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Difference between revisions of "1997 PMWC Problems/Problem I4"

(New page: ==Problem== Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle? Image:1997 PMWC individual 4.png ==Solu...)
 
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Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?
 
Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?
  
−
[[Image:1997 PMWC individual 4.png]]
+
<asy>
 +
/* File unicodetex not found. */
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 +
import graph; size(5.12cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = -0.43, xmax = 4.69, ymin = -0.49, ymax = 2.22; /* image dimensions */
 +
draw((0.23,0.23)--(0.59,0.23)--(0.41,0.54)--cycle);
 +
draw((1.27,1.04)--(1.63,1.04)--(1.45,1.35)--cycle);
 +
draw((1.45,0.23)--(1.52,0.3)--(1.59,0.3)--(1.56,0.38)--(1.59,0.46)--(1.52,0.46)--(1.45,0.54)--(1.38,0.46)--(1.31,0.46)--(1.35,0.38)--(1.31,0.3)--(1.38,0.3)--cycle);
 +
draw((1.8,1.04)--(1.87,1.11)--(1.94,1.11)--(1.91,1.19)--(1.94,1.27)--(1.87,1.27)--(1.8,1.35)--(1.73,1.27)--(1.66,1.27)--(1.7,1.19)--(1.66,1.11)--(1.73,1.11)--cycle);
 +
draw((1.27,1.85)--(1.63,1.85)--(1.45,2.16)--cycle);
 +
draw((1.73,0.23)--(1.88,0.23)--(1.88,0.37)--(1.73,0.37)--cycle);
 +
draw((0.52,1.04)--(0.66,1.04)--(0.66,1.18)--(0.52,1.18)--cycle);
 +
draw((1.64,1.85)--(1.79,1.85)--(1.79,1.99)--(1.64,1.99)--cycle);
 +
draw((1.82,1.85)--(1.96,1.85)--(1.96,1.99)--(1.82,1.99)--cycle);
 +
/* draw figures */
 +
draw((0,0)--(2.18,0));
 +
draw((0.41,0)--(0.41,0.23));
 +
draw((1.63,0)--(1.63,0.23));
 +
draw((0.95,-0.15)--(1.09,0));
 +
draw((1.23,-0.15)--(1.09,0));
 +
draw((1.23,-0.15)--(0.95,-0.15));
 +
draw((0.23,0.23)--(0.59,0.23));
 +
draw((0.59,0.23)--(0.41,0.54));
 +
draw((0.41,0.54)--(0.23,0.23));
 +
draw((0.08,0.23)--(0.73,0.23));
 +
draw((1.28,0.23)--(1.98,0.23));
 +
draw((1.31,0.46)--(1.35,0.38));
 +
draw((1.59,0.3)--(1.56,0.38));
 +
draw((1.59,0.46)--(1.56,0.38));
 +
draw((1.31,0.3)--(1.35,0.38));
 +
draw((1.31,0.46)--(1.38,0.46));
 +
draw((1.38,0.46)--(1.45,0.54));
 +
draw((1.52,0.3)--(1.45,0.23));
 +
draw((1.59,0.3)--(1.52,0.3));
 +
draw((1.52,0.46)--(1.45,0.54));
 +
draw((1.59,0.46)--(1.52,0.46));
 +
draw((1.31,0.3)--(1.38,0.3));
 +
draw((1.38,0.3)--(1.45,0.23));
 +
draw((0,0.81)--(2.18,0.81));
 +
draw((0.41,0.81)--(0.41,1.04));
 +
draw((1.63,0.81)--(1.63,1.04));
 +
draw((0.95,0.66)--(1.09,0.81));
 +
draw((1.23,0.66)--(1.09,0.81));
 +
draw((1.23,0.66)--(0.95,0.66));
 +
draw((0.08,1.04)--(0.73,1.04));
 +
draw((1.28,1.04)--(1.98,1.04));
 +
draw((1.72,1.04)--(1.89,1.04));
 +
draw(circle((0.25,1.21), 0.17));
 +
draw((1.45,0.23)--(1.52,0.3));
 +
draw((1.52,0.3)--(1.59,0.3));
 +
draw((1.52,0.46)--(1.45,0.54));
 +
draw((1.38,0.46)--(1.31,0.46));
 +
draw((1.31,0.3)--(1.38,0.3));
 +
draw((1.38,0.3)--(1.45,0.23));
 +
draw((1.8,1.04)--(1.87,1.11));
 +
draw((1.87,1.11)--(1.94,1.11));
 +
draw((1.94,1.11)--(1.91,1.19));
 +
draw((1.91,1.19)--(1.94,1.27));
 +
draw((1.94,1.27)--(1.87,1.27));
 +
draw((1.87,1.27)--(1.8,1.35));
 +
draw((1.8,1.35)--(1.73,1.27));
 +
draw((1.73,1.27)--(1.66,1.27));
 +
draw((1.66,1.27)--(1.7,1.19));
 +
draw((1.7,1.19)--(1.66,1.11));
 +
draw((1.66,1.11)--(1.73,1.11));
 +
draw((1.73,1.11)--(1.8,1.04));
 +
draw((0,1.62)--(2.18,1.62));
 +
draw((0.41,1.62)--(0.41,1.85));
 +
draw((1.63,1.62)--(1.63,1.85));
 +
draw((0.95,1.47)--(1.09,1.62));
 +
draw((1.23,1.47)--(1.09,1.62));
 +
draw((1.23,1.47)--(0.95,1.47));
 +
draw((0.23,1.85)--(0.59,1.85));
 +
draw((0.08,1.85)--(0.73,1.85));
 +
draw((1.28,1.85)--(1.98,1.85));
 +
draw(circle((0.41,2.02), 0.17));
 +
draw((1.73,0.23)--(1.88,0.23));
 +
draw((1.88,0.23)--(1.88,0.37));
 +
draw((1.88,0.37)--(1.73,0.37));
 +
draw((1.73,0.37)--(1.73,0.23));
 +
/* dots and labels */
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
//Credit to dasobson for the diagram</asy>
  
