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Difference between revisions of "2002 PMWC Problems"

(New page: == Problem I1 == Solution == Problem I2 == Solution == Problem I3 == Solution ==...)
 
(→‎Problem T9)
 
(8 intermediate revisions by 2 users not shown)
Line 48: Line 48:
  
 
== Problem I13 ==
 
== 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>
  
 
[[2002 PMWC Problems/Problem I13|Solution]]
 
[[2002 PMWC Problems/Problem I13|Solution]]
Line 69: Line 126:
 
== Problem T3 ==
 
== 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>
 
[[2002 PMWC Problems/Problem T3|Solution]]
 
[[2002 PMWC Problems/Problem T3|Solution]]
  
Line 85: Line 164:
 
== Problem T7 ==  
 
== 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>
 
[[2002 PMWC Problems/Problem T7|Solution]]
 
[[2002 PMWC Problems/Problem T7|Solution]]
  
Line 92: Line 193:
  
 
== Problem T9 ==
 
== 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>
 
[[2002 PMWC Problems/Problem T9|Solution]]
 
[[2002 PMWC Problems/Problem T9|Solution]]
  
 
== Problem T10 ==
 
== 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>
 
[[2002 PMWC Problems/Problem T10|Solution]]
 
[[2002 PMWC Problems/Problem T10|Solution]]

Latest revision as of 15:07, 23 April 2014

Problem I1

Solution

Problem I2

Solution

Problem I3

Solution

Problem I4

Solution

Problem I5

Solution

Problem I6

Solution

Problem I7

Solution

Problem I8

Solution

Problem I9

Solution

Problem I10

Solution

Problem I11

Solution

Problem I12

Solution

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]

Solution

Problem I14

Solution

Problem I15

Solution

Problem T1

Solution

Problem T2

Solution

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 T4

Solution

Problem T5

Solution

Problem T6

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 T8

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

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