2000 PMWC Problems
Contents
- 1 Problem I1
- 2 Problem I2
- 3 Problem I3
- 4 Problem I4
- 5 Problem I5
- 6 Problem I6
- 7 Problem I7
- 8 Problem I8
- 9 Problem I9
- 10 Problem I10
- 11 Problem I11
- 12 Problem I12
- 13 Problem I13
- 14 Problem I14
- 15 Problem I15
- 16 Problem T1
- 17 Problem T2
- 18 Problem T3
- 19 Problem T4
- 20 Problem T5
- 21 Problem T6
- 22 Problem T7
- 23 Problem T8
- 24 Problem T9
- 25 Problem T10
Problem I1
Problem I2
Problem I3
Problem I4
Problem I5
Problem I6
Problem I7
Problem I8
![[asy] for(int i = 0; i < 4; ++i){ if(i < 2){ draw((i,0)--(i,3)); }else{ draw((i,0)--(i,2)); } } for(int i = 0; i < 4; ++i){ if(i < 3) draw((0,i)--(3,i)); else draw((0,i)--(1,i)); } filldraw(Circle((0,3),0.3),white); filldraw((2-0.3*cos(pi/2),1+0.3*sin(pi/2))--(2-0.3*cos(7pi/6),1+0.3*sin(7pi/6))--(2-0.3*cos(11pi/6),1+0.3*sin(11pi/6))--cycle,white); //Credit to chezbgone2 for the diagram[/asy]](http://latex.artofproblemsolving.com/6/0/a/60ab895b88a2a762e82ee7d8a90c842f44f8e0f4.png)
Problem I9
Problem I10
Problem I11
Problem I12
Problem I13
![[asy] void unitsquare(pair p){ draw(p--(p.x+1,p.y)--(p.x+1,p.y+1)--(p.x,p.y+1)--cycle); return; } fill(buildcycle(arc((2,0.5),(2,0),(2,1)),(2,1)--(0,1)--(2.5,0.5)),gray); unitsquare((0,0)); draw((1,0)--(2,0)--(2,1)--(1,1)); draw(arc((2,0.5),(2,0),(2,1))); label("$A$",(0,1),NW); label("$B$",(0,0),SW); label("$C$",(1,0),S); label("$D$",(1,1),N); label("$E$",(2,0),S); label("$F$",(2.5,0.5),E); label("$G$",(2,1),N); //Credit to chezbgone2 for the diagram[/asy]](http://latex.artofproblemsolving.com/4/8/f/48fbdb9ab948aa92a7aa6061eefbedb0e0a04382.png)
Problem I14
Problem I15
Problem T1
Problem T2
![[asy] import three; currentprojection=orthographic(1,1/2,0.4); draw(surface((5,1,1)--(5,1,2)--(5,2,2)--(5,2,1)--cycle),black,nolight); draw(surface((5,1,3)--(5,1,4)--(5,2,4)--(5,2,3)--cycle),black,nolight); draw(surface((5,3,1)--(5,3,2)--(5,4,2)--(5,4,1)--cycle),black,nolight); draw(surface((5,3,3)--(5,3,4)--(5,4,4)--(5,4,3)--cycle),black,nolight); draw(surface((5,2,2)--(5,2,3)--(5,3,3)--(5,3,2)--cycle),black,nolight); draw(surface((1,5,1)--(1,5,2)--(2,5,2)--(2,5,1)--cycle),black,nolight); draw(surface((1,5,3)--(1,5,4)--(2,5,4)--(2,5,3)--cycle),black,nolight); draw(surface((3,5,1)--(3,5,2)--(4,5,2)--(4,5,1)--cycle),black,nolight); draw(surface((3,5,3)--(3,5,4)--(4,5,4)--(4,5,3)--cycle),black,nolight); draw(surface((2,5,2)--(2,5,3)--(3,5,3)--(3,5,2)--cycle),black,nolight); draw(surface((1,1,5)--(1,2,5)--(2,2,5)--(2,1,5)--cycle),black,nolight); draw(surface((1,3,5)--(1,4,5)--(2,4,5)--(2,3,5)--cycle),black,nolight); draw(surface((3,1,5)--(3,2,5)--(4,2,5)--(4,1,5)--cycle),black,nolight); draw(surface((3,3,5)--(3,4,5)--(4,4,5)--(4,3,5)--cycle),black,nolight); draw(surface((2,2,5)--(2,3,5)--(3,3,5)--(3,2,5)--cycle),black,nolight); for(int i = 0; i <= 5; ++i) draw((i,0,5)--(i,5,5)); for(int i = 0; i <= 5; ++i) draw((0,i,5)--(5,i,5)); for(int i = 0; i <= 5; ++i) draw((5,i,0)--(5,i,5)); for(int i = 0; i <= 5; ++i) draw((5,0,i)--(5,5,i)); for(int i = 0; i <= 5; ++i) draw((i,5,0)--(i,5,5)); for(int i = 0; i <= 5; ++i) draw((0,5,i)--(5,5,i)); //Credit to chezbgone2 for the diagram[/asy]](http://latex.artofproblemsolving.com/a/f/7/af72e820083cf8f480f414f7f1359273186cf40a.png)
Problem T3
![[asy] fill(buildcycle((0,2)--(1,0),(2,2)--(0,1),(2,0)--(1,2),(0,0)--(2,1)),gray); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((0,2)--(1,0)); draw((2,2)--(0,1)); draw((2,0)--(1,2)); draw((0,0)--(2,1)); label("A",(0,2),NW); label("B",(2,2),NE); label("C",(2,0),SE); label("D",(0,0),SW); label("P",(1,2),N); label("Q",(2,1),E); label("R",(1,0),S); label("S",(0,1),W); //Credit to chezbgone2 for the diagram[/asy]](http://latex.artofproblemsolving.com/b/a/a/baa5783b2f8d6973e8fe88bcc8ecbfb56857698e.png)
Problem T4
[asy] import geometry; draw((0,0)--(3,0)--(3,2)--(0,2)--cycle); draw((0,1)--(1.5,2)); draw((2.3,0)--(2.3,0.7)--(3,0.7)); draw(ellipse((1.5,1),0.6,0.5)); //Credit to chezbgone2 for the diagram[/asy] Solution
