Download k 9 2 15 x - < . 1 f x y f x f y + = + f f

WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
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. A line through points  k , 4  and  6, 9  is perpendicular to the line with an equation
3 x  2 y  5  0 . Determine the value of k .
4. The degree measures of the angles of a triangle are in the ratio 8:11:13. One of these
angles is selected at random. Determine the probability that the degree measure of this angle
is an integer. Express your answer as a common fraction reduced to lowest terms.
5. Determine the least integer solution for the inequality 9  2 x  15 .
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. For all x and y , f  x  y   f  x   f  y   1 . f 1  2 . Determine the value of f  3 .
8. Ten students in a class took a test. Two of the students scored a 79. Two others scored 83.
Four of the rest had scores that averaged 92. The class average for these ten students was an
88. Determine the average score of the remaining two students.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
ALGEBRA I
PAGE 2 OF 3
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. For some value of k , 13kx 12 x 9  1 has exactly one real solution for x . Determine the
ordered pair  k , x  . Express your answer as the ordered pair  k , x  .
2
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. Dave notices his odometer in his car reads 686,686 miles, which is a palindromic number
(reads the same forwards and backwards.) If Dave could drive at an uninterrupted 55 miles
per hour, determine the number of hours Dave would need to drive until the odometer
reached the first palindromic mileage reading greater than seven hundred thousand miles.
Express your answer as an exact decimal.
14. When a woman drives directly from home to work at an average speed of 26.25 miles per
hour, she arrives 2 minutes late. When she drives the same route at an average speed of 35
miles per hour, she arrives 2 minutes early. Determine the number of miles in the length of
this route.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
15. For all valid replacements of a and b , a  b 
ALGEBRA I
PAGE 3 OF 3
3a  b
. Determine the value of 4   6  9  .
b
16. The base 10 number 218 is represented by 431 in base b . (That is 218  431b ). Then
253b  k where k is a base 10 integer. Determine the value of k .
17. Jack began mowing a field for 30 minutes. Jill joined him and together they finished the
field in another hour. Jill can mow the same field by herself in 2 hours and 30 minutes.
Determine the number of minutes additionally that Jack would have had to mow the field if
he had mowed alone. Express your answer to the nearest whole minute.
18. Determine the first (leading) digit in the sum 11  22  33  4 4  55    999999  10001000 .
1  x 
19. When simplified,
1  x 
1 1
1 1
1
1
 kx  w . Determine the ordered pair  k , w  . Express your
answer as the ordered pair  k , w  .
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 RAA
Algebra I
Correct X
2 pts. ea. =
Name
School
ANSWERS
(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
 3
specified in the question. No units of measurement are required.
 0,  or
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
295
11.
144
13.5 OR
1
3
27
1
OR 13
2
2
(Must be this reduced
common fraction.)
2
94
9
14.
7
17.
18.
("students" optional.)
19.
20.
4
(Must be this
decimal.)
("miles" optional.)
5
136
120
("minutes" optional.)
1 OR ONE
 2, 1
(Must be this
ordered pair.)
63 ("people" optional.)
 0.0.75 
also accepted
71
242.2
16.
8
1
 3

OR  4,  OR  4,1 
2
 2

13.
15.
32
200
12.
 4,1.5 

60 also accepted as correct
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
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. Point A  2,1 is reflected over the line y  x to point A ' . Point B
y-axis
has coordinates  8, 4  . Determine the area of A ' BA . Express your
answer as an exact decimal.
B(8,4)
A(-2,1)
x-axis
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
C
O
D
5. A is an acute angle such that A  5 x  10 . Then k  x  w is the restriction on x .
Determine the ordered pair  k , w  . Express your answer as this ordered pair.
6. A right rectangular box is 60 centimeters wide, 75 cm. long, and 45 cm. high. Determine the
length, in centimeters, of the longest stick that can be placed inside this box. Express your
answer as a decimal rounded to 4 significant digits.
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  .
A
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
GEOMETRY
PAGE 2 OF 3
A
8. AB  FC  DE . AB  11 and FC  3 . Determine the length of DE .
Express your answer as a common or improper fraction reduced to lowest
terms.
B
F
C
E
9. Three congruent circles are externally tangent to each other and each has
radius 6. The common external tangents, two circles at a time, intersect
to form a triangle. Determine the exact area of this triangle.
D
10. Consider the statement "You can go outside after you eat your supper." Which statement(s)
below is/are logically equivalent to the given statement? Express your answer as the capital
letter(s) corresponding to your choice(s).
A) If you eat your supper, then you can go outside.
B) If you don't eat your supper, then you can't go outside.
C) If you can't go outside, then you didn't eat your supper.
D) If you can go outside, then you ate your supper.
11. Two intersecting spheres have diameters 30 and 40 and the centers of the spheres are 25 units
apart. The area of the circle formed by the intersection of the spheres has numeric area k .
Determine the exact value of k .
12. In the diagram, ACB  90 , AD  5 , DC  10 , CB  20 , and
EB  17 . Determine the area of ADE .
A
D
E
C
B
2
1
 1  

