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Subcontests
(15)
8
2
Hide problems
Table from triangle
In triangle
A
B
C
ABC
A
BC
,
B
C
=
23
BC = 23
BC
=
23
,
C
A
=
27
CA = 27
C
A
=
27
, and
A
B
=
30
AB = 30
A
B
=
30
. Points
V
V
V
and
W
W
W
are on
A
C
‾
\overline{AC}
A
C
with
V
V
V
on
A
W
‾
\overline{AW}
A
W
, points
X
X
X
and
Y
Y
Y
are on
B
C
‾
\overline{BC}
BC
with
X
X
X
on
C
Y
‾
\overline{CY}
C
Y
, and points
Z
Z
Z
and
U
U
U
are on
A
B
‾
\overline{AB}
A
B
with
Z
Z
Z
on
B
U
‾
\overline{BU}
B
U
. In addition, the points are positioned so that
U
V
‾
∥
B
C
‾
\overline{UV} \parallel \overline{BC}
U
V
∥
BC
,
W
X
‾
∥
A
B
‾
\overline{WX} \parallel \overline{AB}
W
X
∥
A
B
, and
Y
Z
‾
∥
C
A
‾
\overline{YZ} \parallel \overline{CA}
Y
Z
∥
C
A
. Right angle folds are then made along
U
V
‾
\overline{UV}
U
V
,
W
X
‾
\overline{WX}
W
X
, and
Y
Z
‾
\overline{YZ}
Y
Z
. The resulting figure is placed on a level floor to make a table with triangular legs. Let
h
h
h
be the maximum possible height of a table constructed from triangle
A
B
C
ABC
A
BC
whose top is parallel to the floor. Then
h
h
h
can be written in the form
k
m
n
\tfrac{k \sqrt{m}}{n}
n
k
m
, where
k
k
k
and
n
n
n
are relatively prime positive integers and
m
m
m
is a positive integer that is not divisible by the square of any prime. Find
k
+
m
+
n
k + m + n
k
+
m
+
n
.[asy] unitsize(1 cm);pair translate; pair[] A, B, C, U, V, W, X, Y, Z;A[0] = (1.5,2.8); B[0] = (3.2,0); C[0] = (0,0); U[0] = (0.69*A[0] + 0.31*B[0]); V[0] = (0.69*A[0] + 0.31*C[0]); W[0] = (0.69*C[0] + 0.31*A[0]); X[0] = (0.69*C[0] + 0.31*B[0]); Y[0] = (0.69*B[0] + 0.31*C[0]); Z[0] = (0.69*B[0] + 0.31*A[0]);translate = (7,0); A[1] = (1.3,1.1) + translate; B[1] = (2.4,-0.7) + translate; C[1] = (0.6,-0.7) + translate; U[1] = U[0] + translate; V[1] = V[0] + translate; W[1] = W[0] + translate; X[1] = X[0] + translate; Y[1] = Y[0] + translate; Z[1] = Z[0] + translate;draw (A[0]--B[0]--C[0]--cycle); draw (U[0]--V[0],dashed); draw (W[0]--X[0],dashed); draw (Y[0]--Z[0],dashed); draw (U[1]--V[1]--W[1]--X[1]--Y[1]--Z[1]--cycle); draw (U[1]--A[1]--V[1],dashed); draw (W[1]--C[1]--X[1]); draw (Y[1]--B[1]--Z[1]);dot("
A
A
A
",A[0],N); dot("
B
B
B
",B[0],SE); dot("
C
C
C
",C[0],SW); dot("
U
U
U
",U[0],NE); dot("
V
V
V
",V[0],NW); dot("
W
W
W
",W[0],NW); dot("
X
X
X
",X[0],S); dot("
Y
Y
Y
",Y[0],S); dot("
Z
Z
Z
",Z[0],NE); dot(A[1]); dot(B[1]); dot(C[1]); dot("
U
U
U
",U[1],NE); dot("
V
V
V
",V[1],NW); dot("
W
W
W
",W[1],NW); dot("
X
X
X
",X[1],dir(-70)); dot("
Y
Y
Y
",Y[1],dir(250)); dot("
Z
Z
Z
",Z[1],NE); [/asy]
Complex roots of a polynomial [2011.II.8]
Let
z
1
,
z
2
,
z
3
,
…
,
z
12
z_1,z_2,z_3,\dots,z_{12}
z
1
,
z
2
,
z
3
,
…
,
z
12
be the 12 zeroes of the polynomial
z
12
−
2
36
z^{12}-2^{36}
z
12
−
2
36
. For each
j
j
j
, let
w
j
w_j
w
j
be one of
z
j
z_j
z
j
or
i
z
j
i z_j
i
z
j
. Then the maximum possible value of the real part of
∑
j
=
1
12
w
j
\displaystyle\sum_{j=1}^{12} w_j
j
=
1
∑
12
w
j
can be written as
m
+
n
m+\sqrt{n}
m
+
n
where
m
m
m
and
n
n
n
are positive integers. Find
m
+
n
m+n
m
+
n
.
