Download Division 1A/2A - ICTM Math Contest

WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
ALGEBRA I
PAGE 1 OF 3
1. Let x  5 and y  3 . Determine the value of 2 x3  5 y 2 .
2. Let x and y be integers such that 17  x  20 and 7  y  9 . Determine the smallest
possible value of the product  xy  .
3. There are 30 students on a school bus. 24 students either play in the band or sing in the
chorus. 6 play in the band but do not sing in the chorus. 14 sing in the chorus and play in the
band. One of these students is selected at random. Determine the probability that student
sings in the chorus but does not play in the band. Express your answer as a common fraction
reduced to lowest terms.
4. Determine the sum of the smallest and largest solutions for the inequality 2 x  5  11 .
2 

 3  1  x  kx  w

5. The rational expression 
where k , w , p , and q are integers and k  0 .
 3
 p  qx
 x  1  1


Determine the sum  k  w  p  q  .
6. The product of two positive numbers is 768, and their sum is 7 times their positive
difference. Determine the larger of the two numbers.
7. Start with a number that is an improper fraction. Multiply by 2. Add 16 to your result.
Divide that result by 2. From that new result, subtract the improper fraction that you started
with. Determine the final answer.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
ALGEBRA I
PAGE 2 OF 3
8. Six monkeys all eating at the same rate can eat 8 bananas in 4 minutes. At this same rate,
determine how many minutes it will take 8 monkeys to eat 12 bananas.
9. The total cost of prom this year was $17,850. The cost was to be shared equally by those
who planned to attend. Regrettably, at the last minute, 10 people could not attend, raising the
cost for each person by $4.25. Determine the number of students who actually attended
prom.
10. A palindromic number is a positive integer that reads the same backwards and forwards, such
as 64746. Craig is challenged to produce the largest palindromic number possible, subject to
the requirements that no digit is 0, at least four different digits are used, and the sum of all of
the digits in the number is 25. Craig is successful with his offering of N. Determine the sum
of the first five digits of N. (The first five digits when reading the digits of N from left to
right.)
11. Determine the sum of all positive values of k such that x 2  kx  24 can be factored into two
binomial factors with integer coefficients.
12. Let k be the largest integer which leaves the same remainder when each of the three
numbers 163, 305, and 518 are divided by k . Determine the value of k .
13. Determine all real numbers x such that  3 x 2  2 x   1 .
6x
14. Points A  3,8  and B  5, 2  lie in a plane. Point C lies in quadrant one of the same plane,
on the perpendicular bisector of AB , and 15 units from the midpoint of AB . Determine the
coordinates of point C . Express your answer as an ordered pair  x, y  .
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
ALGEBRA I
PAGE 3 OF 3
15. On the first quiz of a new unit, Terri worked 4 out of 5 problems correctly. On the second
quiz, she was able to answer twice as many questions correctly, but earned only half as good
a grade. Determine the number of questions on the second quiz. (There is no partial credit
involved in the scoring.)
16. For all valid replacements of x and y , x  y 
2   4   9  .
x2  2 y
. Determine the value of
y
17. (All ages are whole numbers of years.) Ten years ago, Paul was twice as old as Jerry was
2
then and three times as old as Mary was then. In 8 years, Mary will be
as old as Paul is
3
then. Determine Jerry's age now.
18. Let f  x   x3 and g  x  
x
. Determine the value of f  g  4   .
2
19.  4, 20  and  k , 8  are points that lie on a line with slope 2. Determine the value of k .
20. In the town of Chambana, Illinois, the total population is 120,840 people. Exactly one-third
of the population have exactly two-letter initials, at least seven-fifteenths of the remainder of
the population have exactly three-letter initials, and the rest of the population have exactly
four-letter initials. Assuming the normal English alphabet, determine the minimum number
of people in Chambana, Illinois that have the same exact initials.
2016 RA
Name
Algebra I
Correct X
2 pts. ea. =
ANSWERS
School
(Use full school name – no abbreviations)
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1.
2.
295
144
2
15
3.
8.
9.
10.
16.
8
4.5 OR
9
1
OR 4
2
2
200
9
13.
15.
32
6.
12.
14.
5
5.
7.
(Must be this reduced
common fraction.)
5
4.
60
11.
("minutes"
optional.)
("students" optional.)
17.
18.
19.
20.
1,
71
1
3
OR 1,
10,17 
20
20
19
1 answers, either
 order.)
3
(Must have both
(Must be this
ordered pair.)
("questions" optional.)
("years old" optional.)
8
10
63
("people" optional.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
GEOMETRY
PAGE 1 OF 3
1. Two of the faces of a solid cube block were painted orange, and the rest of the faces were
painted blue. Determine the number of faces of this cube that were painted blue.
2. A triangle has vertices at A  0, 0  , B  6, 0  , and C  0,8  . Determine the length of segment
BC .
3. One of the angles of an isosceles trapezoid has a degree measure of 42. Determine the
largest possible sum of the degree measures of two of the other angles of the trapezoid.
4. In the diagram (not drawn to scale), points A , B , and D lie on
circle O , CD is tangent at point D , and A , B and C are
collinear. AB is a diameter, CD  12 , and CB  4 . Determine the
area of COD .
B
O
C
A
D
5. An equation of the line that passes through the point  3, 4  with a slope of 2 can be written
as y  kx  w . Determine the sum  k  w  .
6. AC is a diameter of Circle O . Point B lies on the circle such that BAC  60 and
AB  10 . Determine the exact length of the radius of Circle O .
7. A triangle has sides with lengths of 5, 6, and 8. The area of this triangle can be expressed as
k w
in simplified and reduced radical form where k , w , and f are positive integers.
f
Determine the value of  k  w  f  .
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
GEOMETRY
PAGE 2 OF 3
2
x  15 . The supplement of this angle is 36 less than three
3
times the complement of this angle. Find the degree measure of the original angle.
8. An angle has degree measure
9. In ABC , AB  13 , AC  5 , and the median from A to BC has length 6. Determine the
numeric area of ABC .
10. A quadrilateral is formed by an equilateral triangle sharing
a side with a leg of a right triangle. A side of the
equilateral triangle measures 2.7 units while the
hypotenuse of the right triangle measures 4.9 units.
Determine the area of this quadrilateral. Express your
answer as a decimal rounded to 4 significant digits.
11. In the diagram, PW and PZ are secants to the circle such
  200 .
that ZY  WX and the degree measure of WZY
Determine the degree measure of P .
4.9
2.7
W
X
Z
12. In the diagram, ACB  90 , AD  5 , DC  10 , CB  20 , and
EB  17 . Determine the numeric area of ADE .
Y
A
D
P
E
C
13. The numeric area of a rectangle is 24. The numeric perimeter of this rectangle is 20.
Determine the exact length of a diagonal of this rectangle.
B
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
14. In the right triangle shown,  x  y  z   2  k  w  p  in simplified
and reduced radical form with k , w , and p positive integers. Determine
the sum  k  w  p  .
15. Trapezoid ABCD is inscribed in a circle. A  40 and 
AB  220 .