13. Three points  3 , 2  ,  5, 2  , and  k , 2  are collinear. Determine the value of k .
5
2
 2  

Express your answer as a common or improper fraction reduced to lowest terms.
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  .
z
x
y
2
10
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
GEOMETRY
PAGE 3 OF 3
15. Quadrilateral ABCD is inscribed in Circle O as shown, (but
not drawn to scale.) AB  2 , BC  4 , CD  5 , and DA  3 .
The ratio of the area of FAB to EAD is k : w where k and
w are relatively prime positive integers, Determine the ratio
k : w . Express your answer as that ratio k : w .
D
C
E
O
A
B
16. To estimate the height of a vertical climbing wall, you place a mirror horizontally on the
ground so that its center is 85 feet from the base of the wall and then keep walking backward
from the mirror so that the mirror is between you and the wall until you can see the top of the
wall centered in the mirror. At this point, you are 6.5 feet from the center of the mirror and
your eyes are 5 feet vertically from the ground. Assuming the ground is flat, determine the
height of the wall in feet. Express your answer as a decimal rounded to the nearest
hundredth.
17. The four vertices of quadrilateral ABCD has coordinates A  0, 0  , B  b, a  , C  k , w  and
D  a, b  . Determine the coordinates of point C so that quadrilateral ABCD is a
parallelogram. Express your answer as the ordered pair  k , w  where k and w are
expressed in terms of a and b .
18. Two sides of a right triangle have measure 12 and 13. Determine the absolute value of the
difference between the largest and smallest possible measures of the third side. Express your
answer as a decimal rounded to four significant digits.
19. Kite ABCD has right angles DAB and DCB . EB  8 and
AD  3 . Determine the exact numeric area for kite ABCD .
D
3
A
E
C
8

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  .
B
F
2016 RAA
Name
Geometry
Correct X
School
2 pts. ea. =
ANSWERS
(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.
4
("faces" optional.)
(Must be this
exact decimal.)
19.5
276
("degrees" or "  " optional.)
96
 2, 20 
(Must be this
ordered pair.)
(Must be this
exact decimal.)
106.1
406
33
8

(Must be this reduced
improper fraction.)
144 3  216 OR 72 2 3  3


OR 216  144 3 OR 72 3  2 3
A, C

(Must have both
capital letters,
either order.)
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
144
16
57
8
(Must be this reduced
improper fraction.)
41
4 : 27
65.38
(Must be this
exact ratio and in
this form.)
("feet" or "ft."
optional.)
 a  b, a  b  OR  a  b, b  a  ordered pair.)
OR  b  a, b  a  OR  b  a, a  b 
(Must be this
12.69
(Must be this
exact decimal.)
18 2
(Must be this
exact answer.)
 4,7 
(Must be this
ordered pair.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
ALGEBRA II
PAGE 1 OF 3
1. Let k  1 be a positive real number. Determine all value(s) of k such that log k 5  log5 k .
Express your answer(s) as an integer or common or improper fraction reduced to lowest
terms.
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  .
2 x  12
. Determine the exact value of f 2016 10  . (That is, 2015 function
x2
compositions of f  x  or f f  f 10   with 2016 " f " s .)
5. Let f  x  


6. A parabola has directrix y  3 and endpoints of the latus rectum are  2, 1 and  6, 1 .
An equation for this parabola can be written either as 4 p  y  k    x  h  or
4 p  x  h    y  k  . Determine the sum  h  k  p  .
2
2
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 AA
ALGEBRA II
PAGE 2 OF 3
8. $1200 is invested at an Annual Percentage Rate of 6% compounded monthly. Determine the
amount of interest earned during the 121st month of this investment. Express your answer in
dollars rounded to the nearest cent.
9. Three monkeys randomly toss 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.
10. A ball is dropped from a height of 100 feet and it rebounds to
forever. Determine the total number of feet the ball travels.
11. Let  2 x  7  
 x2  x 6

3
of its height each time
5
 1 . Determine all possible value(s) for x such that this equation is true.
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. f  x  2  is quadratic function with roots of 5 and 3 . The graph of f  x  passes through
 2, 45  .
Then f  x  can be written as f  x   ax 2  bx  c where a , b , and c are integer
coefficients and. Determine the sum  a  b  c  .
14. Determine the value of k such that f  x  
28 x  123
is its own inverse.
7x  k
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
ALGEBRA II
PAGE 3 OF 3
15. Joe invests $12000 divided into three funds. Fund A earns 2%, Fund B 3.5% and Fund C
5%, all annual percentage rates and compounded annually. At the end of 1 year, Joe earned
$480 total interest on his investments. Joe had D dollars invested in Fund B. The number of
dollars invested in Fund A is then w  M  kD where M is some number of dollars.
Determine the product  Mk  .
16. Let y vary directly as the square root of x and inversely as z 2 . When y is tripled and z
quadrupled, x is multiplied by k . Determine the value of k .
17. P  x  is a polynomial function. When P  x  is divided by  x  3 , the remainder is 1.
When P  x  is divided by  x  1 , the remainder is 3. Determine the remainder when P  x 
is divided by  x 2  2 x  3 . Express your answer as a polynomial expression in terms of x
with integer or common or improper coefficients reduced to lowest terms.
18. An unfair weighted coin results in heads being tossed 60% of the time. The coin is tossed 4
times and the results recorded consecutively as H or T (for heads or tails.) Determine the
probability there is exactly one occurrence of two consecutive tosses the same. Express your
answer as common fraction reduced to lowest terms.
19. Bob's boat travels at a constant speed in still water. The current in the river has a constant
speed also. Bob makes a round trip, driving the boat upstream from the ramp to his cabin
and then returning to the ramp. The average speed of this trip is 36% of his still water speed.
The speed of the current is k % of his still water speed. Determine the value of k . Do not
use the % symbol as part of your answer.
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 RAA
Algebra II
Correct X
Name
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.
(Must have all 4 sol'ns,
1 (Must have both answers,
either order, with reduced
in any order.)
5 common fraction.) 11.
1.
(Must be the whole
word, capitalization
optional.
5,
2.
3.
4.
5.
6.
7.
8.
9.
10.
4, 3, 2,3
YES
12.
4
3 2
(Must be this
exact answer.)
OR 2  3
10
17 OR 1710 OR 17ten
325
400
14.
15.
3
10.92
13.
(Must be this exact
decimal, "dollars"
"$" optional.)
16.
17.
18.
("outcomes" optional.)
("feet" or
"ft." optional.)
19.
20.
137
36
4
2000
2304
1
5
x
2
2
228
625
80
11
("dollars or
"$" optional.)
(Must be this polynomial
expression with rational
coefficients.)
(Must be this reduced
common fraction.)
(Must be this integer,
with no % symbol used.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PRECALCULUS
PAGE 1 OF 3
1. Determine the value of the indicated vector inner (dot) product
 5,3 6,8  .
5 x3  3x 2  7 x  3
.
x 0
2x  3
2. Determine the value of lim
kw p
1
with  in quadrant I. Then the exact value for cos   cot  
in
2
q
simplified and reduced radical form with k , w , p , and q integers. Determine the sum
3. Let tan  
k  w  p  q .
4. Determine the value of the sum
 (2
5
2
k
 1) .
4
10
5. A sine curve has a maximum value of 3 at x   , and a minimum value of 3 at x  
3
3
with exactly one zero between these two values for x . Determine the standard equation
y  A sin  B  x  C    D for this curve with x measured in radians and where A  0 is the
least possible value. Express your answer as the sum  A  B  C  D  written as a common
or improper fraction reduced to lowest terms.
3
6. Let f  x   x3  , g  x   2 x x , and h  x   sin 2 x . Determine the value of
2
    
