Difference between revisions of "2001 PMWC Problems"
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== 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
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
![[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]](http://latex.artofproblemsolving.com/e/b/0/eb0d9c0dece28d46bf6400fdf9173aa1bf8539f2.png)
![[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]](http://latex.artofproblemsolving.com/5/4/8/5482fd531ae125ed0b79c02b99faa9f896503c46.png)
![[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]](http://latex.artofproblemsolving.com/d/f/6/df6c2e26b3639776b2389bfa4424478c1728fce5.png)
![[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]](http://latex.artofproblemsolving.com/5/d/5/5d5b80810d44f0051320360cff87bda70545cf63.png)
![[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]](http://latex.artofproblemsolving.com/3/1/5/31561b4717595123e7443f561f199477d23ec086.png)