Determine all possible degree measures for arc CD .
GEOMETRY
PAGE 3 OF 3
z
x
y
2
10
16. Decagons A and B are regular. A side of decagon B has length equal to the perimeter of
decagon A . The area of decagon B is equal to k times the area of decagon A . Determine
the exact value of k .
17. An array of 12 points is set up in 3 evenly spaced rows and 4 evenly spaced
columns (forming 6 adjacent congruent squares if connected horizontally
and vertically.) One point is chosen at random from each row. Determine
the probability the 3 points chosen will be collinear. Express your answer
as a common fraction reduced to lowest terms.
18. A triangle has sides of lengths 4, 9, and 12. Determine the type of triangle that has these side
lengths. Answer by writing the whole word "acute", "right", "obtuse", or "non-existent". Be
sure to write the whole word.
19. Quadrilateral ABCD has coordinate points A  0, 0  , B  5, 2  , C  k , w  and D  2,5  .
Determine the coordinates of point C so that quadrilateral ABCD is a parallelogram.
Express your answer as the ordered pair  k , w  .

20. In ABC , the altitude from A to BC intersects BC between B and C and the foot of the
altitude has coordinates  2,3 . An equation of the perpendicular bisector of BC is
2 x  y  4 and an equation of the median from A is 3 x  4 y  16 . Determine the
coordinates of A . Express your answer as an ordered pair  x, y  .
2016 RA
Geometry
Correct X
2 pts. ea. =
Name
ANSWERS
School
(Use full school name – no abbreviations)
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1.
2.
3.
4.
5.
6.
4
10
276
27
15.
16.
17.
("degrees" or "  " optional.)
30
8.677
13.
14.
10
8.
10.
("degrees" or "  " optional.)
4
406
11.
12.
96
7.
9.
("faces" optional.)
18.
19.
(Must be this decimal.)
20.
20
("degrees" or "  " optional.)
16
2 13
(Must be this
exact answer.)
41
20
(Must be this answer only,
"degrees" or "  " optional.)
100
1
8
obtuse
(Must be this reduced
common fraction.)
(Must be the whole word,
capitalization optional.)
 7,7 
 4,7 
(Must be this
ordered pair.)
(Must be this
ordered pair.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
1. Let f  x   x3 . Determine the value of f 1  8  .
ALGEBRA II
PAGE 1 OF 3
2. (Yes or No) Let k represent an irrational number and let w represent an irrational number.
Determine if it is possible for the product  kw  to represent a rational number. Express your
answer as the whole word "yes" or "no".
3. Determine the value of n such that n !  24 .
4. Let i  1 . The roots for x of x 4  7 x3  ax 2  cx  24  0 are p ,  p  1 , and bi where
p is a positive integer and b  0 . Determine the exact value of the sum  p  b  .
5. Let
x3
 x  1 . Determine the least value for x such that this inequality is true.
1 x
6. Sam rolls two fair, standard cubical dice. Determine the probability the sum of the numbers
on the upper faces will be equal to or greater than 5. Express your answer as a common
fraction reduced to lowest terms.
7. Let w be the symbol for the digit twelve in a base system. Let k be a positive integer that
is a number base. Let x represent a base 10 number such that x  16k and x3  281wk
(where w is the units digit of the number). Determine the value of k . Express your answer
in base ten.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
ALGEBRA II
PAGE 2 OF 3
8. Let log b 5  x and log b 3  y . Then log b 3 135b  kx  wy  p where k , w , and p are real
numbers. Determine the value of the sum  k  w  p  .
9. Three monkeys randomly toss a total of 24 indistinguishable ping pong balls into three
containers labeled A, B, C. Each ball is equally likely to end up in any one of the containers
and all balls go into one of the containers. Determine the number of distinct outcomes that
are possible. The order in which the balls are placed in the containers does not matter, just
the outcome.
10. Let k be the numeric value of the area of the region enclosed by 2 x  1  y  3  8 .
Determine the value of k .
11. Let p  x   x 2 , q  x   2 x  4 , f  x    p  q  x  , and g  x    q  p  x  . Determine the
value of x such that f  x   2 g  x  .
k p w
in simplified and reduced radical form
q
and k , w , p , and q are positive integers. Determine the value of the sum  k  w  p  q  .
12. Let x  33  33  33  . Then x 
13. In the given diagram (not drawn to scale), P lies on a parabola

with vertex V , focus F , directrix QD , and latus rectum LR .
Determine which of the following statement(s) is/are FALSE.
Express your answer as the capital letter of the false
statement(s). (Note: R is a point on the parabola and does not
lie on PD . Points P and V are distinct.)
A) FV  VQ
B) VP  PD
C) FP  PD
D) LF  QF
14. Determine the value of k such that
2
6 x 1
L
k
3 52 
 x  x 1 for all real values of x .
6
6
F
P
R
V
Q
D
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
15. Determine all real value(s) for x such that 2 x  5  4  3 x  1 .
ALGEBRA II
PAGE 3 OF 3
16. Let f  x   ax 2  abx  b a with a  0 and b  0 . Determine the minimum value for
f  x  in terms of a and b . Express your answer as a single rational expression using
integer coefficients and reduced to lowest terms.
17. Determine all possible real solutions for x when 2 x 3  x  5  0 . Express your answer(s) as
a decimal(s) rounded to four significant digits.
18. n  2016! has k trailing zeros when expanded and written as an integer. (Trailing zeros are
the zeros to the right of the last non-zero digit in the integer representation read from left to
right.) Determine the value of k .
19. Consider the conic represented by 3 y 2  4 x 2  432  0 . Determine the exact eccentricity of
this conic.
20. A group of 2n mathematicians consisting of exactly n men and of exactly n women are
having a dinner meeting. One of the mathematicians, Cindy, notes that there are k distinct
arrangements possible if the group were to sit at a round table with no distinguishing marks
and if men and women must alternate. Another of the mathematicians, Tom, suggests that
the group find the minimum value for n such that there would be more than 100, 000k
distinct arrangements possible if the group were to sit at a round table with no distinguishing
marks and if men and women did not necessarily alternate. Determine that minimum value
of n .
2016 RA
Name
Algebra II
Correct X
2 pts. ea. =
ANSWERS
School
(Use full school name – no abbreviations)
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2
YES
(Must be the whole
word, capitalization
optional.
4
3 2
(Must be this
exact answer.)
OR 2  3
2
5
6
(Must be this reduced
common fraction.)
17 OR 1710 OR 17ten
5
2
OR 1 OR 1.6
3
3
325
64
11.
12.
13.
14.
15.
16.
17.
18.
("outcomes" optional.)
("square units" or
"sq. un." or
un optional.)
2
19.
20.