f  g  h     . Express your answer as a common or improper fraction reduced to lowest
   6 
terms.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PRECALCULUS
PAGE 2 OF 3
7. Let f  x    x  a  x  b   x  c  with 0  a  b  c . Determine the solution for f  x   0 .
2
Express your answer(s) in interval notation. (For example, the interval  k , w  represents
k  x  w .)
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. Determine the numeric area of a triangle with sides 20 and 16 and the included angle
between those two sides having measure 150 .
11. In ABC , 2sin A  3cos B  4 and 3sin B  2 cos A  3 . Determine the degree measure of
C .
12. Points 1,3, 1 ,  2,1,5 , and  3, 2, 6  lie on a plane. A standard equation for that plane can
be written as Ax  By  Cz  D  0 where A , B , C , and D are relatively prime integers and
A  0 . Determine the sum  A  B  C  D  .
13. A parabola has the equation of 2 x 2  12 x  3 y . The equation of the directrix of this
parabola can be expressed in the form x  k or y  w . Determine the value of k or w ,
whichever is appropriate, as an exact decimal. Express your answer as the equation x  k or
y  w.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2016 DIVISION AA
14.

PRECALCULUS
PAGE 3 OF 3

3  5 is one zero of a fourth-degree polynomial with integer coefficients and leading
coefficient 1 when written in decreasing degree of x . Determine the constant term for this
polynomial.
15. Determine the number of distinct arrangements of the letters in the word MATHEMATICS.
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( )
1  24tn
for n  1 be a sequence. Determine the value of t2016 .
24  3tn
Express your answer as a common or improper fraction reduced to lowest terms.
18. Let t1  77 and tn 1 
19. Determine the number of points of intersection for the curves y  1 
x is measured in radians.
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.
x
sin x
and y 
where
4
x
C(x,y)
A(0,0)
P
B(26,0)
2016 RAA
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.
("degrees" or
"  " optional.)
6
1.
1
2.
22
3.
4.
5.
6.
7.
8.
9.
10.
23
6
(Or any value of the form
97
64
 c, a 
6
12
80
90
12.
52
13.
56
23
 4n
6
11.
where n is an integer)
(Must be this
improper fraction.)
(Must be only this
ordered pair used as
interval notation.)
14.
15.
16.
17.
18.
19.
20.
y  6.375
(Must be this equation
in y using this exact
decimal value.)
4
4989600
("arrangements" and
comma usage optional.)
97
21
1847
207
2
 739 110 
,


 78 13 
(Must be this reduced
improper fraction.)
("points of intersection"
or "points" optional.)
(Must be this ordered
pair with reduced
improper fraction entries.)
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 1 of 3
NO CALCULATORS
1. Determine the number of distinct arrangements using all the letters of the word ILLINOIS
exactly once.
2. The diagonals of a rhombus are length 6 and 10. Determine the perimeter of this rhombus.
A
3. AD and BC are common internal tangent segments
to Circles E and F as shown. EB  3 , GC  3 , and
CF  2 . Determine the sum of the lengths of these
two common internal tangents.
E
3
3
C
2
G
B
F
D
4. A farmer's rectangular field is to be bounded on one side by a river and fenced in on the other
three sides. The farmer only has 200 meters of fencing. Determine, in square meters, the
maximum area of the field he can contain using the river and fence.
 . A , D , E , and B are
5. A is the center of arc BC
collinear as are A , F , and C .
DEF  FEC  CEB . AFD  EFC .
FD  AB . Determine the degree measure of ABC .
Express your answer as an integer or exact decimal.
D E
A
F
6. A solid cube whose sides have length 8 has a vertex cut off, leaving an equilateral triangle
trace with sides of length 1. The surface area of the now 7-faced solid can be written in the
kw p
form
when simplified and reduced radical form with k , w , p , and q integers.
q
Determine the value of the sum  k  w  p  q  .
7. The sum of two numbers is 10. The product of these same two numbers is 7. Determine the
sum of the reciprocals of these two numbers. Express your answer as a common or improper
fraction reduced to lowest terms.
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
B
C
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 2 of 3
NO CALCULATORS
x
8. The figure shows a square and two similar, non-square rectangles,
where y  x . The ratio of the area of the larger non-square rectangle to
x
k w
that of the smaller non-square rectangle may be written as
in
p
simplified and reduced radical form where k , w , and p are positive
x
x
x
integers. Determine the sum  k  w  p  .
9. A right triangle has legs of 12 and 35. The altitude to the
hypotenuse divides the hypotenuse into two parts k and
w . Determine the geometric mean of k and w . Express
your answer as an improper fraction reduced to lowest
terms.
y
y
35
12
k
w
10. Seven good friends eat at the same restaurant. All of them are eating there today but only
one of them eats there every day. The second eats there every 2 days, the third every 3 days,
the 4th every 4 days, the fifth every 5 days, the 6th every 6 days, and the 7th every 7 days.
Determine the number of days from today before they will all be eating there again on the
same day.
11. Determine the value of k such that k  17 
1
1
8
8 
.
12. Determine the base 3 representation for the base 4 number 1334 .
M
13. In right EBM with right angle at B , points G and O trisect MB while
points L and U trisect EB . ML  8 and OE  6 . Determine the length
of ME .
G
O
E
U
L
B
14. A circle is circumscribed about a square and at the same time the circle is inscribed in
another square. Determine the probability that a point selected in the interior of the larger
square is also a point in the interior of the smaller square. Express your answer as a common
fraction reduced to lowest terms.
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
NO CALCULATORS
PAGE 3 of 3
A
B
10 x  19
A
B
Cx  D

