Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2001 PMWC Problems"

(→‎Problem T9)
(→‎Problem I14)
 
(One intermediate revision by the same user not shown)
Line 85: Line 85:
 
== Problem I11 ==
 
== Problem I11 ==
  
 +
<asy>
 +
size(8cm);
 +
defaultpen(fontsize(11));
 +
pair A,B,C,D;
 +
A=origin;
 +
B=(0.523713039560712, 0.150288579021811);
 +
C=(0.564402523837359, 1.185901582062919);
 +
D=(-0.640531236354945, 1.247592090482351);
 +
pair Ap,Bp,Cp,Dp;
 +
Ap=rotate(180,B)*A;
 +
Bp=rotate(180,C)*B;
 +
Cp=rotate(180,D)*C;
 +
Dp=rotate(180,A)*D;
 +
draw(A--Ap--Bp--Cp--Dp--D--C--B);
 +
draw(Dp--Ap);
 +
draw(Bp--C);
 +
draw(Cp--D);
 +
label("$A$",A,SW);
 +
label("$B$",B,S);
 +
label("$C$",C,E);
 +
label("$D$",D,N);
 +
label("$A'$",Ap,unit(Ap));
 +
label("$B'$",Bp,unit(Bp));
 +
label("$C'$",Cp,W);
 +
label("$D'$",Dp,unit(Dp));
 +
//Credit to dasobson for the diagram</asy>
 
[[2001 PMWC Problems/Problem I11|Solution]]
 
[[2001 PMWC Problems/Problem I11|Solution]]
  
Line 97: Line 123:
 
== Problem I14 ==
 
== Problem I14 ==
  
 +
<asy>
 +
size(9cm);
 +
real horiscale=11/8;
 +
for(int i = 0; i < 4; ++i)
 +
{
 +
for(int j = 0; j < 3; ++j)
 +
{
 +
draw((i*horiscale,j)--((i+1)*horiscale,j)--((i+1)*horiscale,j+1)--(i*horiscale,j+1)--cycle);
 +
if((i == 0 && j == 1) || (j == 0 && i > 0) || (i == 2 && j == 2))
 +
{
 +
label( "$\bullet$",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 );
 +
}
 +
else
 +
{
 +
label( "$\circ$",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 );
 +
}
 +
}
 +
}
 +
draw(origin--(-1.3*3,0)--(-1.3*3,3)--(0,3)--cycle);
 +
label("Hotel",(-1.3*3/2,5/2));
 +
label("This side up",(-1.3*3/2,3/2));
 +
label("(This is an example of one possibility)", ((-1.3*3+1.3*4)/2,-.4),fontsize(11));
 +
//Credit to dasobson for the diagram</asy>
 
[[2001 PMWC Problems/Problem I14|Solution]]
 
[[2001 PMWC Problems/Problem I14|Solution]]
  

Latest revision as of 20:24, 21 April 2014

Problem I1

Solution

Problem I2

Solution

Problem I3

[asy] void wdot(pair p) { fill(circle(p,.025),white); draw(circle(p,.025),black); return; } import patterns; real eta=pi/2; pair A,B,C,D,E,F,G,O; A=origin; B=(1,0); C=(1,1); D=(0,1); O=(A+B+C+D)/4; draw(A--B--C--D--cycle); E=(O.x+cos(-eta/4),O.y+sin(-eta/4)); G = rotate(90,O)*E; F=O+(G-O)+(E-O); draw(O--G--F--E--cycle); add("crosshatch",crosshatch(.6mm)); pair a1, a2; a1 = intersectionpoint(O--G,D--C); a2 = intersectionpoint(B--C,O--E); fill(O--a1--C--a2--cycle,pattern("crosshatch")); wdot(F); wdot(G); wdot(O); wdot(a1); wdot(a2); wdot(A); wdot(B); wdot(C); wdot(D); wdot(E); label("$A$",A,SW); label("$B$",B,SW); label("$C$",C,E); label("$D$",D,W); label("$E$",E,S); label("$F$",F,E); label("$G$",G,NE); label("$O$",O,W); //Credit to dasobson for the diagram[/asy] Solution