1
1
OR
OR 0.5 OR .5
2
2
137
B
(Must be this capital
letter only.)
2
8 , 0
(Must have both answers,
either order.)
1.235
(Must be this
decimal only.)
(Must be this single, reduced
rat'l expression using
integer coeff.s.)
3b a
3 ab
OR
4
4
502
("trailing zeros" or
"zeros" optional.)
(Must be an exact answer.)
7
1
OR
7 OR 0.5 7 OR .5 7
2
2
11
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
PRECALCULUS
PAGE 1 OF 3
1. Determine the value of the indicated vector inner (dot) product
 5,3 6,8  .
 4 x5  3x 2  5 
2. Determine the value of lim  5
.
x  8 x  2 x 4  x


3
3. Angle A lies in quadrant IV with tan A   . Determine the value of sin  2 A . Express
4
your answer as a common or improper fraction reduced to lowest terms.
4. Determine the value of the sum
 (2
5
2
k
 1) .


5. Determine the value(s) for x such that log 5 5log5  log 5 x 5   1 . Express your answer(s) as
common or improper fraction(s) reduced to lowest terms.
F
E
6. Consider the diagram as shown, but not necessarily drawn to
scale. The degree measures of the arcs are in the ratio

 : CD
 : DA
  2 : 4 : 6 : 3 , DE  EF , and DE  8 as
AB : BC
marked in the diagram. Determine the length of DF . Express
your answer as a decimal rounded to four significant digits.
8
D
3x
A
2x
6x
B
4x
C
7. The expression  e x  e  x  is expanded and written in decreasing degree of e x . Determine
7
the sixth term of this expansion. Express your answer as the entire sixth term.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
PRECALCULUS
PAGE 2 OF 3
8. In a geometric sequence, the last term is 1458, the common ratio is 3 , and the sum of the
terms is 1094. Determine the second term of this geometric sequence.
9. Let a , b , c , and d be positive real numbers. Let a  b  c  d  10 , let c  d , and let
k w
a 2  b 2  c 2  d 2  30. The largest possible value of c can be expressed as
in
f
simplified and reduced radical form where k , w , and f are positive integers. Determine
the value of  k  w  f  .
10. ( i  1 and acisb  a cis  b    a  cos  b   i sin  b   ). In polar form, the product
1  i  3  3i   kcis  w  where
Determine the product  kw  .
k  0 and w is measured in radians with 0  w  2 .
 

11. ( i , j , and k are the usual standard normal unit vectors.) Determine the exact norm of
  

v  2i  3 j  7 k .
12. Let  be in degrees with 0    360 . Determine all value(s) for  such that
sin   sin  cos   0 .
13. A parabola has the equation of 2 x 2  12 x  3 y . The equation of the axis of symmetry of
this parabola can be expressed in the form x  k or y  w . Determine the value of k or w ,
whichever is appropriate. Express your answer as the equation x  k or y  w .
14. A forest ranger in Nebraska (assume flat ground) stands on the top landing of a fire tower
where a telescope is mounted exactly 70 meters above the ground. The ranger spots a hiker
at an angle of depression of 3.4 . He then rotates the telescope to his left and spots a second
hiker at an angle of depression of 2.5 . He then determines the angle between the two lines
of sight from telescope to hikers is 17.6 . Determine the distance, in meters, between the
two hikers. Express your answer as a decimal rounded to four significant digits.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION A
PRECALCULUS
PAGE 3 OF 3
 x  15t  30
15. The parametric equations in the system 
can be written as the direct relationship
 y  3t  7
kx  w
y
in simplified and reduced form with integers k , w , and p . Determine the value
p
of the sum  k  w  p  .
16. The roots for x in x3  ax 2  bx  105  0 are each two more than the roots for x in
x3  dx 2  ex  f  0 . a , b , d , e , and f are constant real numbers. Determine the value
of the sum  4d  2e  f  .
17. For all real values of  such that cos( )  sin( )  0 ,
Determine the value of  7 k  28w  .
cos3 ( )  sin 3 ( )
 k  w  sin  2   .
cos( )  sin( )
18. Let f  x   4 x 2  2 x  3 and g  x   2 x  4 . Determine the value of g  f  4   .
19. In ABC and using standard notation convention with side a opposite A , etc., A  55 ,
b  12 and c  18 . Determine the numeric area of ABC . Express your answer as a
decimal rounded to four significant digits.
20. In the diagram with coordinates as shown, AC  51 and
BC  55 . The ratio of the area of PAB to the area of
PAC to the area of PBC is 1:2:3. Determine the
ordered pair that represents point P. Express your answer
as an ordered pair  k , w  with each member of the
ordered pair expressed as a common or improper fraction
reduced to lowest terms.
C(x,y)
A(0,0)
P
B(26,0)
2016 RA
Pre-Calculus
Correct X
2 pts. ea. =
Name
ANSWERS
School
(Use full school name – no abbreviations)
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
(Must be this
exact value.)
6
1.
1
OR 0.5 or .5
2
2.