 
==Solution==
 
==Solution==
−
<math>circle=triangle+2square</math>
+
<math>\circ=\triangle+2</math> <math>\square</math>
  
−
<math>circle+square=triangle+star</math>
+
<math>\circ+\square=\triangle+\star</math>
  
−
<math>triangle+2square+square=triangle+star</math>
+
<math>\triangle+2</math> <math>\square+\square=\triangle+\star</math>
  
−
<math>3square=star</math>
+
<math>3</math> <math>\square=\star</math>
  
−
<math>triangle=3square+square=4square</math>
+
<math>\triangle=3</math> <math>\square+\square=4</math> <math>\square</math>
  
−
<math>circle=4square+2square=\boxed{6}square</math>
+
<math>\circ=4</math> <math>\square+2</math> <math>\square=\boxed{6}</math> <math>\square</math>
  
−
==See also==
+
==See Also==
 +
{{PMWC box|year=1997|num-b=I3|num-a=I5}}

Latest revision as of 07:49, 24 July 2017

Problem

Each of the three diagrams in the image show a balance of weights using different objects. How many squares will balance a circle?

[asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5.12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -0.43, xmax = 4.69, ymin = -0.49, ymax = 2.22; /* image dimensions */ draw((0.23,0.23)--(0.59,0.23)--(0.41,0.54)--cycle); draw((1.27,1.04)--(1.63,1.04)--(1.45,1.35)--cycle); draw((1.45,0.23)--(1.52,0.3)--(1.59,0.3)--(1.56,0.38)--(1.59,0.46)--(1.52,0.46)--(1.45,0.54)--(1.38,0.46)--(1.31,0.46)--(1.35,0.38)--(1.31,0.3)--(1.38,0.3)--cycle); draw((1.8,1.04)--(1.87,1.11)--(1.94,1.11)--(1.91,1.19)--(1.94,1.27)--(1.87,1.27)--(1.8,1.35)--(1.73,1.27)--(1.66,1.27)--(1.7,1.19)--(1.66,1.11)--(1.73,1.11)--cycle); draw((1.27,1.85)--(1.63,1.85)--(1.45,2.16)--cycle); draw((1.73,0.23)--(1.88,0.23)--(1.88,0.37)--(1.73,0.37)--cycle); draw((0.52,1.04)--(0.66,1.04)--(0.66,1.18)--(0.52,1.18)--cycle); draw((1.64,1.85)--(1.79,1.85)--(1.79,1.99)--(1.64,1.99)--cycle); draw((1.82,1.85)--(1.96,1.85)--(1.96,1.99)--(1.82,1.99)--cycle); /* draw figures */ draw((0,0)--(2.18,0)); draw((0.41,0)--(0.41,0.23)); draw((1.63,0)--(1.63,0.23)); draw((0.95,-0.15)--(1.09,0)); draw((1.23,-0.15)--(1.09,0)); draw((1.23,-0.15)--(0.95,-0.15)); draw((0.23,0.23)--(0.59,0.23)); draw((0.59,0.23)--(0.41,0.54)); draw((0.41,0.54)--(0.23,0.23)); draw((0.08,0.23)--(0.73,0.23)); draw((1.28,0.23)--(1.98,0.23)); draw((1.31,0.46)--(1.35,0.38)); draw((1.59,0.3)--(1.56,0.38)); draw((1.59,0.46)--(1.56,0.38)); draw((1.31,0.3)--(1.35,0.38)); draw((1.31,0.46)--(1.38,0.46)); draw((1.38,0.46)--(1.45,0.54)); draw((1.52,0.3)--(1.45,0.23)); draw((1.59,0.3)--(1.52,0.3)); draw((1.52,0.46)--(1.45,0.54)); draw((1.59,0.46)--(1.52,0.46)); draw((1.31,0.3)--(1.38,0.3)); draw((1.38,0.3)--(1.45,0.23)); draw((0,0.81)--(2.18,0.81)); draw((0.41,0.81)--(0.41,1.04)); draw((1.63,0.81)--(1.63,1.04)); draw((0.95,0.66)--(1.09,0.81)); draw((1.23,0.66)--(1.09,0.81)); draw((1.23,0.66)--(0.95,0.66)); draw((0.08,1.04)--(0.73,1.04)); draw((1.28,1.04)--(1.98,1.04)); draw((1.72,1.04)--(1.89,1.04)); draw(circle((0.25,1.21), 0.17)); draw((1.45,0.23)--(1.52,0.3)); draw((1.52,0.3)--(1.59,0.3)); draw((1.52,0.46)--(1.45,0.54)); draw((1.38,0.46)--(1.31,0.46)); draw((1.31,0.3)--(1.38,0.3)); draw((1.38,0.3)--(1.45,0.23)); draw((1.8,1.04)--(1.87,1.11)); draw((1.87,1.11)--(1.94,1.11)); draw((1.94,1.11)--(1.91,1.19)); draw((1.91,1.19)--(1.94,1.27)); draw((1.94,1.27)--(1.87,1.27)); draw((1.87,1.27)--(1.8,1.35)); draw((1.8,1.35)--(1.73,1.27)); draw((1.73,1.27)--(1.66,1.27)); draw((1.66,1.27)--(1.7,1.19)); draw((1.7,1.19)--(1.66,1.11)); draw((1.66,1.11)--(1.73,1.11)); draw((1.73,1.11)--(1.8,1.04)); draw((0,1.62)--(2.18,1.62)); draw((0.41,1.62)--(0.41,1.85)); draw((1.63,1.62)--(1.63,1.85)); draw((0.95,1.47)--(1.09,1.62)); draw((1.23,1.47)--(1.09,1.62)); draw((1.23,1.47)--(0.95,1.47)); draw((0.23,1.85)--(0.59,1.85)); draw((0.08,1.85)--(0.73,1.85)); draw((1.28,1.85)--(1.98,1.85)); draw(circle((0.41,2.02), 0.17)); draw((1.73,0.23)--(1.88,0.23)); draw((1.88,0.23)--(1.88,0.37)); draw((1.88,0.37)--(1.73,0.37)); draw((1.73,0.37)--(1.73,0.23)); /* dots and labels */ clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ //Credit to dasobson for the diagram[/asy]

Solution

$\circ=\triangle+2$ $\square$

$\circ+\square=\triangle+\star$

$\triangle+2$ $\square+\square=\triangle+\star$

$3$ $\square=\star$

$\triangle=3$ $\square+\square=4$ $\square$

$\circ=4$ $\square+2$ $\square=\boxed{6}$ $\square$

See Also

1997 PMWC (Problems)
Preceded by
Problem I3 		Followed by
Problem I5
I: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
T: 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
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