1,0)); whitebox(c-(0,1)); if(i==j) { whitebox(c-(1,0)+(0,1)); whitebox(c-(0,1)+(1,0)); } else { whitebox(c+(1,0)+(0,1)); whitebox(c-(1,0)-(0,1)); } } } //Credit to dasobson for the diagram[/asy] Solution

Problem I4

Solution

Problem I5

Solution

Problem I6

[asy] size(5cm); pair C=origin, B=(0,1), A= (1,1),D=(4/3,0); pair E=intersectionpoint(A--C,B--D); draw(B--C--A--B--D--C); draw(A--D); label("$A$",A,NE); label("$B$",B,NW); label("$C$",C,SW); label("$D$",D,SE); label("$E$",E,S*2); //Credit to dasobson for the diagram[/asy] Solution

Problem I7

Solution

Problem I8

Solution

Problem I9

Solution

Problem I10

[asy] size(8cm); pair A=(2,0),B=(2,1),C=(0,1),D=origin,E=(1,1),F=(B+A)/2,H=(1.0/2,0); draw(A--B--C--D--cycle); filldraw(D--E--B--F--H--cycle,gray(.8)); label("$A$",A,SE);label("$B$",B,NE);label("$C$",C,NW);label("$D$",D,SW);label("$E$",E,N);label("$F$",F,E);label("$H$",H,S); //Credit to dasobson for the diagram[/asy] Solution

Problem I11

Solution

Problem I12

Solution

Problem I13

Solution

Problem I14

[asy] size(14cm); path[] a = arc((0,0),1,0,180)^^((-1,0)--(1,0)); draw(a); draw(shift((5,0))*scale(2)*a); draw(shift((12,0))*scale(3)*a); label("I",origin,S*2.5); label("II",(5,0),S*2.5); label("III",(12,0),S*2.5); //Credit to dasobson for the diagram[/asy] Solution

Problem I15

Solution

Problem T1

Solution

Problem T2

Solution

Problem T3

Solution

Problem T4

[asy] for(int i = -4; i < 5; ++i){ for(int j = -4; j < 5; ++j){ if(abs(i)+abs(j)==4) label("S",(i,j)); else if(abs(i)+abs(j)==3) label("H",(i,j)); else if(abs(i)+abs(j)==2) label("T",(i,j)); else if(abs(i)+abs(j)==1) label("A",(i,j));}} label("M",(0,0)); //Credit to chezbgone2 for the diagram[/asy] Solution

Problem T5

Solution

Problem T6

Solution

Problem T7

Solution

Problem T8

Solution

Problem T9

Solution

Problem T10

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(1cm); draw((-1.38,0)--(1.38,0)--(0,2.39)--cycle); draw((-1.38,0)--(1.38,0)--(0,-2.39)--cycle); /* draw figures */ draw((-1.38,0)--(1.38,0)); draw((1.38,0)--(0,2.39)); draw((0,2.39)--(-1.38,0)); //Credit to dasobson for the diagram[/asy]

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(1.5cm); draw((-1.38,0)--(1.38,0)--(0,2.39)--cycle); draw((2.76,2.39)--(1.38,0)--(0,2.39)--cycle); draw((1.38,0)--(4.14,0)--(2.76,2.39)--cycle); /* draw figures */ draw((-1.38,0)--(1.38,0)); draw((1.38,0)--(0,2.39)); draw((0,2.39)--(-1.38,0)); //Credit to dasobson for the diagram[/asy]

[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); draw((7,0)--(9,0)--(8,1.73)--cycle); draw((9,0)--(11,0)--(10,1.73)--cycle); draw((8,1.73)--(10,1.73)--(9,3.46)--cycle); draw((12,1.73)--(14,1.73)--(13,3.46)--cycle); draw((12,1.73)--(14,1.73)--(13,3.46)--cycle); draw((15,3.46)--(14,1.73)--(13,3.46)--cycle); draw((12,1.73)--(14,1.73)--(13,3.46)--cycle); draw((14,1.73)--(16,1.73)--(15,3.46)--cycle); draw((16,1.73)--(15,0)--(14,1.73)--cycle); draw((1,1.73)--(3,1.73)--(2,0)--cycle); draw((1,1.73)--(3,1.73)--(2,0)--cycle); draw((4,0)--(3,1.73)--(2,0)--cycle); draw((3,1.73)--(5,1.73)--(4,0)--cycle); draw((6,0)--(5,1.73)--(4,0)--cycle); //Credit to dasobson for the diagram[/asy]

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ size(12cm); label( "$=$" , ((-0.87+-2)/2,0),fontsize(15)); label( "$=$" , ((13)/2,0),fontsize(15)); label( "$=$" , (19/2,0),fontsize(15)); draw((-3,0)--(-2,0)--(-2.5,0.87)--cycle); draw((-3,0)--(-2,0)--(-2.5,-0.87)--cycle); draw((0,0.5)--(0,-0.5)--(0.87,0)--cycle); draw((0,0.5)--(0,-0.5)--(-0.87,0)--cycle); draw((4,0)--(5,0)--(4.5,0.87)--cycle); draw((5,0)--(6,0)--(5.5,0.87)--cycle); draw((5,0)--(5.5,0.87)--(4.5,0.87)--cycle); draw((6,0)--(5,0)--(5.5,-0.87)--cycle); draw((7,0)--(8,0)--(7.5,0.87)--cycle); draw((8,0)--(7,0)--(7.5,-0.87)--cycle); draw((8,0)--(7.5,-0.87)--(8.5,-0.87)--cycle); draw((8,0)--(8.5,-0.87)--(9,0)--cycle); draw(shift(0,0)*((11,-1)--(11,0)--(10.13,-0.5)--cycle)); draw(shift(0/2,0)*((11,0)--(11,-1)--(11.87,-0.5)--cycle)); draw(shift(0/2,0)*((11,0)--(11.87,-0.5)--(11.87,0.5)--cycle)); draw(shift(0/2,0)*((11,0)--(11.87,0.5)--(11,1)--cycle)); //Credit to dasobson for the diagram[/asy] Solution

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