3.
1
5
5.
7.
8.
9.
10.
(Must be this reduced
common fraction.)
56
4.
6.
24
24
OR
25
25

25.89
(Must be this reduced
common fraction.)
(Must be this
exact decimal.)
21
21
OR
OR 21e 3 x
3x
3x
e
e
6
12
9
11.
12.
13.
14.
15.
16.
17.
62
0, 180
x  3
597.9
(Must be this
equation in x.)
(Must be this exact
decimal, "m." or
"meters" optional.)
59
97
21
18.
114
19.
88.47
20.
(Must have both answers,
either order, "  " or
"degrees" optional.)
 739 110 
,


 38 13 
(Must be this decimal,
"square units" or
"sq. un." optional.)
(Must be this ordered
pair with reduced
improper fraction entries.)
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 1 OF 3
NO CALCULATORS
1. It takes 6 complete turns of a crank to raise a window 2 inches. At this rate, determine the
number of complete turns of the crank it would take to raise the window 5 inches.
2. The degree measures of the angles in a convex pentagon are in the ratio 1: 3 : 4 : 5 : 5 . One of
the angles is chosen at random. Determine the probability the degree measure of the chosen
angle is greater than 90 . Express your answer as a common fraction reduced to lowest
terms.
3. The sum of two numbers is 6.5 and the product of the same two numbers is 9. Determine
the sum of the squares of these two numbers. Express your answer as a common or improper
fraction reduced to lowest terms.
4. Quad ABCD is a parallelogram with side lengths as shown.
Determine the values of x and y . Express your answer as
the ordered pair  x, y  .
3x+6
2y+2
A
B
y+4
D
C
3y-9
5. In right ABC with right angle at C , A  B and AC  3 . Determine the perimeter of
ABC .
A
6. In the concentric circles with centers at E , AB  40 , CD  24 ,
CD  EA , and AB is tangent to the smaller circle at point C .
Determine the length of CE .
C
F
E
D
B
7. Determine the area of the region bounded by x  1 , y  6 , x  2 y  14 , and 2 x  y  3 .
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 2 OF 3
NO CALCULATORS
8. Determine all possible value(s) for x such that
2x 1  4  7 .
9. A triangle has side lengths 9, 11, and 3 x  5 . Determine the number of integral values for x
such that this triangle exists.
10. Parallelogram ABCD has AB  8 , AD  12 , and B  120 . Determine the area of this
parallelogram.
11. A rectangular solid box has dimensions 3  4  5 . An ant starts at one corner and walks along
the faces of the box until it reaches the directly opposite corner. (That is, from one vertex to
the opposite vertex of a diagonal of the solid.) Determine the shortest distance the ant could
walk.
12. Three neighbors John, Paul, and George have identical adjacent, large back yards. It takes
John 60 minutes to mow his yard alone, Paul 54 minutes to mow his yard alone, and George
36 minutes to mow his yard alone. One week, they start at the same time and work together
to mow all three yards. All mow until all three yards are finished. Determine the time in
hours they mow together. Express your answer as a common or improper fraction reduced to
lowest terms.
13. A survey of 65 English majors at a large university produced the following results: 19 of the
students read biographies, 18 read science fiction, 50 read the classics, 13 read biographies
and science fiction, 11 read science fiction and the classics, 13 read biographies and the
classics, and 9 students read all three types. Determine the number of students that read none
of these three types of books.
14. Square SQUA is circumscribed about Circle O and TRI is
inscribed in Circle O . R  30 and IT  8 . Determine the
numeric area of Square SQUA .
NO CALCULATORS
S
O
U
NO CALCULATORS
R
A
T
I
Q
NO CALCULATORS
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
NO CALCULATORS
PAGE 3 OF 3
15. Two sides of a triangle have length 5 and 12, and the third side is also an integral length. The
triangle is obtuse and scalene. Determine the sum of all possible lengths for the third side.
4
1

and passes through points  , 2  and  3, k  . Determine the
3
3

value of k . Express your answer as an integer or common or improper fraction reduced to
lowest terms.
16. A certain line has slope
17. In Fantasy Land, the Cubs and the Sox are playing in the World Series where the first team to
win 4 games is the champion. Each team is equally likely to win a game. The Cubs win 2 of
the first 3 games, the Sox winning the other. Determine the probability the Sox will win the
series. Express your answer as a common fraction reduced to lowest terms.
18. In a 4 digit base ten number reading left to right, with non-zero first digit and the units
considered the fourth digit, the second and third digits are equal. The second digit is one
more than twice the fourth digit, the first digit is 3 less than the fourth digit, and the sum of
the digits is 5 more than twice the third digit. Determine this four digit number.
19. In the base ten alphanumeric equation 
Determine the value of the digit D .
D
A
I , each letter represents a unique digit.
D D D
20. The number 5.325 , where the "25" is the repeating block of digits, can be expressed as an
improper fraction reduced to lowest terms. Determine this fraction. Express your answer as
the improper fraction reduced to lowest terms.
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
2016 RA
Fr/So 8 Person
Correct X
School
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1.
15 ("turns" or "cranks" optional.)11.
2.
12.
3.
4.
5.
6.
7.
8.
9.
10.
3
5
97
4
 3,11
(Must be this reduced
common fraction.)
(Must be this reduced
improper fraction.)
(Must be this ordered pair.)
(Must be one of
these exact
6  3 2 OR 3 2  6 OR
answers.)
3 2  2 OR 3 2  2




15
11
5 (Must be this value only.)
5
48 3
("integers" or
"values" optional.)
(Must be this exact answer.)
13.
74 (Must be this exact answer.)
27
34
6 ("students" optional.)
14.
256
15.
72
16.
17.
18.
19.
20.
(Must be this reduced
common fraction, "hrs"
or "hours" optional.)
14
9
5
16
(Must be this reduced
improper fraction.)
(Must be this reduced
common fraction.)
1994
3
2636
495
(Must be this reduced
improper fraction.)
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 1 OF 3
NO CALCULATORS
 1
1. Let k  lim  1   . Determine the value of k .
n 
 n
n
2. Let A  1, 2,3, 4,5, 6, 7,8,9 . Determine the number of subsets of A that do not contain a
prime number.
3. Let x  0 such that 3
x 3 y
1
 
9
3 x 2 y
. Determine the value of
x
3y .
4. Three workers working at the same rate can harvest 40 acres of land in 2 days. Determine
the number of workers working at the same rate that it would take to harvest 100 acres in 3
days.
5. Let sin x  2 cos x  sin x with x measured in radians and 0  x  2 . The sum of all
solutions for x in this equation is k . Determine the value of k . Express your answer as
an integer or common or reduced fraction reduced to lowest terms.
6. Let a  b  a  b . Let a  b  a 2  b 2 . Determine the exact value of  3  4    2  5 .
7. The graph of y 2  8 y  8 x  8  0 is a parabola. The coordinates of the focus of this parabola
are  k , w  . Determine this ordered pair  k , w  . Report as your answer the ordered pair
 k , w .
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
NO CALCULATORS
PAGE 2 OF 3
8. i  1 . When 10 x 2  x  24  0 is solved over the set of complex numbers, the solution(s)
a  bi c
in simplified and reduced radical form where a , b ,
d
c , and d are integers with b  0 , c  0 and d  0 . Determine the sum  a  b  c  d  .
may be expressed in the form
k
5 
3 
9. Let 0    x5    . Determine all real solution(s) for x.
k 0 
  x  
10. All angles are measured in radians. Determine the sum
 
