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Contests
National and Regional Contests
USA Contests
USA - Middle School Tournaments
Math Invitational for Girls
2019 MIG
2019 MIG
Part of
Math Invitational for Girls
Subcontests
(24)
25
1
Hide problems
2019 Individual #25
Each day John's mother sends him to the store with
$
1
\$1
$1
to buy widgets and gadgets, each of which cost a whole number of cents. On the first day John comes back with
4
4
4
widgets,
5
5
5
gadgets, and
35
35
35
cents in change. On the second day, John comes back with
5
5
5
widgets,
4
4
4
gadgets, and
39
39
39
cents in change. On the third day, John comes back with only
c
c
c
cents in change. He hands his mother the change, telling her that he had tripped coming home and broken all the widgets and gadgets. His mother, thinking for a moment, begins yelling at him for lying, as she noticed that there was no way he could have received exactly
c
c
c
cents in change given the price of widgets and gadgets. What is the sum of the digits of the least possible value of
c
c
c
?
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10
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13
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15
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18
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impossible to determine
<span class='latex-bold'>(A) </span>10\qquad<span class='latex-bold'>(B) </span>13\qquad<span class='latex-bold'>(C) </span>15\qquad<span class='latex-bold'>(D) </span>18\qquad<span class='latex-bold'>(E) </span>\text{impossible to determine}
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10
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(
B
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13
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(
C
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15
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18
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impossible to determine
24
1
Hide problems
2019 Individual #24
Regular hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
has area
1
1
1
. Starting with edge
A
B
AB
A
B
and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side
A
B
AB
A
B
, the new point,
G
G
G
, is drawn so
A
G
=
1
2
A
B
AG = \tfrac12 AB
A
G
=
2
1
A
B
. This forms hexagon
G
H
I
J
K
L
GHIJKL
G
H
I
J
K
L
, as shown. What is the area of this new hexagon? [asy] size(120); pair A = (-1/2, sqrt(3)/2); pair B = (1/2, sqrt(3)/2); pair C = (1,0); pair D = (1/2, -sqrt(3)/2); pair EE = (-1/2, -sqrt(3)/2); pair F = (-1,0); pair G = (A+B)/2; pair H = (B+C)/2; pair I = (C+D)/2; pair J = (D+EE)/2; pair K = (EE+F)/2; pair L = (F+A)/2; draw(A--B--C--D--EE--F--cycle); draw(G--H--I--J--K--L--cycle); dot(A^^B^^C^^D^^EE^^F^^G^^H^^I^^J^^K^^L); label("
A
A
A
",A,NW); label("
B
B
B
",B,NE); label("
C
C
C
",C,E); label("
D
D
D
",D,SE); label("
E
E
E
",EE,SW); label("
F
F
F
",F,W); label("
G
G
G
",G,N); label("
H
H
H
",H,NE); label("
I
I
I
",I,SE); label("
J
J
J
",J,S); label("
K
K
K
",K,SW); label("
L
L
L
",L,NW); [/asy]
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3
5
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(
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5
7
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4
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(
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7
9
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4
5
<span class='latex-bold'>(A) </span>\dfrac35\qquad<span class='latex-bold'>(B) </span>\dfrac57\qquad<span class='latex-bold'>(C) </span>\dfrac34\qquad<span class='latex-bold'>(D) </span>\dfrac79\qquad<span class='latex-bold'>(E) </span>\dfrac45
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5
3
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7
5
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3
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9
7
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5
4
23
1
Hide problems
2019 Individual #23
How many ordered pairs of integers
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfy
x
y
−
6
y
−
4
x
+
20
=
0
xy - 6y - 4x + 20 = 0
x
y
−
6
y
−
4
x
+
20
=
0
?
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1
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3
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4
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6
<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>6
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1
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2
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(
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3
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4
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6
22
1
Hide problems
2019 Individual #22
Scientists perform an experiment on a colony of bacteria with an initial population of
32
32
32
. The scientists expose the bacteria to alternating rounds of light and darkness. They first put the bacteria in a bright environment for one hour before placing it in a dark room for the second hour, and then repeating this process. Because they are vulnerable to light, the population of the bacteria will be halved in one hour of exposure to sunlight. However, in one hour of darkness, the population triples. How many hours will it take for the bacteria’s population to exceed
150
150
150
?