14
2
Hide problems
Octagon midpoints
Let
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
be a regular octagon. Let
M
1
M_1
M
1
,
M
3
M_3
M
3
,
M
5
M_5
M
5
, and
M
7
M_7
M
7
be the midpoints of sides
A
1
A
2
‾
\overline{A_1 A_2}
A
1
A
2
,
A
3
A
4
‾
\overline{A_3 A_4}
A
3
A
4
,
A
5
A
6
‾
\overline{A_5 A_6}
A
5
A
6
, and
A
7
A
8
‾
\overline{A_7 A_8}
A
7
A
8
, respectively. For
i
=
1
,
3
,
5
,
7
i = 1, 3, 5, 7
i
=
1
,
3
,
5
,
7
, ray
R
i
R_i
R
i
is constructed from
M
i
M_i
M
i
towards the interior of the octagon such that
R
1
⊥
R
3
R_1 \perp R_3
R
1
⊥
R
3
,
R
3
⊥
R
5
R_3 \perp R_5
R
3
⊥
R
5
,
R
5
⊥
R
7
R_5 \perp R_7
R
5
⊥
R
7
, and
R
7
⊥
R
1
R_7 \perp R_1
R
7
⊥
R
1
. Pairs of rays
R
1
R_1
R
1
and
R
3
R_3
R
3
,
R
3
R_3
R
3
and
R
5
R_5
R
5
,
R
5
R_5
R
5
and
R
7
R_7
R
7
, and
R
7
R_7
R
7
and
R
1
R_1
R
1
meet at
B
1
B_1
B
1
,
B
3
B_3
B
3
,
B
5
B_5
B
5
,
B
7
B_7
B
7
respectively. If
B
1
B
3
=
A
1
A
2
B_1 B_3 = A_1 A_2
B
1
B
3
=
A
1
A
2
, then
cos
2
∠
A
3
M
3
B
1
\cos 2 \angle A_3 M_3 B_1
cos
2∠
A
3
M
3
B
1
can be written in the form
m
−
n
m - \sqrt{n}
m
−
n
, where
m
m
m
and
n
n
n
are positive integers. Find
m
+
n
m + n
m
+
n
.
Permutations [2011.II.14]
There are
N
N
N
permutations
(
a
1
,
a
2
,
…
,
a
30
)
(a_1,a_2,\dots,a_{30})
(
a
1
,
a
2
,
…
,
a
30
)
of
1
,
2
,
…
,
30
1,2,\dots,30
1
,
2
,
…
,
30
such that for
m
∈
{
2
,
3
,
5
}
m\in\{2,3,5\}
m
∈
{
2
,
3
,
5
}
,
m
m
m
divides
a
n
+
m
−
a
n
a_{n+m}-a_n
a
n
+
m
−
a
n
for all integers
n
n
n
with
1
≤
n
<
n
+
m
≤
30
1\leq n <n+m\leq 30
1
≤
n
<
n
+
m
≤
30
. Find the remainder when
N
N
N
is divided by 1000.
15
2
Hide problems
Cubic, Integer Solutions
For some integer
m
m
m
, the polynomial
x
3
−
2011
x
+
m
x^3-2011x+m
x
3
−
2011
x
+
m
has the three integer roots
a
a
a
,
b
b
b
, and
c
c
c
. Find
∣
a
∣
+
∣
b
∣
+
∣
c
∣
|a|+|b|+|c|
∣
a
∣
+
∣
b
∣
+
∣
c
∣
.
Polynomial Probability [2011.II.15]
Let
P
(
x
)
=
x
2
−
3
x
−
9
P(x)=x^2-3x-9
P
(
x
)
=
x
2
−
3
x
−
9
. A real number
x
x
x
is chosen at random from the interval
5
≤
x
≤
15
5\leq x \leq 15
5
≤
x
≤
15
. The probability that
⌊
P
(
x
)
⌋
=
P
(
⌊
x
⌋
)
\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}
⌊
P
(
x
)
⌋
=
P
(⌊
x
⌋)
is equal to
a
+
b
+
c
−
d
e
\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}
e
a
+
b
+
c
−
d
, where
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
and
e
e
e
are positive integers and none of
a
,
b
,
a,b,
a
,
b
,
or
c
c
c
is divisible by the square of a prime. Find
a
+
b
+
c
+
d
+
e
a+b+c+d+e
a
+
b
+
c
+
d
+
e
.