 
 
 
sin  0   sin    sin 2    sin 3      sin 2016   .
4
4
4
4
11. Determine the length of the hypotenuse of a 30  60  90 that has numeric area 25 3 .
12. The expression  x 4  8 x 3  34 x 2  kx  w  is a perfect square. Determine the values of k and
w . Express your answer as the ordered pair  k , w  .
13. Let x 
log 4 27
. Then k  16 x . Determine the value of k .
log 2 9
14. A Casino charges $5 to play a dice game. The patron then rolls 2 fair, standard cubical dice.
If the sum of the face up numbers is 2 or 12, the patron receives $18. If the sum of the
numbers is 3 or 11, the patron receives $9. If the sum of the numbers is 4 or 10, the patron
receives $3. If the sum of the numbers is 5, 6, 8, or 9, the patron receives $1. If the sum of
the numbers is 7, the patron receives nothing. Determine the Casino's expected profit, in
dollars, for each roll in this game.
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 3 OF 3
NO CALCULATORS
7
0
4
1 3
0
15. Determine the value(s) for x such that x
5 2  158 .
16. At Euler High School, there are 1400 graduating seniors this year. Of these seniors, 893
studied limits, 625 studied trigonometry, and 338 studied both limits and trigonometry during
their high school career. Determine the probability that a randomly chosen graduating senior
studied limits or trigonometry. Express your answer as a common fraction reduced to lowest
terms.
17. Let 3 x  8 y  103 . Determine the number of ordered pairs  x, y  that are solutions to this
equation and such that x and y are positive integers.
18. Determine the numerical coefficient of the term involving x3 when  x  3 is expanded and
completely simplified.
6
19. The general form for a conic's equation is Ax 2  Bxy  Cy 2  Dx  Ey  F  0 where the
coefficients are relatively prime integers and A  0 . A certain conic has center  5,3 , a
vertex at  5, 1 and eccentricity of
5
. Determine the general form of the equation for this
4
conic section. Report as your answer the sum  A  B  C  D  E  F  .
8x  5
A
B


. Determine the values of A and B . Report as your answer
x  3 x  10 x  2 x  5
the ordered pair  A, B  .
20. Let
2
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
2016 RA
Jr/Sr 8 Person
Correct X
School
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1.
e
2.
32
3.
2187
4.
5.
6.
7.
8.
9.
10.
5
7
2
5  29 OR
 1, 4 
11.
("subsets" optional.)
13.
("workers" optional.)
(Must be this reduced
improper fraction.)
29  5
(Must be this
ordered pair.)
981
3
12.
(Must be this
solution only.)
0 OR zero
14.
15.
16.
17.
10 2
 72,81
8
2 OR 2.00
8
59
70
4
18.
540
19.
684
20.
(Must be this
ordered pair.)
 3,5
("dollars" or
"$" optional.)
(Must be this
value only.)
(Must be this reduced
common fraction.)
("solutions" or
"ordered pairs" optional.)
(Must be this
ordered pair.)
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 1 of 3
Round answers to four significant digits and write in standard notation unless otherwise
specified in the question. Except where noted, angles are in radians. No units of measurement
are required. (NOTE: DO NOT USE SCIENTIFIC NOTATION UNLESS SPECIFIED IN
THE QUESTION)
5 x  7 y  35
1. Determine the exact solution(s) to the system 
. Express your answer(s) as
7 x  9 y  143
ordered pair(s)  x, y  .
2. At the same time, two ladybugs leave two different points which are 240 meters apart. They
fly directly back and forth between the two points without stopping at rates of 10 meters per
second and 5 meters per second respectively. Determine the exact number of seconds that
will elapse before they pass for the first time.
3.
x2
Determine the value of the expression  2 x    3 x14  when x  1.024 .
4. Let n be a positive integer. Determine the greatest value for n such that ln  n   log  n   3 .
5. The midpoints of the sides of a square with perimeter 32 are connected to form a new square.
This process continues forever. Determine the sum of the perimeters of all the squares.
Express your answer as a decimal rounded to four significant digits.
6. Let D lie on AB in right ABC with right angle at C such that CD  AB . AD  4 and
BD  25 . Let k be the arithmetic mean, w the geometric mean, and p the harmonic mean
between AD and DB . Determine the sum  k  w  p  .
7. RST is a right triangle with a side of length 15 and the length of the other two sides are
integers. Determine the exact sum of all possible perimeters for RST .
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 2 of 3
8. For n  2016 , n ! ends in k trailing zeros (the zeros to the right of the last non-zero digit in
the expansion of n ! ). Determine the exact sum of all possible integers n such that n ! ends
in k trailing zeros.
9. A hybrid sequence of terms ak is formed by terms numbered k which are congruent to 1
mod 3 forming an arithmetic sequence with common difference 0.01 , terms numbered k
which are congruent to 2 mod 3 being squares of the previous term in the hybrid sequence,
and terms numbered k which are congruent to 0 mod 3 are the product of the two previous
terms in the hybrid sequence. Let a1  1.61 . Determine the exact decimal sum of the 9th and
11th terms.
10. A lottery game uses 100 ping pong balls, each numbered uniquely with an integer from 1 to
100 inclusive in an urn. The player buys a ticket selecting 6 of those integers as a possible
winning combination. The Lottery then selects 6 ping pong balls at random from the urn
without replacement. If the player's numbers match the ping pong ball numbers in any order,
the player wins. Determine the probability the player wins. Express your answer in
scientific notation.
11. Let k and w be integers satisfying both inequalities w  k 2  2 and w  3  k 2 . Determine
the number of distinct possible ordered pairs  k , w  that exist.
 4 1
1 2 
3 1
12. Let A  
, B
and C  


