Difference between revisions of "2002 PMWC Problems"
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== 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]] | ||
Revision as of 20:25, 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
Problem I1
Problem I2
Problem I3
Problem I4
Problem I5
Problem I6
Problem I7
Problem I8
Problem I9
Problem I10
Problem I11
Problem I12
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]](http://latex.artofproblemsolving.com/2/7/e/27e0876f4b1e6ec0efae197f55861712553f3787.png)
![[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]](http://latex.artofproblemsolving.com/4/8/7/487631644b0889fd5a8470976b2370bc66ee677f.png)
![[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]](http://latex.artofproblemsolving.com/6/5/c/65cc5caffec60bb29b14fbfb442365b653b054dd.png)
![[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]](http://latex.artofproblemsolving.com/f/2/4/f241fddede509bd43613f2278e9233c700ef5747.png)
![[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]](http://latex.artofproblemsolving.com/4/f/4/4f4db103daaee0fe3fa11fa0bcfae6607803b121.png)
