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Canadian Senior Mathematics Contest
2018 Canadian Senior Mathematics Contest
B1
B1
Part of
2018 Canadian Senior Mathematics Contest
Problems
(1)
CSMC 2018 Part B Problem 1
Source:
11/24/2018
Alexandra draws a letter A which stands on the
x
x
x
-axis. [*]The left side of the letter A lies along the line with equation
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
. What is the
x
x
x
-intercept of the line with equation
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
? [*]The right side of the letter A lies along the line
L
2
L_2
L
2
and the leter is symmetric about the
y
y
y
-axis. What is the equation of line
L
2
L_2
L
2
? [*]Determine the are of the triangle formed by the
x
x
x
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
y
=
c
y=c
y
=
c
, as in Figure 2. The area of the shaded region inside the letter A and above the line with equation
y
=
c
y=c
y
=
c
is
4
9
\frac49
9
4
of the total area of the region above the
x
x
x
axis and between the left and right sides. Determine the value of
c
c
c
. 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("
y
=
3
x
+
6
y=3x+6
y
=
3
x
+
6
",(-2.874280000573916,3.508459668295191),SE*labelscalefactor); label("
L
2
L_2
L
2
",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor); label("
O
O
O
",(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("
O
O
O
",(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("
y
=
c
y=c
y
=
c
",(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]
CSMC
CSMC 2018