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between
4
and
5
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(
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between
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and
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between
6
and
7
<
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(
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between
7
and
8
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between
8
and
9
<span class='latex-bold'>(A) </span>\text{between }4\text{ and }5\qquad<span class='latex-bold'>(B) </span>\text{between }5\text{ and }6\qquad<span class='latex-bold'>(C) </span>\text{between }6\text{ and }7\qquad<span class='latex-bold'>(D) </span>\text{between }7\text{ and }8\qquad<span class='latex-bold'>(E) </span>\text{between }8\text{ and }9
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between
4
and
5
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>
(
B
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<
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between
5
and
6
<
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(
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between
6
and
7
<
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−
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>
(
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<
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between
7
and
8
<
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x
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(
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between
8
and
9
21
1
Hide problems
2019 Individual #21
The first
32
32
32
perfect squares,
1
1
1
,
4
4
4
,
9
9
9
,
16
16
16
,
25
25
25
,
…
\ldots
…
,
961
961
961
,
1024
1024
1024
are combined together into one large number by appending their digits in succession, forming the number
N
=
1491625
…
9611024
N = 1491625\ldots9611024
N
=
1491625
…
9611024
. How many digits does
N
N
N
have?
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84
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85
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86
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87
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88
<span class='latex-bold'>(A) </span>84\qquad<span class='latex-bold'>(B) </span>85\qquad<span class='latex-bold'>(C) </span>86\qquad<span class='latex-bold'>(D) </span>87\qquad<span class='latex-bold'>(E) </span>88
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84
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(
B
)
<
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85
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(
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86
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(
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<
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87
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(
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88
20
1
Hide problems
2019 Individual #20
Given that two real numbers
x
x
x
and
y
y
y
satisfy
x
2
−
6
x
y
+
9
y
2
+
∣
x
−
3
∣
=
0
x^2 - 6xy + 9y^2 + |x - 3| = 0
x
2
−
6
x
y
+
9
y
2
+
∣
x
−
3∣
=
0
, calculate
x
+
y
x + y
x
+
y
.
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)
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1
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(
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(
C
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4
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(
D
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16
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(
E
)
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impossible to determine
<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>4\qquad<span class='latex-bold'>(D) </span>16\qquad<span class='latex-bold'>(E) </span>\text{impossible to determine}
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(
A
)
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1
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p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
impossible to determine
19
1
Hide problems
2019 Individual #19
Let
S
(
n
)
S(n)
S
(
n
)
denote the sum of digits of an integer
n
n
n
(For example,
S
(
17
)
=
1
+
7
=
8
S(17) = 1 + 7 = 8
S
(
17
)
=
1
+
7
=
8
). If a positive two digit integer is randomly selected, what is the probability
S
(
S
(
n
)
)
≥
8
S(S(n)) \ge 8
S
(
S
(
n
))
≥
8
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
11
45
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
13
45
<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>\dfrac19\qquad<span class='latex-bold'>(C) </span>\dfrac29\qquad<span class='latex-bold'>(D) </span>\dfrac{11}{45}\qquad<span class='latex-bold'>(E) </span>\dfrac{13}{45}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
9
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
9
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
45
11
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