 . Determine matrix X such that the matrix
 2 3
3 4 
4 2
equation AX  C  BX is true. Report as your answer the exact sum of the elements of
matrix X .
13. Determine the number of real values for x such that 1  x 
1
1
1
1 x
.
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 3 of 3
14. The expressions x 2  9 x ,  x  6 , and x 2  7 x taken in some order form an arithmetic
progression of real numbers. Determine the sum of all possible terms of these arithmetic
progressions.
15. In order to calculate the distance from point B to point A across a river, an explorer moves
1147 feet straight down the river bank from point B to point C (on the same side of the river
as point B ). The three points now form ABC and the explorer can measure
ABC  107.0 and ACB  14.7 . Determine the distance, measured in feet, from point
B to point A .
16. Let ABC be such that AC  30 , BC  40 , and ACB  33 . Determine the numeric area
of ABC .
 20 16 3
17. Let matrix A   10 24 9  . Determine the exact value of the determinant of A .
 30 32 21
18. Determine the numerical coefficient of the 8th term when  0.3 x 2  4 y 4  is expanded and
19
completely simplified and written in decreasing degree of x .
19. The degree measure of the smallest angle in a triangle with sides of 9, 10, and 14 can be
written in the form k w ' p " where k and w are integers. Determine the measure of this
angle. Express your answer in the form k w ' p " rounded to the nearest second.
20. Let k  20  16  20  16  . Determine the real value(s) of k .
2016 RA
Calculator Team
Correct X
School
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly. Round answers to four significant digits and
write in standard notation unless otherwise specified in the question. Except
where noted, angles are in radians. No units of measurement are required.
(Must be this
("ordered pairs" or
ordered pair.)
"solutions" optional.)
1.
2.
3.
14, 5 
16
8.867
4.
200
5.
109.3
6.
7.
("seconds" or
"sec." optional.)
(Must be this
decimal.)
18.3 OR 18.30
8.
10085
9.
7.020347
(Must be this
exact decimal.)
10
14.
15.
16.
466
8.389  10
12.
13.
(Comma usage
optional.)
10.
11.
14
1
0 OR none OR zero
126 OR 126.0
342.1
326.8
17.
6480
18.
438.7
19.
3950'17"
20.
4.958
(Must be in
scientific notation.)
("feet" or
"ft." optional.)
(Must be this answer
in this form.)
(Must be this decimal
only.)
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
1.
PAGE 1 OF 2
Given 3k  2  k  3  9 and 4  w  5  2w  42 . Determine the sum  k  w  .
2.
Given that ABC  DEF ,  A  10 x  8  ,  C   7 y   ,  D   8 x  16   and
3.
For all values x and for k  0 and w  0 , let  2 x  k   ax 2  bx  16 , and
4.
Given k  50  27 , w  75  98 , and m   2k  w  . Determine the value of m .
 F   5 y  20   . Determine the degree measure of  E .
 3x  w 
2
2
 dx 2  ex  9 . Determine the value of the expression  a  b  d  e  .
5.
Two sides of a right triangle have lengths 8 and 10. Determine the possible length(s) of
the third side of this triangle.
6.
A math team is ordering pizza for practice. 20 of the team members like pepperoni pizza and 16
like sausage pizza. There are 30 students on the team where all like at least one of pepperoni and
sausage. One of the team members is selected at random. Determine the probability the selected
team member likes both pepperoni and sausage. Express your answer as a common fraction
reduced to lowest terms.
7.
10 lines lie in a plane in such a way that no two lines are parallel and no point of
intersection contains more than 2 lines. Determine the largest number of regions into
which these 10 lines divide the plane.
8.
Ann and Mark each buy a certain item for the same price. Ann then sells hers for a 10%
profit, and Mark sells his for a 10% loss. Ann's sale price is $10 more than Mark's sale
price. Determine the number of dollars in the original price they each paid for the item.
9.
10.
Let f  x   4 x  7 and g  a   3a  2 . Determine the value(s) of k for which
f  k  3  g  k  1 .
In the given diagram, l1  l2 . The
ratios of angle degree measures are
a : c  1: 2 and b : e  5 : 6 .
Determine the value of d .
a
b
e
c
d
l1
l2
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA QUESTIONS 11-12
PAGE 2 OF 2
11.
A cube is inscribed in a sphere with radius 5. Determine the numeric volume of the part
of the sphere that is outside the cube. Report your answer as a decimal rounded to the
nearest hundredth.
12.
Let k represent the number of degrees in one exterior angle of a regular decagon and w
represent the number of diagonals of that decagon. Let p represent the number of
degrees in one exterior angle of a regular octagon and q represent the number of
diagonals of that octagon. Determine the value of  k  w  p  q 
ICTM Math Contest
Freshman – Sophomore
2 Person Team
Division A
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 1
NO CALCULATORS ALLOWED
1. Given 3k  2  k  3  9
and 4  w  5  2w  42.
Determine the sum
 k  w .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 2
NO CALCULATORS ALLOWED
2. Given that
ABC  DEF ,
 A  10 x  8 ,
 C   7 y  ,
 D  8 x  16  and
 F  5 y  20 .
Determine the degree
measure of  E .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 3
NO CALCULATORS ALLOWED
3. For all values x and for
k  0 and w  0, let
2
2
 2 x  k   ax  bx  16,
and
2
2
3x  w   dx  ex  9.
Determine the value of
the expression
 a  b  d  e .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 4
NO CALCULATORS ALLOWED
4. Given k  50  27 ,
w  75  98 , and
m   2 k  w .
Determine the value
of m.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 5
NO CALCULATORS ALLOWED
5. Two sides of a right
triangle have lengths 8
and 10. Determine the
possible length(s) of the
third side of this triangle.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 6
CALCULATORS ALLOWED
6. A math team is ordering pizza for
practice. 20 of the team members
like pepperoni pizza and 16 like
sausage pizza. There are 30
students on the team where all
like at least one of pepperoni and
sausage. One of the team
members is selected at random.
Determine the probability the
selected team member likes both
pepperoni and sausage pizza.
Express your answer as a common