 2
. Then

 2
. Determine the values of C
x  3 x  4 x  x  12
x  3 x  4 x  x  12
and D . Express your answer as the ordered pair  C , D  .
15. Let
16. Determine the number of positive even integers of three digits that can be formed using the
digits 1, 2, 3, 4, 5, 6, and 7 if no digit is repeated.
17. A rectangular solid prism has sides of length 6, 8, and 24 and is inscribed in a sphere. The
surface area of this sphere is k . Determine the value of k .
18. A regular octagon is inscribed in a circle with diameter 8. A square is inscribed in the same
circle so that the vertices of the square are also vertices of the octagon. Determine the exact
area inside the octagon but outside the square.
19. When the Cousins Club decided to plan their summer barbecue, they found that 14 members
could meet in June only, 10 members could meet in July only, and 12 members could meet in
August only. Also, 3 members could meet only in July or August, 2 members could meet
only in June or August, and 2 members could meet only in June or July. There were 22
cousins who could meet in June. Determine the number of cousins who could meet in July.
20. The linear function f  x   6 x  72 has the value of its x-intercept doubled and the value of
its y-intercept tripled to create a new linear function g  x  . Determine the slope of the graph
of g  x  .
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
2016 RAA
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.
3360
2.
4 34
3.
15
4.
5.
6.
7.
8.
5000
82.5
(Must be this exact
answer.)
("square meters"
or " m 2 " optional.)
(Must be this exact
decimal, "  " or
"degrees" optional.)
1541
10
7
(Must be this reduced
improper fraction.)
9.
10.
420
12.
13.
14.
15.
16.
17.
4
1011 OR 10113 OR 1011three
3 10
1
2
 4, 37 
90
676
32 2  32 OR 32
10
420
37
11.

(Must be this reduced
common fraction.)
(Must be this
ordered pair.)
("integers" optional.)


2  1 OR 32 1  2





32  32 2 OR 32 1  2 OR 32  2  1
(Must be this reduced
improper fraction.)
("days" optional.)
18.
19.
20.
19
9
("cousins" or
"members" optional.)
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
NO CALCULATORS
PAGE 1 OF 3


1. Let f  x   2.8  3.5cos 2.3  x   . Determine the maximum value of this function.
4

Express your answer as an exact decimal.
2. Let k  0.23 in base 3. Determine the value of k in base 10.
3. There are 76 marbles in a jar, either orange or blue. Ella draws one marble at random and
does not replace it in the jar. The marble is blue. The probability the next marble Ella draws
3
will be blue is . Determine the ratio of orange marbles to blue marbles that were in the jar
5
before Ella drew any marbles. Express your answer as a ratio k : w where k and w are
relatively prime positive integers.
4. Let f  x   x 2  2 x and g  x   2 x  6 . Determine the largest value for x such that
f  g  x   g  f  x  .
13
1 
5. Let A   log 2  2k  . Determine the value of A .
4 
k 0
x 3
. Express your answer as an integer or common or
x 9
improper fraction reduced to lowest terms.
6. Determine the value of lim
x 9
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
NO CALCULATORS
PAGE 2 OF 3
3 2 5 
7. Let A   x 4 1 . Determine the average of the values for x such that det A  0 .
 1 x 2 
Express your answer as a common or improper fraction reduced to lowest terms.
8. Henrik and Will need to create 150 ounces of a 50% acidic solution. They have 80% acidic
solution and 30% acidic solution available to combine. Determine the number of ounces of
the 30% acidic solution that will be used.
8
12 

9. Let k  sin  Arc tan  Arc cos
 . Determine the value of k . Express your answer as a
15
13 

common or improper fraction reduced to lowest terms. (Note: This notation uses the
convention arcsin x represents the inverse sine relation while Arc sin x represents the
inverse sine function, etc.)
10. Let  be in radian measure such that 
2 cos 2   k sin   3  0 are

6
and

2

2
 

2
. The solutions to the equation
. Determine the value of k .
 sin 27 cos 27  sin 27 cos 63   k w 
11. Let 


