CSMC 2018 Part B Problem 1
Source:
November 24, 2018
CSMCCSMC 2018
Problem Statement
Alexandra draws a letter A which stands on the -axis.
[*]The left side of the letter A lies along the line with equation . What is the -intercept of the line with equation ?
[*]The right side of the letter A lies along the line and the leter is symmetric about the -axis. What is the equation of line ?
[*]Determine the are of the triangle formed by the axis and the left and right sides of the letter A.
[*]Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line , as in Figure 2. The area of the shaded region inside the letter A and above the line with equation is of the total area of the region above the axis and between the left and right sides. Determine the value of .
Figure 1
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
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 = -4.8408113739622465, xmax = 5.491811096383217, ymin = -3.0244242161812847, ymax = 8.241467380517944; /* image dimensions */
pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, EndArrow(6), above = true);
yaxis(ymin, ymax, EndArrow(8.25),above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw((0,6)--(-2,0), linewidth(2));
draw((-2,0)--(2,0), linewidth(2));
draw((2,0)--(0,6), linewidth(2));
label("",(-2.874280000573916,3.508459668295191),SE*labelscalefactor);
label("",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor);
label("",(0,0),SW*labelscalefactor);
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
Figure 2
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
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 = -4.707487213054563, xmax = 5.6251352572909, ymin = -3.4577277391312538, ymax = 7.808163857567977; /* image dimensions */
pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882); draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--(0,6)--cycle, linewidth(2));
Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, EndArrow(6), above = true);
yaxis(ymin, ymax, EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */
/* draw figures */
draw((0,6)--(-2,0), linewidth(2));
draw((-2,0)--(2,0), linewidth(2));
draw((2,0)--(0,6), linewidth(2));
label("",(0,0),SW*labelscalefactor);
draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2));
draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2));
draw((1.1148845961134441,2.6553462116596678)--(0,6), linewidth(2));
draw((0,6)--(-1.114884596113444,2.6553462116596678), linewidth(2));
fill((0,6)--(-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--cycle,black);
label("",(1.4920862691527148,3.1251527056856054),SE*labelscalefactor);
/* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* yes i used geogebra fight me*/
[/asy]