fraction reduced to lowest terms.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 7
CALCULATORS ALLOWED
7. 10 lines lie in a plane in
such a way that no two
lines are parallel and no
point of intersection
contains more than 2
lines. Determine the
largest number of regions
into which these 10 lines
divide the plane.
8. Ann and Mark each buy
a certain item for the
same price. Ann then
sells hers for a 10%
profit, and Mark sells his
for a 10% loss. Ann's
sale price is $10 more
than Mark's sale price.
Determine the number of
dollars in the original
price they each paid for
the item.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 8
CALCULATORS ALLOWED
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 9
CALCULATORS ALLOWED
9. Let f  x   4 x  7 and
let g  a   3a  2.
Determine the value(s)
of k for which
f  k  3  g  k  1.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 10
CALCULATORS ALLOWED
10. In the given diagram,
l1  l2 . The ratios of angle
degree measures are
a : c  1: 2 and b : e  5 : 6.
Determine the value of d .
a
b
e
c
d
l1
l2
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 11
CALCULATORS ALLOWED
11. A cube is inscribed
in a sphere with radius
5. Determine the
numeric volume of the
part of the sphere that
is outside the cube.
Report your answer as
a decimal rounded to
the nearest hundredth.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 12
CALCULATORS ALLOWED
12. Let k represent the number
of degrees in one exterior
angle of a regular decagon
and let w represent the
number of diagonals of that
decagon. Let p represent the
number of degrees in one
exterior angle of a regular
octagon and q represent the
number of diagonals of that
octagon. Determine the value
of  k  w  p  q .
2016 RA
School
Fr/So 2 Person Team
Total Score (see below*) =
ANSWERS
(Use full school name – no abbreviations)
NOTE: Questions 1-5 only
are NO CALCULATOR
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
Answer
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
14
62
11
Score
(to be filled in by proctor)
("degrees" or "  " optional.)
17 2  3 OR 3  17 2
6, 2 41
1
5
56
50
20
100
(Must have both answers, either order.)
(Must be this reduced common fraction.)
("regions" optional.)
("dollars" or "$" optional.)
(Allow "degrees" or "  " )
TOTAL SCORE:
(*enter in box above)
Extra Questions:
11.
12.
13.
14.
15.
331.15
26
(Must be this decimal.)
* Scoring rules:
Correct in 1st minute – 6 points
Correct in 2nd minute – 4 points
Correct in 3rd minute – 3 points
PLUS: 2 point bonus for being first
In round with correct answer
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
PAGE 1 OF 2
1. Let x 2  4 x  12 and y 2  y  12 . Determine the largest possible value for the sum  x  y  .
2. Let 36, k , and 20 in that order be terms in an arithmetic sequence. Let 12, w , and 48 in that
order be terms in a geometric sequence with w  0 . Then the terms k , w , and p in that
order form another arithmetic sequence. Determine the value of p .
3. The point P is located on the x  axis and the point Q is located on the y  axis . Each
point is 5 units from the point  3, 4  . Determine the largest possible length of PQ .
4. The function f  x   ax 2  bx  c has a maximum at f 1  12 . The function
g  x   dx 2  ex  f has a minimum at g 1   3 . Determine the value of the sum
a  b  c  d  e  f  .
5. If a  b  0 , simplify a  b  a  b  b  a  b  a to an expression in a and b that does
not involve absolute value symbols.
5  x
6. Let f  x   
 x  3
f  x  6 .
for x  2
for x  2
. Determine the sum of all values for x such that
7. Two circles are represented by equations x 2  y 2  2 x  8  0 and x 2  y 2  4 x  2 y  59  0 .
Determine the exact area of the region outside the smaller circle and inside the larger circle.
8. Let log b 2  a , log b 3  c and log b 5  d . Then log b  6!  ka  wc  pd . Determine the sum
k  w  p .
9. Let 20   16 and 16   20 . Determine the value of
x
y
20
16
x  2
2 y 1
.
10. An antenna is mounted on outer edge of the roof of a building. From a point on level ground
30 feet away from that side of the building, the angles of elevation to the bottom and top of
the antenna are 37 and 51 respectively. Determine the height, in feet, of the antenna.
Express your answer as a decimal rounded to the nearest tenth of a foot.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA QUESTIONS 11-12
PAGE 2 OF 2
35
35
and b x 
with b  0 . Determine the value of x as an algebraic
17
17b
expression in terms of a and b .
11. Let b a 
12. One hundred balls numbered with consecutive integers from 1 to 100 inclusive are mixed in
a jar. One ball is selected at random. Determine the probability that the integer on that ball
has an odd number of distinct positive integral factors. Express your answer as a common
fraction reduced to lowest terms.
ICTM Math Contest
Junior – Senior
2 Person Team
Division A
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 1
NO CALCULATORS ALLOWED
1. Let x  4 x  12 and
2
let y  y  12.
2
Determine the largest
possible value for the
sum  x  y .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 2
NO CALCULATORS ALLOWED
2. Let 36, k, and 20 in
that order be terms in an
arithmetic sequence.
Let 12, w , and 48 in that
order be terms in a
geometric sequence with
w  0. Then the terms k,
w , and p in that order
form another arithmetic
sequence. Determine the
value of p.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 3
NO CALCULATORS ALLOWED
3. The point P is located
on the x  axis and the
point Q is located on the
y  axis. Each point is 5
units from the point 3,4.
Determine the largest
possible length of PQ .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 4
NO CALCULATORS ALLOWED
4. The function
2
f  x   ax  bx  c has a
maximum at f 1  12.
The function
2
g  x   dx  ex  f has a
minimum at g 1   3.
Determine the value of the
sum  a  b  c  d  e  f .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 5
NO CALCULATORS ALLOWED
5. If a  b  0, simplify
a b  a b  ba  ba
to an expression in a and b
that does not involve
absolute value symbols.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
6. Let
5  x
f  x  
x