 . Determine the sum  k  w  p  q  .
cos 27  sin 27 sin 63 cos 27   p q 
kw p
1
 3 . Then the largest real solution for x can be written as
in reduced
2
x
q
and simplified radical form with k , w , p , and q integers with q  0 . Determine the value
12. Let x 2 
of the sum  k  w  p  q  .
NO CALCULATORS
NO CALCULATORS
NO CALCULATORS
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 3 OF 3
NO CALCULATORS
13. Determine the sum of all solution(s) for x when 2 x  7  9  4 x  74 . Express your answer
as an integer or common or improper fraction reduced to lowest terms.
14. Determine the remainder when 32016 is divided by 11.
15. All angles are measured in radians. Determine the sum
1
1
1
1
1
1
cos 0  sin    cos  2   sin  3   cos  4     sin  2015   cos  2016  .
4
4
4
4
4
4
16. Determine the sum of the first 19 terms of an arithmetic series whose tenth term is 41.
17. Determine the numeric coefficient of the 6th term of  3m  2n  when expanded and
7
completely simplified in decreasing degree of m .
18. Consider the conic represented by the equation x 2  3 y 2  15 . Determine the exact
eccentricity of this conic. Express your answer as a simplified rational expression.
19. Let g  x   ax11  bx5  cx  3 where a , b , and c are real valued constants. g  2   7 .
Determine the value of g  2  .
20. Let x 2  y 2  6 xy and 0  y  x . Determine the value of
NO CALCULATORS
NO CALCULATORS
x y
.
x y
NO CALCULATORS
2016 RAA
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.
(Must be this
exact decimal.)
6.3
1.
2.
1
3.
15 : 23
4.
6  21 OR
1
6
6.
1
10
7.
9.
(Must be this
exact ratio.)
21  6
63
5.
8.
11.

10.
171
221
90
OR
(Must be this reduced
common fraction.)
(Must be this reduced
common fraction.)
("ounces" or
"oz." optional.)
12.
13.
14.
15.
16.
17.
18.
171(Must be this reduced
common fraction.)
221
19.
3
20.
2
9
2
3
(Must be this reduced
common fraction.)
3
1
779
6048
6
3
13
2
(Must be this exact
and simplified
rational expression.)
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
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)
1. Concave quadrilateral ABCD has vertices A 1.379,5.431 , B  2.543,3.791 ,
C  4.972, 3.145  and D  0.2473,1.895  . Determine the numerical area of this quadrilateral.
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 meet or pass for the second time.
3. The vertices of ABC are A  2, 4  , B  1, 2  and C  7,1 . Determine the length of the
altitude from vertex B to the line containing side AC .
4. The number 2016! ends in k trailing zeros (the zeros to the right of the last non-zero digit in
the expansion of 2016! ). Determine the largest possible positive integer n such that n ! ends
in  k  3 trailing zeros.
5. Ten committees of 8 people each must be chosen from a group of 90 people with no one
serving on more than one committee. Determine the number of possible ways these
committees can be chosen. Express your answer in scientific notation.
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  .
 23   23   y  1!  
b
7. Let   x !  
   a where a is a positive integer. Determine the value of b

 x  0   y 0  y !  
when a is as small as possible.
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 2 of 3
8. Determine the largest possible degree measure for  such that 0    360 and
2 5
sin  
.
5
.
9. Let f  x  
x x  x2
x x 
x is an integer.
2x
. Determine the sum of all real function value(s) f  x  when
10. Let g  x   3 x 2  10 x  1 and f  x   g  2 x  1 . Determine the value of f
 5.
11. Three pumps, A , B , and C , pump at different but constant non-zero rates and are available
to pump water into or out of a swimming pool. With all three pumps working together, they
would fill the empty pool in 2 hours and 6 minutes. If A and B are pumping water into the
empty pool but C is accidently set to pump water out, it would take 20 hours to fill the pool.
Determine the number of hours it would take for C to fill the empty pool alone.
1 

