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Difference between revisions of "2002 PMWC Problems"

(→‎Problem T10)
(→‎Problem I13)
Line 48: Line 48:
  
 
== Problem I13 ==
 
== 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>
  
 
<asy>
 
<asy>

Revision as of 13:55, 20 April 2014

Problem I1

Solution

Problem I2

Solution

Problem I3

Solution

Problem I4

Solution

Problem I5

Solution

Problem I6

Solution

Problem I7

Solution

Problem I8

Solution

Problem I9

Solution

Problem I10

Solution

Problem I11

Solution

Problem I12

Solution

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]

[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]

Solution

Problem I14

Solution

Problem I15

Solution

Problem T1

Solution

Problem T2

Solution

Problem T3

[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] Solution

Problem T4

Solution

Problem T5

Solution

Problem T6

Solution

Problem T7

Solution

Problem T8

Solution

Problem T9

Solution

Problem T10

[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] Solution

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