13
2
Hide problems
Suspended cube
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled
A
A
A
. The three vertices adjacent to vertex
A
A
A
are at heights 10, 11, and 12 above the plane. The distance from vertex
A
A
A
to the plane can be expressed as
r
−
s
t
\tfrac{r-\sqrt{s}}{t}
t
r
−
s
, where
r
r
r
,
s
s
s
, and
t
t
t
are positive integers, and
r
+
s
+
t
<
1000
r+s+t<1000
r
+
s
+
t
<
1000
. Find
r
+
s
+
t
r+s+t
r
+
s
+
t
.
Points in a square [2011.II.13]
Point
P
P
P
lies on the diagonal
A
C
AC
A
C
of square
A
B
C
D
ABCD
A
BC
D
with
A
P
>
C
P
AP>CP
A
P
>
CP
. Let
O
1
O_1
O
1
and
O
2
O_2
O
2
be the circumcenters of triangles
A
B
P
ABP
A
BP
and
C
D
P
CDP
C
D
P
respectively. Given that
A
B
=
12
AB=12
A
B
=
12
and
∠
O
1
P
O
2
=
12
0
∘
\angle O_1 P O_2 = 120^\circ
∠
O
1
P
O
2
=
12
0
∘
, then
A
P
=
a
+
b
AP=\sqrt{a}+\sqrt{b}
A
P
=
a
+
b
where
a
a
a
and
b
b
b
are positive integers. Find
a
+
b
a+b
a
+
b
.
12
2
Hide problems
Some Men, Many Women
Six men and some number of women stand in a line in random order. Let
p
p
p
be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that
p
p
p
does not exceed 1 percent.
Delegates around table [2011.II.12]
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
.
11
2
Hide problems
Remainders when divided by 1000
Let
R
R
R
be the set of all possible remainders when a number of the form
2
n
2^n
2
n
,
n
n
n
a nonnegative integer, is divided by
1000
1000
1000
. Let
S
S
S
be the sum of all elements in
R
R
R
. Find the remainder when
S
S
S
is divided by
1000
1000
1000
.
Determinants of a Matrix [2011.II.11]
Let
M
n
M_n
M
n
be the
n
×
n
n\times n
n
×
n
matrix with entries as follows: for
1
≤
i
≤
n
1\leq i \leq n
1
≤
i
≤
n
,
m
i
,
i
=
10
m_{i,i}=10
m
i
,
i
=
10
; for
1
≤
i
≤
n
−
1
,
m
i
+
1
,
i
=
m
i
,
i
+
1
=
3
1\leq i \leq n-1, m_{i+1,i}=m_{i,i+1}=3
1
≤
i
≤
n
−
1
,
m
i
+
1
,
i
=
m
i
,
i
+
1
=
3
; all other entries in
M
n
M_n
M
n
are zero. Let
D
n
D_n
D
n
be the determinant of matrix
M
n
M_n
M
n
. Then
∑
n
=
1
∞
1
8
D
n
+
1
\displaystyle \sum_{n=1}^{\infty} \dfrac{1}{8D_n+1}
n
=
1
∑
∞
8
D
n
+
1
1
can be represented as
p
q
\frac{p}{q}
q
p
, where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p+q
p
+
q
.Note: The determinant of the
1
×
1
1\times 1
1
×
1
matrix
[
a
]
[a]
[
a
]
is
a
a
a
, and the determinant of the
2
×
2
2\times 2
2
×
2
matrix
[
a
b
c
d
]
=
a
d
−
b
c
\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]=ad-bc
[
a
c
b
d
]
=
a
d
−
b
c
; for
n
≥
2
n\geq 2
n
≥
2
, the determinant of an
n
×
n
n\times n
n
×
n
matrix with first row or first column
a
1
a
2
a
3
…
a
n
a_1\ a_2\ a_3 \dots\ a_n
a
1
a
2
a
3
…
a
n
is equal to
a
1
C
1
−
a
2
C
2
+
a
3
C
3
−
⋯
+
(
−
1
)
n
+
1
a
n
C
n
a_1C_1 - a_2C_2 + a_3C_3 - \dots + (-1)^{n+1} a_nC_n
a
1
C
1
−
a
2
C
2
+
a
3
C
3
−
⋯
+
(
−
1
)
n
+
1
a
n
C
n
, where
C
i
C_i
C
i
is the determinant of the
(
n
−
1
)
×
(
n
−
1
)
(n-1)\times (n-1)
(
n
−
1
)
×
(
n
−
1
)
matrix found by eliminating the row and column containing
a
i
a_i
a
i
.
10
2
Hide problems
Obtuse triangle from n-gon
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular
n
n
n
-gon determine an obtuse triangle is
93
125
\tfrac{93}{125}
125
93
. Find the sum of all possible values of
n
n
n
.
