Difference between revisions of "2001 PMWC Problems"
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== Problem T9 == | == 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> | ||
[[2001 PMWC Problems/Problem T9|Solution]] | [[2001 PMWC Problems/Problem T9|Solution]] | ||
Revision as of 16:44, 20 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] 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)