45
13
18
1
Hide problems
2019 Individual #18
A class of
10
10
10
children is divided into
5
5
5
pairs of partners. Each pair of partners sits next to each other and works together during class. One day, the teacher decides he wants to divide the class into two groups. In order to make sure the students work with new people, he makes sure not to put any student in the same group as his or her partner. How many different ways can he divide the class into these two groups?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
32
<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>5\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>16\qquad<span class='latex-bold'>(E) </span>32
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
32
16
1
Hide problems
2019 Individual #16
For some constant
b
b
b
, the graph of
y
=
x
2
+
b
2
+
2
b
x
−
b
+
2
y = x^2 + b^2 + 2bx - b + 2
y
=
x
2
+
b
2
+
2
b
x
−
b
+
2
has only one
x
x
x
intercept. What is the value of
b
b
b
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
10
<span class='latex-bold'>(A) </span>1\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>4\qquad<span class='latex-bold'>(D) </span>8\qquad<span class='latex-bold'>(E) </span>10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
10
15
1
Hide problems
2019 Individual #15
Alice, Bob, and Catherine decide to have a race. Alice runs at a speed of
3
3
3
feet per second, and Bob runs at a speed of
5
5
5
feet per second. In the end, Bob finishes the same amount of time before Catherine as Catherine finishes before Alice. What was Catherine's speed, in feet per second?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
15
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
17
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
9
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
impossible to determine
<span class='latex-bold'>(A) </span>\dfrac{15}4\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>\dfrac{17}4\qquad<span class='latex-bold'>(D) </span>\dfrac92\qquad<span class='latex-bold'>(E) </span>\text{impossible to determine}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
15
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
17
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
impossible to determine
14
1
Hide problems
2019 Individual #14
A cue ball is shot at a
45
45
45
degree angle from the upper right corner of a billiard table with dimensions
4
ft
4\text{ ft}
4
ft
by
5
ft
5\text{ ft}
5
ft
, as shown. How many times does the ball bounce before hitting another corner? Assume that when the ball bounces, its path is perfectly reflected. The final impact in the corner does not count as a bounce. [asy] size(120); draw((0,0)--(5,0)--(5,4)--(0,4)--cycle); label("
5
5
5
",(0,0)--(5,0),S); label("
4
4
4
",(0,0)--(0,4),W); filldraw(circle((0.4,3.6),0.4),black); draw((0,4)--(1.5,2.5),EndArrow); draw((1.5,2.5)--(4,0)--(5,1), dashed); draw(arc((0,4),1.25,315,270)); label(scale(0.8)*"
4
5
∘
45^\circ
4
5
∘
",(0.2,2.8),SE); [/asy]
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
7
<span class='latex-bold'>(A) </span>3\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>5\qquad<span class='latex-bold'>(D) </span>6\qquad<span class='latex-bold'>(E) </span>7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
7
13
1
Hide problems
2019 Individual #13
What is the remainder when
1
+
10
+
19
+
28
+
⋯
+
91
1 + 10 + 19 + 28 + \cdots + 91
1
+
10
+
19
+
28
+
⋯
+
91
is divided by
9
9
9
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
8
<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>2\qquad<span class='latex-bold'>(C) </span>3\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
8
12
1
Hide problems
2019 Individual #12
Calculate the product
1
3
×
2
4
×
3
5
×
⋯
×
18
20
×
19
21
\tfrac13 \times \tfrac24 \times \tfrac35 \times \cdots \times \tfrac{18}{20} \times \tfrac{19}{21}
3
1
×
4
2
×
5
3
×
⋯
×
20
18
×
21
19
.
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<span class='latex-bold'>(A) </span>\dfrac{1}{210}\qquad<span class='latex-bold'>(B) </span>\dfrac{1}{190}\qquad<span class='latex-bold'>(C) </span>\dfrac{1}{21}\qquad<span class='latex-bold'>(D) </span>\dfrac{1}{20}\qquad<span class='latex-bold'>(E) </span>\dfrac{1}{10}
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1
11
1
Hide problems
2019 Individual #11
An integer
N
N
N
which satisfies exactly three of the four following conditions is called two-good.
~
(I)
N
N
N
is divisible by
2
2
2
(II)
N
N
N
is divisible by
4
4
4
(III)
N
N
N
is divisible by
8
8
8
(IV)
N
N
N
is divisible by
16
16
16
~
How many integers between
1
1
1
and
100
100
100
, inclusive, are two-good?
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<span class='latex-bold'>(A) </span>6\qquad<span class='latex-bold'>(B) </span>7\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>9\qquad<span class='latex-bold'>(E) </span>10
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10
10
2
Hide problems
2019 Individual #10
John defines the function
f
(
x
)
=
(
x
−
3
)
(
x
−
9
)
+
8
f(x) = (x-3)(x-9) + 8
f
(
x
)
=
(
x
−
3
)
(
x
−
9
)
+
8
. What is the value of
f
(
3
)
f(3)
f
(
3
)
?