12. Determine the exact numerical coefficient for the term involving x16 when  x 4  2  is
x 

expanded and completely simplified in decreasing degree of x .
10
13. Let f  x   4 x 5  3 x 4  72 x3  54 x 2  320 x  240 . Let k be the largest positive zero and w
be the smallest positive zero for f  x  . Compute the value of k w .
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 3 of 3
14. A right circular cylinder has a base radius of x  4 , height of 2 x  3 , and numeric volume 20.
Determine the sum of all possible value(s) for x .
15. A regular pentagon PENTA has perimeter 10. An isosceles triangle is created by drawing
two segments from vertex P of the pentagon to the midpoints of two of the non-consecutive
sides EN and TA of the pentagon and the segment joining those midpoints.. Determine the
perimeter of this isosceles triangle.
16. The previous census determined that babies were born at the rate of 103 boys for every 100
girls. A family has three children. At this rate of birth, determine the probability the three
children are two of one sex and one of the other.
17. Determine the standard deviation for the set of numbers 8, 9, 6,8,5, 6, 7, 6,9,8,8,5 .
18. Let k   3  2 
10
i 1
1 i 
. Determine the value of k .
19. On an analog clock face, the hour hand is 6 inches long and the minute hand is 7.5 inches
long, both attached at the center of the circular clock face. Determine the positive difference
in number of inches between the arc length distances the tips of the hands travel between
1:00 pm and 6:00 pm, inclusive.
20. Let k  20  15  20  15  . Determine the value(s) of k .
2016 RAA
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.
1.
2.
3.
4.
5.
8.559
48 OR 48.00
3.258
3.606  10
(Must be this answer
85 and in this form.)
31.40
7.
82.75
8.
116.6
9.
2.894
143.6
12.
13.
2029
6.
10.
("seconds" or
"sec." optional.)
11.
14.
15.
(Must be this
decimal with
trailing zero necessary.)
16.
("degrees" or
"  " optional.)
4.693
210
(Must be this
exact answer.)
2.371
4.716
7.612
0.7498 OR .7498
17.
1.382
18.
5.994
19.
219.9
20.
("hours" or "hrs" optional.)
4.946
("inches" or
"in." optional.)
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
PAGE 1 OF 2
1.
Given 4k  3  k  1  5  k  2   3 and 3  2w  1  13  4  w  5  . Determine the sum
2.
Given ABC  DEF . If  A   3 x  29   ,  B   3 y  2   ,  D   5 y  3  and
 k  w .
 E   x  24   , determine the degree measure of  F .
3.
Two of the five lines 2 x  6 y  11 , 3 x  y  9 , x  3 y  15 , 4 x  12 y  7 , and
9 x  3 y  20 are selected at random . Determine the probability that the selected lines are
perpendicular. Express your answer as a common fraction reduced to lowest terms.
4.
One square has a diagonal of length 20. A second square has a diagonal of length 24.
Determine the exact positive difference between the perimeters of these squares.
5.
Determine the sum of all integers that are solutions to both x 2  2 x  15  0 and
x2 4
6.
In the “triskadec” measurement system, one “triskafoot” is equivalent to 13 “triskainches”.
Determine the number of cubic “triskafeet” in a rectangular solid with length 40
“triskainches”, width 10 “triskainches” and height 70 “triskainches”. Express your answer
as a decimal rounded to the nearest tenth.
7.
A circle is inscribed in a right triangle with legs of lengths 9 and 12. A second circle is
inscribed in an equilateral triangle with sides of length 6. Determine the exact absolute
value of the difference between the areas of the two circles.
8.
Rich drove a distance of 100 miles on a trip. On a second trip, he drove a distance of 400
4
miles. His second trip took
the number of hours the first trip took. His average speed
3
on the second trip was k times his average speed on the first trip. Determine the value of
k.
9.
Determine the number of rectangles in a standard 8  8 chess
board that are not squares.
10.
Determine the largest integer x for which both
integers.
32
72
and
represent positive
x3
x5
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
30
EXTRA QUESTIONS 11-12
PAGE 2 OF 2
11.
In a circle with radius
12.
Jack and Jill can mow the golf course together in 6 hours and 40 minutes. Jack and Joe
can mow the same golf course together in 7 hour and 30 minutes. Jill and Joe can mow
2
the same golf course together in 8 hours and 34 minutes. Determine how long, in
7
hours, it will take for all three together to mow the same golf course.

units, let L represent the length of a minor arc of a circle
formed by a central angle with measure 60 . In the coordinate plane, let D represent the
distance between points  5,  2  and 10,10  . Determine the sum  L  D  .
ICTM Math Contest
Freshman – Sophomore
2 Person Team
Division AA
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 1
NO CALCULATORS ALLOWED
1. Given
4k  3  k  1  5  k  2   3
and
3  2w  1  13  4  w  5.
Determine the sum
 k  w .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 2
NO CALCULATORS ALLOWED
2. Given ABC  DEF .
If  A  3 x  29 ,
 B   3 y  2  ,
 D  5 y  3  and
 E   x  24  ,
determine the degree
measure of  F .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 3
NO CALCULATORS ALLOWED
3. Two of the five lines
2 x  6 y  11, 3 x  y  9,
x  3 y  15, 4 x  12 y  7,
and 9 x  3 y  20 are
selected at random .
Determine the probability
that the selected lines are
perpendicular. Express
your answer as a common
fraction reduced to lowest
terms.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 4
NO CALCULATORS ALLOWED
4. One square has a
diagonal of length 20.
A second square has a
diagonal of length 24.
Determine the exact
positive difference
between the perimeters
of these squares.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 5
NO CALCULATORS ALLOWED
5. Determine the sum of all
integers that are solutions
2
to both x  2 x  15  0
and x  2  4
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 6
CALCULATORS ALLOWED
6. In the “triskadec”
measurement system, one
“triskafoot” is equivalent to
13 “triskainches”.
Determine the number of
“cubic triskafeet” in a
rectangular solid with length
40 “triskainches”, width 10
“triskainches” and height 70
“triskainches”. Express your
answer as a decimal rounded
to the nearest tenth.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 7
CALCULATORS ALLOWED
7. A circle is inscribed in a
right triangle with legs of
lengths 9 and 12. A
second circle is inscribed
in an equilateral triangle
with sides of length 6.
Determine the exact
absolute value of the
difference between the
areas of the two circles.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 8
CALCULATORS ALLOWED
8. Rich drove a distance of 100
miles on a trip. On a second
trip, he drove a distance of
400 miles. His second trip
4
took the number of hours
3
the first trip took. His average
speed on the second trip was
k times his average speed on
the first trip. Determine the
value of k .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 9
CALCULATORS ALLOWED
9. Determine
the number
of rectangles
in a standard
8  8 chess
board that are not
squares.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 10
CALCULATORS ALLOWED
10. Determine the largest
integer x for which both
32
72
and
represent
x3
x5
positive integers.
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
EXTRA LARGE PRINT QUESTION 11
CALCULATORS ALLOWED
11. In a circle with radius
30