Problem I4

Solution

Problem I5

Solution

Problem I6

Solution

Problem I7

Solution

Problem I8

Solution

Problem I9

Solution

Problem I10

Solution

Problem I11

[asy] size(8cm); defaultpen(fontsize(11)); pair A,B,C,D; A=origin; B=(0.523713039560712, 0.150288579021811); C=(0.564402523837359, 1.185901582062919); D=(-0.640531236354945, 1.247592090482351); pair Ap,Bp,Cp,Dp; Ap=rotate(180,B)*A; Bp=rotate(180,C)*B; Cp=rotate(180,D)*C; Dp=rotate(180,A)*D; draw(A--Ap--Bp--Cp--Dp--D--C--B); draw(Dp--Ap); draw(Bp--C); draw(Cp--D); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,E); label("$D$",D,N); label("$A'$",Ap,unit(Ap)); label("$B'$",Bp,unit(Bp)); label("$C'$",Cp,W); label("$D'$",Dp,unit(Dp)); //Credit to dasobson for the diagram[/asy] Solution

Problem I12

Solution

Problem I13

Solution

Problem I14

[asy] size(9cm); real horiscale=11/8; for(int i = 0; i < 4; ++i) { for(int j = 0; j < 3; ++j) { draw((i*horiscale,j)--((i+1)*horiscale,j)--((i+1)*horiscale,j+1)--(i*horiscale,j+1)--cycle); if((i == 0 && j == 1) || (j == 0 && i > 0) || (i == 2 && j == 2)) { label( "$\bullet$",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 ); } else { label( "$\circ$",((i*horiscale,j)+((i+1)*horiscale,j)+((i+1)*horiscale,j+1)+(i*horiscale,j+1))/4 ); } } } draw(origin--(-1.3*3,0)--(-1.3*3,3)--(0,3)--cycle); label("Hotel",(-1.3*3/2,5/2)); label("This side up",(-1.3*3/2,3/2)); label("(This is an example of one possibility)", ((-1.3*3+1.3*4)/2,-.4),fontsize(11)); //Credit to dasobson for the diagram[/asy] Solution

Problem I15

Solution

Problem T1

Solution

Problem T2

Solution

Problem T3

[asy] pair A = (0,4); pair B = (7,0); void drawvert(int xstart, int len) { draw((xstart,A.y)--(xstart,A.y-len)); } void drawhori(pair start, int len) { draw(start--(start.x+len,start.y)); } //draw verticals int[] verts = {4,4,4,2,4,3,4,4}; for(int i = 0; i < verts.length; ++i) { drawvert(i,verts[i]); } //draw horizontals drawhori(A,7); drawhori(A - (0,1),7); drawhori(A - (0,2),4); drawhori((B.x-1,2),1); drawhori((0,1),2); drawhori((4,1),3); drawhori(origin,7); //labels label("$A$",A,NW); label("$B$",B,SE); //Credit to dasobson for the diagram[/asy] Solution

Problem T4

Solution

Problem T5

Solution

Problem T6

Solution

Problem T7

Solution

Problem T8

Solution

Problem T9

[asy] void wdot(pair p) { fill(circle(p,.025),white); draw(circle(p,.025),black); return; } draw(circle(origin,1)); pair z = expi(3pi/10); pair[] rotations; int max = 7; for(int i = 0; i < max; ++i) { rotations.push(rotate(-360*i/7)*z); } int i = 1; for(pair x : rotations) { wdot(x); label("$" + string(i) + "$", x , unit(x)); ++i; } //Credit to dasobson for the diagram[/asy] Solution

Problem T10

Solution

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2001_PMWC_Problems&oldid=61695"