5-QUESTION
CHALLENGE 6
Name
Calculators may not be used.
1.��������
1
0
4
8
1
6
2
3
1
There are 16 non-overlapping equilateral triangles (unit
triangles) in the figure shown. Each new number to be
written in an empty unit triangle is the product of the three
closest numbers in the row directly below it. What number
will be written in the shaded unit triangle at the top? students The ratio of girls to boys in a volleyball club at Ash Middle School is 7 to 4. 2.��������
There are 42 girls in the club. What is the total number of students in the
club?
inches
3.��������
A B
8”
Happy
Birthda
y!
6”
A greeting card is 6 inches wide and 8 inches tall. Point A
is 3 inches from the fold, as shown. As the card is opened
to an angle of 45 degrees, through how many more inches
than point A does point B travel? Express your answer as a
common fraction in terms of �.
4.�������� Three standard dice are rolled. What is the probability that the sum of
the numbers on the tops of the three dice is 17 or greater? Express your
answer as a common fraction. 5.�������� On a graph, a lattice point is an ordered pair (x, y)
with integers x and y. Exactly 15 lattice points lie
strictly in the interior of the triangular region (not
on the triangle) with vertices (0, 0), (N, 0) and (N, N), where N > 0. What is the value of N? Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
s
n
tio
5-QUESTION
CHALLENGE 6
lu
o
S
Name
Calculators may not be used.
0
1.��������
1
0
8
1
4
6
2
3
1
There are 16 non-overlapping equilateral triangles (unit
triangles) in the figure shown. Each new number to be
written in an empty unit triangle is the product of the three
closest numbers in the row directly below it. What number
will be written in the shaded unit triangle at the top?
We are given the entire bottom row of entries: 1, 0, 4, 1, 2, 3, 1. If we fill in all of the
entries, we see that the second row from the bottom is 0, 0, 8, 6, 6. The third row
from the bottom is 0, 0, 288. Finally the top row - or the last entry - is 0. Working
with just the 0 from the bottom row, students may be able to see that the top entry
must be 0.
66 students The ratio of girls to boys in a volleyball club at Ash Middle School is 7 to 4. 2.��������
There are 42 girls in the club. What is the total number of students in the
club?
The ratio of girls to boys to total students is 7:4:(7 + 4) or 7:4:11. If there are
42 girls, then the 7 representing the girls in the ratio must be multiplied by 6 to
get the actual number of girls. Therefore, the actual numbers of girls, boys and
total students is 42:24:66 when the entire initial ratio is multiplied by 6. The total
number of students is 66.
3π
4 inches
3.��������
A B
8”
Happy
Birthda
y!
6”
A greeting card is 6 inches wide and 8 inches tall. Point A
is 3 inches from the fold, as shown. As the card is opened
to an angle of 45 degrees, through how many more inches
than point A does point B travel? Express your answer as a
common fraction in terms of �.
Point A is traveling along the circumference of a circle with a diameter of 6 inches.
This circumference is 6� inches. Point B is traveling along the circumference
of a circle with a diameter of 12 inches. This circumference is 12� inches. Both
points travel 45 degrees, which is 45 ÷ 360 = 81 of the circles’ circumferences. The
difference is then ( 81 )(12�) – (1/8)(6�) = ( 81 )(12� – 6�) = ( 81 )(6�) = 34π inches.
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges
1
54
4.��������
Three standard dice are rolled. What is the probability that the sum of
the numbers on the tops of the three dice is 17 or greater? Express your
answer as a common fraction. If three standard dice are rolled, each die has six possible outcomes. Together,
there are 6 × 6 × 6 = 216 possible outcomes for the three dice. The only way we
will get a sum of 17 or greater is if we get the following outcomes: 6,6,6; 5,6,6;
4
1
= 54
6,5,6; or 6,6,5. Therefore, 4 of the 216 outcomes work, which is 216
.
7
5.��������
On a graph, a lattice point is an ordered pair (x, y)
with integers x and y. Exactly 15 lattice points lie
strictly in the interior of the triangular region (not
on the triangle) with vertices (0, 0), (N, 0) and (N, N), where N > 0. What is the value of N? With vertices (0, 0), (N, 0) and (N, N), we see that
one side of the triangle is along the positive x-axis, one side is along the line y = x and the
other side is a vertical segment at x = N. If N = 1
or 2, there are no strictly interior lattice points.
When N = 3, there is 1. When N = 4, there are 3.
When N = 5, there are 6. (Did you notice these
are the triangular numbers?) When N = 6, there
are 10. Finally, when N = 7, there are 15 strictly
interior lattice points.
(N, N)
(0, 0)
(N, 0)
Copyright MATHCOUNTS Inc. 2013. All rights reserved. The National Math Club: 5 Question Challenges