units, let L represent the
length of a minor arc of a
circle formed by a central
angle with measure 60.
In the coordinate plane, let D
represent the distance
between points  5,  2  and
10,10 . Determine the sum
 L  D .
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
EXTRA LARGE PRINT QUESTION 12
CALCULATORS ALLOWED
12. Jack and Jill can mow the
golf course together in 6
hours and 40 minutes. Jack
and Joe can mow the same
golf course together in 7 hour
and 30 minutes. Jill and Joe
can mow the same golf
course together in 8 hours
2
and 34 minutes. Determine
7
how long, in hours, it will
take for all three together to
mow the same golf course.
2016 RAA
Fr/So 2 Person Team
School
ANSWERS
(Use full school name – no abbreviations)
Total Score (see below*) =
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.
3
Score
(to be filled in by proctor)
95
("degrees" or "  " optional.)
8 2
(Must be this exact answer.)
2
5
12
(Must be this reduced common fraction.)
12.7
(Must be this decimal, :"triskafeet" optional.)
1092
(Must be this integer, comma usage and "rectangles" optional.)
6
3
(Must be this exact answer.)
13
TOTAL SCORE:
(*enter in box above)
Extra Questions:
11.
12.
13.
14.
15.
23
5
("hours" optional.)
* 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 AA
PAGE 1 OF 2
1. The point P is located on the x  axis and the point Q is located on the y  axis . Each
point is 10 units from the point  6, 8  . Determine the largest possible length of PQ .
2. Let 10log3log 2  2k and  4 w  8w2   2 2 w . Determine the sum  k  w  .
3. In TRI , TR  8 and RI  10 . Determine the value of the expression
answer as a common or improper fraction reduced to lowest terms.
sin T
. Express your
sin I
4. On a number line, the distance between 5 and a number x is 1 less than the distance between
 2 and x . Determine the value of x .
5. i  1 . Let 1i  2i 2  3i 3  4i 4  a  bi where a and b are real numbers. Let
4 3 16  3 3 2  2 3 54  c 3 d in simplified and reduced radical form and where d is a positive
integer. Determine the value  a  2b  3c  4d  .
6. Let P be the point that is the vertex of the parabola y  3x 2  24 x  50 . Let Q be the point
of intersection of the lines with equations 4 x  3 y  2 and y  3 x  1 . Determine the
length of PQ .
1
7. The n term of 6 sequences are described as follows: tn  5n  2 , tn  8    , tn  3  2n  ,
2
2n  3
n3
tn 
, tn 
, and tn  8  2n . Two of these sequences are selected at random.
7
n2
Determine the probability that both are arithmetic sequences. Express your answer as a
common fraction reduced to lowest terms.
n
th
8. Determine the sum of all integers that are solutions for x to both log 3  7  2 x   2 and
x 2  10 x  24 .
 3  x

9. Let f  x    2 x  1
 x 2  3x  14

for x  5
for  5  x  5 . Determine the sum of all x for which f  x   4 .
for x  5
10. Two standard and fair 6-sided die with faces numbered uniquely with integers 1 to 6
inclusive are tossed. Determine the probability that the sum of the two numbers showing is
greater than the product of the same two numbers. Express your answer as a common
fraction reduced to lowest terms.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA QUESTIONS 11-12
PAGE 2 OF 2
11. Let 1  2  3    2016  k w when in reduced and simplified radical form. Determine
the sum  k  w  .
12. Let cos  2 x   cos x with x measured in radians. The greatest solution for x when this
equation is solved over the interval  2 , 0  is k . Determine the value of k . Express
your answer as a common or improper fraction reduced to lowest terms.
ICTM Math Contest
Junior – Senior
2 Person Team
Division AA
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 1
NO CALCULATORS ALLOWED
1. The point P is located
on the x  axis and the
point Q is located on the
y  axis. Each point is 10
units from the point
6, 8. Determine the
largest possible length of
PQ .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
2. Let 10
LARGE PRINT QUESTION 2
NO CALCULATORS ALLOWED
log3 log 2
and
 2k
 4 8   2
w
w2
2w
.
Determine the sum
 k  w .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 3
NO CALCULATORS ALLOWED
3. In TRI , TR  8 and
RI  10.
Determine the value of
sin T
the expression
.
sin I
Express your answer as a
common or improper
fraction reduced to lowest
terms.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 4
NO CALCULATORS ALLOWED
4. On a number line, the
distance between 5 and
a number x is 1 less
than the distance
between  2 and x .
Determine the value of
this number x .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 5
NO CALCULATORS ALLOWED
5. i  1. Let
2
3
4
1i  2i  3i  4i  a  bi
where a and b are real
numbers. Let
3
3
3
3
4 16  3 2  2 54  c d
in simplified and reduced
radical form and where d
is a positive integer.
Determine the value
 a  2b  3c  4d .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 6
CALCULATORS ALLOWED
6. Let P be the point that
is the vertex of the
parabola
2
y  3 x  24 x  50.
Let Q be the point of
intersection of the lines
with equations
4 x  3 y  2 and
y  3 x  1.
Determine the length
of PQ .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 7
CALCULATORS ALLOWED
7. The n term of 6 sequences are
described as follows:
th
tn  5n  2,
t n  3  2 ,
n
1
tn  8    ,
 2
2n  3
tn 
,
7
n3
tn 
, and tn  8  2n.
n2
n
Two of these sequences are selected
at random. Determine the
probability that both are arithmetic
sequences. Express your answer as
a common fraction reduced to
lowest terms.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 8
CALCULATORS ALLOWED
8. Determine the sum of
all integers that are
solutions for x to both
log 3  7  2 x   2 and
2
x  10 x  24.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
9. Let
 3  x

f  x   2x  1
 x 2  3x  14

LARGE PRINT QUESTION 9
CALCULATORS ALLOWED
for x  5
for  5  x  5
for x  5
Determine the sum of all x
for which f  x   4.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION AA
LARGE PRINT QUESTION 10
CALCULATORS ALLOWED
10. Two standard and fair
6-sided die with faces
numbered uniquely with
integers 1 to 6 inclusive are
tossed. Determine the
probability that the sum of the
two numbers showing is
greater than the product of the
same two numbers. Express
your answer as a common
fraction reduced to lowest
terms.
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 11
CALCULATORS ALLOWED
11. Let
1  2  3    2016  k w
when in reduced and
simplified radical form.
Determine the sum
 k  w .
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2016 REGIONAL DIVISION A
EXTRA LARGE PRINT QUESTION 12
CALCULATORS ALLOWED
12. Let cos  2 x   cos x
with x measured in
radians. The greatest
solution for x when this
equation is solved over the
interval  2 ,0 is k .
Determine the value of k.
Express your answer as a
common or improper
fraction reduced to lowest
terms.
2016 RAA
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
Score
(to be filled in by proctor)
20
1.
1
2.
5
4
3.
2
4.
(Must be this reduced improper fraction.)
33
5.
5
6.
7.
1
5
(Must be this reduced common fraction.)
11
36
(Must be this reduced common fraction.)
65
2
8.
9.
10.
TOTAL SCORE:
(*enter in box above)
Extra Questions:
11.
12.
13.
14.
15.

