March 31, 1999
Wocomal Varsity Meet
ROUND I Number theory
ALL ANSWERS MUST BE IN SIMPLEST EXACT FORM
1. Explore powers of 3. Determine the units digit of 3 2 2 .
2. 11.11,+101.101,-?,
3. The greatest common divisor of a pair of numbers is 315 and the least common multiple of
the pair is 3 2 • 53 • 7 2 . Find both pairs of whole numbers which satisfy these conditions
ANSWERS
Opt) I- (2pts) 2.
ȣ.
(3 pts) 3
Assabet Valley, St. John's, Shrewsbury
March 31,1999
Wocomal Varsity Meet
ROUND II: Algebra 1 - open
ALL ANSWERS MUST BE IN SIMPLEST EXACT FORM
2
1
i—
1. Evaluate this expression if a = -9;g = — ;i = 8;o = -6\p = —;r = -0.8;^ = vl21;r = -12 :
go-patriots
2. In the equation 2x + 3y = 24, the variables x and y are non-negative integers. Find the
number of ordered pairs (x,y) which satsify this equation.
3. A bank teller was to change a $20 bill into quarters and dimes. She made an error and
interchanged the number of quarters and the number of dimes, thereby paying out $9 more
than she should have. How many dimes should she have paid out?
ANSWERS
(lpt) 1.
(2pts) 2.
(3pts) 3.
Bromfield, Tahanto, Westborough
March 31, 1999
Wocomal Varsity Meet
ROUND El: Geometry - open
ALL ANSWERS MUST BE IN SIMPLEST EXACT FORM
1. Find the value of c:
2. In the squares shown, the vertices A, B, and C
lie in a striaght line. Find the value of length x.
3. The vertices of an equilateral triangle are the midpoints of the sides of a larger equilateral
triangle. A circle is inscribed in the smaller triangle. Find the exact ratio of the area of the
circle to the area of the larger triangle. Do not approximate it or any radicals involved.
ANSWERS
(lpt) 1.
(2pts) 2.
(3 pts) 3.
Hudson, Shrewsbury, Tahanto
March 31, 1999
Wocomal Varsity Meet
ROUND IV: Logs, exponents, radicals
ALL ANSWERS MUST BE IN SIMPLEST EXACT FORM
- /27\T
1. Express in simplest form: 3 • 8 3 • I — j
2. Solve the radical equation -Jx + 6 + x = 14
3. Solve for x: log2 x • log4 x • log6 x = log2 x • log 4 x + log2 x • log,, x + log 4 x • log6 x
ANSWERS
(lpt) 1. _
(2pts) 2.
(3pts) 3.
Bromfield, Tahanto, Westborough
March 31,1999
Wocomal Varsity Meet
ROUND V: Trigonometry - open
ALL ANSWERS MUST BE IN SIMPLEST EXACT FORM, EXCEPT NUMBER 1
1 An airplane leaves the runway climbing at 18° on a straight path with a constant speed of
285 feet per second. Find the altitude of the plane after one minute, to the nearest foot.
2. A sector of a circle has area 25 sq cm and central angle 0.5 radian. Find the radius and
arc length Include units!
3. Solve for x: sin[(arc cos x ) - 3 0 ° ] = x. (arc cos x is the same as cos"1 x or inverse cos x)
ANSWERS
Opt) i.
£1
(2pts) 2. Ced'tvs
(3 pts) 3.
Holy Name, Shrewsbury, Westborough
arc
March 31,1999
Wocomal Varsity Meet
TEAM ROUND Topics of previous rounds and open
ALL ANWERS MUST BE IN SIMPLEST EXACT FORM UNLESS SPECIFIED
OTHERWISE IN THE PROBLEM
2 POINTS EACH
1 The three digit number 2A3 is added to the number 326 to give the three digit number
5B9 If 5B9 is divisible by 9, then A+B must equal what?
2 At the present time, the sum of the ages of a father and his son is 33 years Find the smallest
whole number of years until the father's age is 4 times the son's age
3 A circle, containing point A and tangent to the x-axis, rolls without slipping along the x-axis,
starting with point A at the origin If after 3.5 revolutions point A is at (JT, y)9 find y
4 If x > y > 0, list by letter those conclusions which are always true "a" may be any real
number
A
x + a>y + a
D 1/x > 1/y
B logj x > log, y
E ax > ay
1
c
2
x(iogl0 o. i) > xiog10 o. i)
F ^
> Vy
5 For 0 <> x <> n, at how many points do the graphs of y = sin(60x) and y = tan(30x)
intersect?
6 Write the next term in this geometric sequence
Jl,
3
j2 , %fl,
7 If the arithmetic mean of z and y is 6 and their geometric mean is 5, write a quadratic
equation in x2 + bx + c = 0 form that has z and y as its roots
8 How long, to the nearest day, is 10» seconds?
9 How many points (x,y) on or inside the circle x2 + y1 = 50 have nothing but integer
coordinates?
Bancroft, Bromfield, Hudson, South, Worcester Academy
Wocomal Varsity Meet Answers
March 31, 1999
Round 1
1.
Team Round
1 pt
# thry
2. 2 pts
9.315 « if
3/5,55/25
W
3. 3 pts
2#>5; 1115
G
1.
%de^t
rtwHtr
2.
3.
Round II
1. 1 pt
alg 1
Round III
2. 2 pts
5
3. 3 pts
100
1.
5.
2.
7
91
12.25" 11-;" 22
2. 2 pts
3. 3 pts
1
4.
1 pt
geom
2 pts each
Of
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6.
fcrn%
Round IV
logs
exp
rad
1.
1 pt
2. 2 pts
3. 3 pts
Round V
1. 1 pt
2
*-inc +25 -0
10
1 Or 1 O
and or
522H
trig
2. 2 p t
3. 3 pts
OCecl Mli and Unifr
8.
HI
9.
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