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<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>3\qquad<span class='latex-bold'>(C) </span>8\qquad<span class='latex-bold'>(D) </span>9\qquad<span class='latex-bold'>(E) </span>12
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12
2019 Team #10
40
40
40
people, numbered
1
1
1
through
40
40
40
counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person
1
1
1
, the first person eliminates the closest "alive" person to their right (so Person
1
1
1
eliminates Person
2
2
2
). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person
2
2
2
is eliminated, Person
3
3
3
eliminates Person
4
4
4
). This process continues until there is only
1
1
1
"alive" person remaining. What is the number of the last "alive" person? [asy] usepackage("cancel", "makeroom, thicklines"); usepackage("bm"); size(15cm); picture p; draw(p, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(p, "
"
+
s
t
r
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n
g
(
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−
i
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+
"
" + string(40 - i) + "
"
+
s
t
r
in
g
(
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−
i
)
+
"
", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(p, "
"
+
s
t
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n
g
(
i
+
1
)
+
"
" + string(i + 1) + "
"
+
s
t
r
in
g
(
i
+
1
)
+
"
", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(p, scale(2)*"
⋅
\cdot
⋅
", 6 *dir(180 / n * i)); } draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); add(shift(-20, 0) * p); draw((-11, 0)--(-8,0), EndArrow); picture q; draw(q, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { label(q, "
"
+
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−
i
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+
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" + string(40 - i) + "
"
+
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in
g
(
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−
i
)
+
"
", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i != 1) label(q, "
"
+
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t
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g
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+
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)
+
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" + string(i + 1) + "
"
+
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in
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i
+
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)
+
"
", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(q, scale(2)*"
⋅
\cdot
⋅
", 6 *dir(180 / n * i)); } draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 1; i+=2) { //label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(q); draw((9,0)--(12,0), EndArrow); picture r; draw(r, circle((0,0), 5)); for(int i = 0; i < 4; ++i) { if(i % 2 == 1) label(r, "
"
+
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n
g
(
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−
i
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+
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" + string(40 - i) + "
"
+
s
t
r
in
g
(
40
−
i
)
+
"
", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); if(i % 2 != 1) label(r, "
"
+
s
t
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i
n
g
(
i
+
1
)
+
"
" + string(i + 1) + "
"
+
s
t
r
in
g
(
i
+
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)
+
"
", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80)); } int n = 20; for(int i = 0; i <= n; ++i) { label(r, scale(2)*"
⋅
\cdot
⋅
", 6 *dir(180 / n * i)); } draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow); for(int i = 0; i < 4; i+=2) { label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100)); label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80)); } add(shift(20, 0) * r); [/asy] In the last step here, Person
39
39
39
eliminates Person
40
40
40
. Next turn, Person
1
1
1
eliminates the closest person to his right, Person
3
3
3
.
9
2
Hide problems
2019 Individual #9
Betsy is addicted to chocolate. Every day, she eats
2
2
2
chocolates at breakfast,
3
3
3
chocolates at lunch,
1
1
1
chocolate during her afternoon snack time, and
5
5
5
chocolates at dinner. If she begins eating a bag of
100
100
100
chocolates at breakfast one day, during which meal will she eat the last piece in the bag?
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<span class='latex-bold'>(A) </span>\text{breakfast}\qquad<span class='latex-bold'>(B) </span>\text{lunch}\qquad<span class='latex-bold'>(C) </span>\text{snack time}\qquad<span class='latex-bold'>(D) </span>\text{dinner}\qquad<span class='latex-bold'>(E) </span>\text{impossible to determine}
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impossible to determine
2019 Team #9
Kevin develops a method for shuffling a stack of
10
10
10
cards numbered
1
1
1
through
10
10
10
. He starts with the unshuffled pile, which is in perfect order with
1
1
1
at the top and
10
10
10
at the bottom. He takes the top card off the unshuffled pile and places it in what he calls the shuffled pile. Then, he flips a coin. If the coin is heads, he takes the card at the top of the unshuffled pile and places it at the top of the shuffled pile. If the coin comes up tails, he places the card at the at the bottom of the shuffled pile. He repeats this process for all the remaining cards. What is the probability that at the end of this shuffling, the top card is a prime number? Express your answer as a common fraction.
