Nassau County Interscholastic Mathematics League
Contest #1
Answers must be integers from 0 to 999, inclusive.
2016 – 2017
No calculators are allowed.
Time: 10 minutes
1)
A clock chimes 4 times at 4 o’clock. It takes 6 seconds from the first
chime to the last chime. At the same constant rate, compute the
number of seconds from the first chime to the last chime at 12
o’clock.
2)
Compute the sum of the roots of
4
.
Time: 10 minutes
3)
4)
A linear function is defined by
slope of the linear function
. Compute the
.
The measure of one angle of
hich is inscri ed in circle is 4 . The measures of the
three angles of
form an arithmetic sequence. If the length of
the second longest side of
is 12 and the radius of circle O, in
simplest radical form, is
com te .
Time: 10 minutes
5)
If
and
com
te the ositi e al e of
.
6)
In isosceles
, point M is the midpoint of altitude
to base .
Lines
and
intersect
and
in points H and K,
respectively. The ratio of the area of quadrilateral RHMK to the area
of
may be expressed in simplest form as
. Compute
.
Solutions for Contest #1
1)
When the clock strikes 4 o’clock there are 3 two-second intervals between
strikes. So for the clock to strike
o’clock there are
two-second
intervals. Therefore, it takes 22 seconds for this clock to strike
o’clock.
2)
Re-write the given equation as
4
. o let
. So e ha e
3
3 93 3
3 =9 or 3
=2 or =1. The required sum is
3.
3)
Since
is linear,
. So,
.
A
60
4)
Let the measures of the angles of
in degrees be
4 4
and 4
. Setting their sum equal to 180 yields
. The degree-measures of the angles of
are 47, 60, and
73. If
6 draw radii
and
to form isosceles
with
. Let
bisect
forming two
-6 90° congruent triangles. Since
and
6 radius
4
and the required product is 12.
O
60 60
B
6
D
6
4 30
3
C
120
5)
Start with
. So
. The required value is 5.
6)
Let
and let
plane with S
and
. Then, the equation of
is
. Place
and
is
in the coordinate
here
and the equation of
R
. Solve the system of equations to get the
coordinates of point K:
4
H
K
4
M
. By symmetry, the coordinates of H are
and
. Since
by
is a kite. The area of the kite is
and
and
. The
S
area of
. So, the required ratio is
and the required sum is 7.
Alternatively, use mass points. The masses at S and T are each 1. So, L and R are
each 2, M is 4, and H and K are each 3. So, let
and
. Let
. Therefore,
. The required sum is 7.
L
T