Mathcounts Chapter Competition Solutions Worksheet - Middle School - 2014 Page 9
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12CUBE 1: All opposite faces of the same color.
CUBE 6: No opposite faces of the same
1G ↔ 2G 3B ↔ 4B 5R ↔ 6R
color. Left face blue and right face green.
1R ↔ 2B 3B ↔ 4G 5R ↔ 6G
Note: Each color has is arbitrarily assigned to a pair of
opposite faces and when rotated the result is still CUBE 1.
CUBE 5 and CUBE 6 are different because
CUBE 2: One pair of opposite faces both red.
no rotation of CUBE 5 results in any rotation
Other opposite faces one blue, one green.
of CUBE 6. In fact, if you turn them upside
1G ↔ 2B 3B ↔ 4G 5R ↔ 6R
down (6 on top and 5 on the bottom), the
lateral faces (1-4) of CUBE 5 are now in the
original order of the lateral faces of CUBE 6,
and vice versa. As we have shown, the
number of different cubes is 6 Ans.
CUBE 3: One pair of opposite faces both
8. What is the greatest possible area of a
blue. Other opposite faces one red, one
triangle with vertices on or above the x-axis
green.
and on or below the parabola
1G ↔ 2R 3R ↔ 4G 5B ↔ 6B
y = −(x − ½)
2
+ 3
y = −(x
2
− x + ¼) + 3
y = −x
2
+ x − ¼ + 3
y = −x
2
+ x + 11/4
For a parabola given by the equation
/
y = ax
2
+ bx + c, the axis of symmetry is
CUBE 4: One pair of opposite faces both
−b
x =
. In this case, a = −1 and b = 1. So,
2a
/
green. Other opposite faces one red, one
the parabola is symmetric across the line x =
blue.
1
½, 3).
, and its vertex occurs at the point (
2
1B ↔ 2R 3R ↔ 4B 5G ↔ 6G
Since this parabola opens down the vertex is
also the highest point. Now let’s determine
where this parabola crosses the xaxis. We
will use the quadratic formula to solve
2
+ x + 11/4. Since a = −1, b = 1 and
0 = −x
c = 11/4, we have
CUBE 5: No opposite faces of the same
− ±
2
−
−
1
1
4( 1)( )
11
− ±
−
2
b
b
4
ac
color. Left face green and right face blue.
=
=
4
x
−
2
a
2( 1)
1R ↔ 2B 3G ↔ 4B 5R ↔ 6G
− ±
− −
1
1 ( 11)
− ±
− ±
1
12
1 2 3
1
=
=
=
=
±
3
−
−
−
2
2
2
2
The parabola crosses the x-axis at (½ − 3,
0) and (½ + 3, 0). The triangle shown has
vertices on the parabola and the xaxis at
A
½, 3), B(½ − 3, 0) and C(½ + 3, 0).
(
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