3

LARGE PRINT QUESTION 6
CALCULATORS ALLOWED
for x  2
for x  2
Determine the sum of all
values for x such that
f  x   6.
.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 7
CALCULATORS ALLOWED
7. Two circles are
represented by equations
2
2
x  y  2 x  8  0 and
2
2
x  y  4 x  2 y  59  0.
Determine the exact area
of the region outside the
smaller circle and inside
the larger circle.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 8
CALCULATORS ALLOWED
8. Let log b 2  a ,
log b 3  c and log b 5  d .
Then
log b  6!  ka  wc  pd .
Determine the sum
 k  w  p .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
9. Let 20
16
 y
 x
 20.
LARGE PRINT QUESTION 9
CALCULATORS ALLOWED
 16 and
Determine the value
 x  2
20
of
.
 2 y 1
16
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
LARGE PRINT QUESTION 10
CALCULATORS ALLOWED
10. An antenna is mounted on
the outer edge of the roof of a
building. From a point on
level ground 30 feet away
from that side of the building,
the angles of elevation to the
bottom and top of the antenna
are 37 and 51 respectively.
Determine the height, in feet,
of the antenna. Express your
answer as a decimal rounded
to the nearest tenth of a foot.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 11
CALCULATORS ALLOWED
35
11. Let b 
and
17
a
35
b 
with b  0.
17b
x
Determine the value of x
as an algebraic expression
in terms of a and b.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 12
CALCULATORS ALLOWED
12. One hundred balls
numbered with consecutive
integers from 1 to 100
inclusive are mixed in a jar.
One ball is selected at random.
Determine the probability that
the integer on that ball has an
odd number of distinct
positive integral factors.
Express your answer as a
common fraction reduced to
lowest terms.
2016 RA
Jr/Sr 2 Person Team
School
Total Score (see below*) =
ANSWERS
(Use full school name – no abbreviations)
NOTE: Questions 1-5 only
are NO CALCULATOR
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
Answer
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Score
(to be filled in by proctor)
5
20
10
9
4a
2
55
7
256
14.4
(Must be this expression.)
(Must be this exact answer.)
(Must be this decimal, "feet" optional.)
TOTAL SCORE:
Extra Questions:
11.
12.
13.
14.
15.
(Must be this algebraic
expression.)
a  1 OR 1  a
1
10
(Must be this reduced
common fraction.)
N/A
N/A
N/A
(*enter in box above)
* Scoring rules:
Correct in 1st minute – 6 points
Correct in 2nd minute – 4 points
Correct in 3rd minute – 3 points
PLUS: 2 point bonus for being first
In round with correct answer
ORAL COMPETITION
ICTM REGIONAL 2016 DIVISION A
1.
Given the sequences defined as follows, tell if each is arithmetic, geometric, or neither.
A.
an  3n
B.
an  2  3 
C.
3
a1  4 and an  an1 for n  1
2
D.
n
an  2 n 2
2. An arithmetic sequence has a first term of 2 x  3 and a fifteenth term of 16 x  25 . If the common
difference of this sequence is 5, determine the numeric value of the fifth term.
3. A sequence is defined recursively by a1  7 and an  an1  8 for n  1 .
A. Is this sequence arithmetic or geometric?
B. Give the value of the 4th term of this sequence.
C. Give an explicit definition (defining an in terms of n) for this sequence.
4. A geometric sequence has a3  128 and a8  972 . Determine the product of terms a2 and a9 in
this sequence.
ORAL COMPETITION
ICTM REGIONAL 2016 DIVISION A
EXTEMPORANEOUS QUESTIONS
Give this sheet to the students at the beginning of the extemporaneous question period.
STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution
to these problems. Either or both the presenter and the oral assistant may present the solutions.
1. Due to its soaring popularity, the cast of a new sitcom show are demanding that they receive a 50%
raise in their salary for each year the show stays on the air. If a particular cast member earns
$1,000,000 in the first year, what is her salary in the fifth year?
2.
The first term of an arithmetic sequence is 2, and the tenth term is 20.
A. What is the common difference for this arithmetic sequence?
B. Find the value of the twelfth term.
3.
Given the sequence 3, a, b, 24, determine the values of a and b if:
A. the sequence is arithmetic
B. the sequence is geometric
ORAL COMPETITION
ICTM REGIONAL 2016 DIVISION A
PAGE 1 OF 2
JUDGES’ SOLUTIONS
1.
Given the sequences defined as follows, tell if each is arithmetic, geometric, or neither.
A.
an  3n
B.
an  2  3 
C.
3
a1  4 and an  an1 for n  1
2
D.
n
an  2 n 2
A: arithmetic (3, 6, 9, 12, ...)
Solution:
B: geometric  6, 18, 54, ...
C: geometric  4, 6, 9, ...
D: neither  2, 8, 18, ...
2. An arithmetic sequence has a first term of 2 x  3 and a fifteenth term of 16 x  25 . If the common
difference of this sequence is 5, determine the numeric value of the fifth term.
a15  a1  14d
Solution: 16 x  25  2 x  3  14  5 
14 x  98  x  7
a1  2  7   3  17
Then: a5  a1  4d
a5  17  4  5   37
ORAL COMPETITION
ICTM REGIONAL 2015 DIVISION A
PAGE 2 OF 2
JUDGES’ SOLUTIONS
3. A sequence is defined recursively by a1  7 and an  an1  8 for n  1 .
A. Is this sequence arithmetic or geometric?
Solution: arithmetic , since a common difference (8) is being added each time.
B. Give the value of the 4th term of this sequence.
a2  7  8  15
Solution: a3  15  8  23
a4  23  8  31
C. Give an explicit definition (defining an in terms of n) for this sequence.
Solution: Possible explicit definitions include an  7   n  1 8 and an  8n  1 .
Any equivalent definition in terms of n should be accepted.
4. A geometric sequence has a3  128 and a8  972 . Determine the product of terms a2 and a9 in
this sequence.
Solution: Students may solve for the common ratio:
and find terms 2 and 9: a2 
Then a2  a9  
a8  a3r 5
972  128r 5
7.59375  r 5  r  1.5
a3 256

; a9  1.5a8  1458 .
1.5
3
256
1458   124, 416
3
Alternatively, some students may note that since a2 
So a2  a9   a3  a8   128  972   124, 416
a3
a
and a9  a8r , a2  a9   3  a8 r 
r
r
ORAL COMPETITION
ICTM REGIONAL 2015 DIVISION A
PAGE 1 OF 2
JUDGES’ SOLUTIONS
EXTEMPORANEOUS QUESTIONS
Give this sheet to the students at the beginning of the extemporaneous question period.
STUDENTS: You will have a maximum of 3 minutes TOTAL to solve and present your solution
to these problems. Either or both the presenter and the oral assistant may present the solutions.
1. Due to its soaring popularity, the cast of a new sitcom show are demanding that they receive a 50%
raise in their salary for each year the show stays on the air. If a particular cast member earns
$1,000,000 in the first year, what is her salary in the fifth year?
Solution: 1, 000, 000  1.54  $5, 062,500
2.
The first term of an arithmetic sequence is 2, and the tenth term is 20.
A. What is the common difference for this arithmetic sequence?
Solution:
a10  a1  9d
20  2  9d  d  2
B. Find the value of the twelfth term.
Solution: Since the common difference is 2,
a12  a10  2d
a12  20  2  2   a12  24
Alternately, students may start from the first term, using a12  a1  11d or use a
recursive formula from term 10, adding the common difference twice.
ORAL COMPETITION
ICTM REGIONAL 2015 DIVISION A
JUDGES’ SOLUTIONS
3.
Given the sequence 3, a, b, 24, determine the values of a and b if:
A. the sequence is arithmetic
a4  a1  3d
Solution: 24  3  3d  d  7
The sequence is then 3, 10, 17, 24  a  10, b  17
B. the sequence is geometric
a4  a1r 3
Solution:
24  3r 3
8  r3  r  2
The sequence is then 3, 6, 12, 24  a  6, b  12
PAGE 2 OF 2