8
2
Hide problems
2019 Individual #8
James randomly selects
4
4
4
distinct numbers between
3
3
3
and
10
10
10
, inclusive. What is the probability that all
4
4
4
numbers are prime?
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28
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14
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4
<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>\dfrac1{28}\qquad<span class='latex-bold'>(C) </span>\dfrac1{14}\qquad<span class='latex-bold'>(D) </span>\dfrac17\qquad<span class='latex-bold'>(E) </span>\dfrac14
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2019 Team #8
Greg plays a game in which he is given three random
1
1
1
digit numbers, each between
0
0
0
and
9
9
9
, inclusive, with repeats allowed. He is to put these three numbers into any order. Exactly one ordering of the three numbers is correct, and if he guesses the correct ordering, he wins
$
150
\$150
$150
. What are Greg's expected winnings for this game, given that he randomly guesses one valid ordering when he plays?
7
2
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2019 Individual #7
In one peculiar family, the mother and the three children have exactly the same birthday. Currently, the mother is
37
37
37
years old while each of children are
9
9
9
years old. How old will the mother be when the sum of the ages of the three children equals her age?
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<span class='latex-bold'>(A) </span>14\qquad<span class='latex-bold'>(B) </span>27\qquad<span class='latex-bold'>(C) </span>42\qquad<span class='latex-bold'>(D) </span>57\qquad<span class='latex-bold'>(E) </span>66
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2019 Team #7
How many positive integers less than or equal to
150
150
150
have exactly three distinct prime factors?
6
2
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2019 Individual #6
How many rectangles are in the following figure? [asy] size(80); draw((0,0)--(3,0)--(3,4)--(0,4)--cycle); draw((0,2)--(3,2)); draw((0.75,2)--(0.75,0)); draw((2.25,2)--(2.25,0)); [/asy]
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<span class='latex-bold'>(A) </span>5\qquad<span class='latex-bold'>(B) </span>6\qquad<span class='latex-bold'>(C) </span>7\qquad<span class='latex-bold'>(D) </span>8\qquad<span class='latex-bold'>(E) </span>9
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2019 Team #6
Square
A
B
C
D
ABCD
A
BC
D
has side length
4
4
4
. Side
A
B
AB
A
B
is extended to point
E
E
E
so that
A
E
AE
A
E
has the same length as
A
C
AC
A
C
, as shown below. What is the length of
E
C
EC
EC
? Express your answer as a decimal to the nearest hundredth. [asy] size(80); defaultpen(fontsize(8pt)); pair EE = (4sqrt(2),0); pair A = (0,0); pair B = (4,0); pair C = (4,4); pair D = (0,4); draw(A--B--C--D--cycle); draw(A--EE); draw(C--EE,dotted); label("
A
A
A
",A,SW); label("
B
B
B
",B,S); label("
C
C
C
",C,N); label("
D
D
D
",D,N); label("
E
E
E
",EE,S); [/asy]
5
2
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2019 Individual #5
How many distinct prime factors does the number
36
36
36
have?
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<span class='latex-bold'>(A) </span>2\qquad<span class='latex-bold'>(B) </span>4\qquad<span class='latex-bold'>(C) </span>6\qquad<span class='latex-bold'>(D) </span>9\qquad<span class='latex-bold'>(E) </span>15
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2019 Team #5
3
3
3
builders are scheduled to build a house in
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60
60
days. However, they suffer from a bout of procrastination and thus do nothing for the first
50
50
50
days. Panicked, they realize in order to build the house on time, they must hire more workers and work twice as fast as they would have originally. If the new workers they hire also will work at the doubled rate, how many new workers will they need to hire? Assume each builder works at the same rate as the others and they do not get in each other's way.