2
3
14131
OR
2
3
N/A
N/A
N/A
(Must be this reduced
common fraction,)
* 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 AA
1.
Discuss the product (or composite) of two rotations.
2.
Six reflections are performed in succession. The order is that of the axes as given here:
First: y = x +1
Second: y = x + 3
Third: x = 0
Fourth: x = 5
Fifth: y = x - 4
Sixth: y = x - 6
Discuss the product (or composite) of these six reflections.
3.
Let a transformation T be given by the equations x¢ = y + 4, y¢ = x -10.
(a)
Explain why T is an isometry.
(b)
Analyze this isometry, giving details.
ORAL COMPETITION
ICTM REGIONAL 2016 DIVISION AA
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.
A reflection has the property that if it is composed with itself, the resulting isometry is the
identity. Are there any other isometries that have this property?
2.
What is the composite of a glide-reflection with itself?
3.
Reflections are sometimes called “the building blocks of the isometries,” much like the relation
between the prime numbers and the set of integers. Explain this description.
ORAL COMPETITION
ICTM REGIONAL 2016 DIVISION AA
PAGE 1 OF 2
JUDGES’ SOLUTIONS
1.
Discuss the product (or composite) of two rotations.
Solution:
The product is either a rotation or a translation, depending on whether the sum of the
angles is or is not a multiple of 360° . The following table summarizes the possibilities:
SUM OF ANGLE MEASURES
Multiple of 360°
Not a multiple of 360°
C
E
N
T
E
R
S
Same (A)
Identity
Rotation about A through
sum of given angles
Different
(First A, then B)
Translation with
vector 2AB
Rotation through sum of
given angles. The center is
the intersection of the
perpendicular bisectors of
PP¢ and QQ¢, where P¢
and Q¢ are the images of
two chosen points P and
Q under the product of the
two rotations.
2.
Six reflections are performed in succession. The order is that of the axes as given here:
First: y = x +1
Second: y = x + 3
Third: x = 0
Fourth: x = 5
Fifth: y = x - 4
Sixth: y = x - 6
Discuss the product (or composite) of these six reflections.
Solution:
The first two give the translation T1 with vector < -2, 2 >.
The third and fourth give the translation T2 with vector < 10, 0 >.
The fifth and sixth give the translation T3 with vector < 2, - 2 >.
Since translations commute with each other the product T3T2T1 is the same as T3T1T2 , but
this is just T2 since T1 and T3 are inverses of each other. The net result is T2.
ORAL COMPETITION
ICTM REGIONAL 2015 DIVISION AA
PAGE 2 OF 2
JUDGES’ SOLUTIONS
3.
Let a transformation T be given by the equations x¢ = y + 4, y¢ = x -10.
(a)
Explain why T is an isometry.
(b)
Analyze this isometry, giving details.
Solution:
(a)
One answer: The equations fit the form (iii) of Property XV in the reference,
with a = 90°, h = 4, k = 10 , so T is an opposite isometry.
Second answer: Let P = (r, s) and Q = (u, v) be distinct arbitrary points.
Then P¢ = (s + 4, r -10) and Q¢ = (v + 4, u -10). The distance
P¢Q¢ = [(s + 4) - (v + 4)]2 +[(r -10) - (u -10)]2 = (s - v)2 + (r - u)2 = PQ,
so T preserves distance and is an isometry by definition.
(b)
Solving for fixed points we get x = y + 4 and y = x -10 . These equations give
x = x - 6, so there are no solutions and thus no fixed points and T is neither a
rotation nor a reflection.
Since the equations are not of the form for a translation, T must be a
glide-reflection.
Choose two points, such as P(0, 0) and Q(10,-4). Then P¢ = (4,-10) and
Q¢ = (0, 0). The midpoints M (2,-5) of PP¢ and N(5,-2) of QQ¢ lie on the
axis of the reflection part and the equation of this line is y = x - 7.
The glide-vector is MM ¢ = < -1- 2, - 8 - (-5) > = < -3,-3 > .
(Or use NN ¢ = < 2 - 5, - 5 - (-2) > = < -3,-3 > .)
Note that the glide-vector is parallel to the reflection’s axis, as it should be.
ORAL COMPETITION
ICTM REGIONAL 2015 DIVISION AA
PAGE 1 OF 1
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.
A reflection has the property that if it is composed with itself, the resulting isometry is the
identity. Are there any other isometries that have this property?
Solution: The identity, of course, but also a half-turn (180 degree rotation). No others.
2.
What is the composite of a glide-reflection with itself?
Solution: Since the translation part and the reflection part commute with each other, the
composite is the translation composed with itself, i.e., the translation whose vector is twice
that of the given translation part.
3.
Reflections are sometimes called “the building blocks of the isometries,” much like the relation
between the prime numbers and the set of integers. Explain this description.
Solution: Every isometry is either a reflection or the composite of two or three reflections,
so the reflections are, in some sense, primary.