4
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2019 Individual #4
Allen flips a fair two sided coin and rolls a fair
6
6
6
sided die with faces numbered
1
1
1
through
6
6
6
. What is the probability that the coin lands on heads and he rolls a number that is a multiple of
5
5
5
?
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<span class='latex-bold'>(A) </span>\dfrac1{24}\qquad<span class='latex-bold'>(B) </span>\dfrac1{12}\qquad<span class='latex-bold'>(C) </span>\dfrac16\qquad<span class='latex-bold'>(D) </span>\dfrac14\qquad<span class='latex-bold'>(E) </span>\dfrac13
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2019 Team #4
A
B
AB
A
B
is the diameter of circle
O
O
O
. A random point
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is selected on
O
O
O
so that
A
P
=
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AP = 4
A
P
=
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and
B
P
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3
BP = 3
BP
=
3
. Points
C
C
C
and
D
D
D
are drawn on circle
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O
O
so that
O
C
OC
OC
bisects
A
P
AP
A
P
and
O
D
OD
O
D
bisects
B
P
BP
BP
. What is the degree measure of
∠
C
O
D
\angle COD
∠
CO
D
?
3
2
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2019 Individual #3
Given that
2
x
+
5
−
3
x
+
7
=
8
2x + 5 - 3x + 7 = 8
2
x
+
5
−
3
x
+
7
=
8
, what is the value of
x
x
x
?
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<span class='latex-bold'>(A) </span>{-}4\qquad<span class='latex-bold'>(B) </span>{-}2\qquad<span class='latex-bold'>(C) </span>0\qquad<span class='latex-bold'>(D) </span>2\qquad<span class='latex-bold'>(E) </span>4
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2019 Team #3
Calculate
1
+
2
+
3
+
4
−
5
−
6
−
7
−
8
+
9
+
⋯
−
96
+
97
+
98
+
99
+
100
1+2+3+4-5-6-7-8+9+\cdots-96+97+98+99+100
1
+
2
+
3
+
4
−
5
−
6
−
7
−
8
+
9
+
⋯
−
96
+
97
+
98
+
99
+
100
2
2
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2019 Individual #2
On Monday, Lyndon receives a
80
80
80
on his daily math quiz. After being scolded by his parents, he works harder and gets an
83
83
83
on Tuesday. From Tuesday onward, his score improves by
3
3
3
points each day. What will be Lyndon's score that Friday?
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<span class='latex-bold'>(A) </span>89\qquad<span class='latex-bold'>(B) </span>92\qquad<span class='latex-bold'>(C) </span>95\qquad<span class='latex-bold'>(D) </span>98\qquad<span class='latex-bold'>(E) </span>100
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<
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95
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(
D
)
<
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98
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E
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100
2019 Team #2
A cup with a volume of
8
8
8
fluid ounces is filled at the rate of
0.5
0.5
0.5
ounces per second. However, a hole at the bottom of the cup also drains it at the rate of
0.3
0.3
0.3
ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup?
1
2
Hide problems
2019 Individual #1
Find
2
×
(
2
+
3
)
2 \times (2 + 3)
2
×
(
2
+
3
)
<
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(
A
)
<
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8
<
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x
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(
B
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<
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9
<
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(
C
)
<
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10
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(
D
)
<
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11
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E
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30
<span class='latex-bold'>(A) </span>8\qquad<span class='latex-bold'>(B) </span>9\qquad<span class='latex-bold'>(C) </span>10\qquad<span class='latex-bold'>(D) </span>11\qquad<span class='latex-bold'>(E) </span>30
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A
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8
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(
B
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<
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9
<
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p
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a
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x
−
b
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(
C
)
<
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10
<
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x
−
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′
>
(
D
)
<
/
s
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>
11
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E
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<
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30
2019 Team #1
An ant walks around on the coordinate plane. It moves from the origin to
(
3
,
4
)
(3,4)
(
3
,
4
)
, then to
(
−
9
,
9
)
(-9, 9)
(
−
9
,
9
)
, then back to the origin. How many units did it walk? Express your answer as a decimal rounded to the nearest tenth.