Chords in a Circle [2011.II.10]
A circle with center
O
O
O
has radius 25. Chord
A
B
‾
\overline{AB}
A
B
of length 30 and chord
C
D
‾
\overline{CD}
C
D
of length 14 intersect at point
P
P
P
. The distance between the midpoints of the two chords is 12. The quantity
O
P
2
OP^2
O
P
2
can be represented as
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find the remainder where
m
+
n
m+n
m
+
n
is divided by 1000.
9
2
Hide problems
Trig equation
Suppose
x
x
x
is in the interval
[
0
,
π
/
2
]
[0,\pi/2]
[
0
,
π
/2
]
and
log
24
sin
x
(
24
cos
x
)
=
3
2
\log_{24\sin{x}}(24\cos{x})=\frac{3}{2}
lo
g
24
s
i
n
x
(
24
cos
x
)
=
2
3
. Find
24
cot
2
x
24\cot^2{x}
24
cot
2
x
.
Six nonnegative numbers [2011.II.9]
Let
x
1
,
x
2
,
…
,
x
6
x_1,x_2,\dots ,x_6
x
1
,
x
2
,
…
,
x
6
be nonnegative real numbers such that
x
1
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
=
1
x_1+x_2+x_3+x_4+x_5+x_6=1
x
1
+
x
2
+
x
3
+
x
4
+
x
5
+
x
6
=
1
, and
x
1
x
3
x
5
+
x
2
x
4
x
6
≥
1
540
x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}
x
1
x
3
x
5
+
x
2
x
4
x
6
≥
540
1
. Let
p
p
p
and
q
q
q
be positive relatively prime integers such that
p
q
\frac{p}{q}
q
p
is the maximum possible value of
x
1
x
2
x
3
+
x
2
x
3
x
4
+
x
3
x
4
x
5
+
x
4
x
5
x
6
+
x
5
x
6
x
1
+
x
6
x
1
x
2
x_1x_2x_3+x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2
x
1
x
2
x
3
+
x
2
x
3
x
4
+
x
3
x
4
x
5
+
x
4
x
5
x
6
+
x
5
x
6
x
1
+
x
6
x
1
x
2
. Find
p
+
q
p+q
p
+
q
.
7
2
Hide problems
Nonnegative integer solutions
Find the number of positive integers
m
m
m
for which there exist nonnegative integers
x
0
,
x
1
,
…
,
x
2011
x_0,x_1,\ldots,x_{2011}
x
0
,
x
1
,
…
,
x
2011
such that
m
x
0
=
∑
k
=
1
2011
m
x
k
.
m^{x_0}=\sum_{k=1}^{2011}m^{x_k}.
m
x
0
=
k
=
1
∑
2011
m
x
k
.
Green and red marbles [2011.II.7]
Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is GGRRRGGRG. Let
m
m
m
be the maximum number of red marbles for which Ed can make such an arrangement, and let
N
N
N
be the number of ways in which Ed can arrange the
m
+
5
m+5
m
+
5
marbles to satisfy the requirement. Find the remainder when
N
N
N
is divided by 1000.
6
2
Hide problems
Parabola
Suppose that a parabola has vertex
(
1
4
,
−
9
8
)
\left(\tfrac{1}{4},-\tfrac{9}{8}\right)
(
4
1
,
−
8
9
)
, and equation
y
=
a
x
2
+
b
x
+
c
y=ax^2+bx+c
y
=
a
x
2
+
b
x
+
c
, where
a
>
0
a>0
a
>
0
and
a
+
b
+
c
a+b+c
a
+
b
+
c
is an integer. The minimum possible value of
a
a
a
can be written as
p
q
,
\tfrac{p}{q},
q
p
,
where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p+q
p
+
q
.
Interesting ordered quadruples
Define an ordered quadruple of integers
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
as interesting if
1
≤
a
<
b
<
c
<
d
≤
10
1 \le a<b<c<d \le 10
1
≤
a
<
b
<
c
<
d
≤
10
, and
a
+
d
>
b
+
c
a+d>b+c
a
+
d
>
b
+
c
. How many ordered quadruples are there?
5
2
Hide problems
Arrangements of vertices of nonagon
The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits
1
1
1
through
9
9
9
in such a way that the sum of the numbers on every three consecutive vertices is a multiple of
3
3
3
. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.
Sum of a geometric series [2011.II.5]
The sum of the first 2011 terms of a geometric series is 200. The sum of the first 4022 terms of the same series is 380. Find the sum of the first 6033 terms of the series.
4
2
Hide problems
Triangle and angle bisectors
In triangle
A
B
C
ABC
A
BC
,
A
B
=
125
,
A
C
=
117
AB=125,AC=117
A
B
=
125
,
A
C
=
117
, and
B
C
=
120
BC=120
BC
=
120
. The angle bisector of angle
A
A
A
intersects
B
C
‾
\overline{BC}
BC
at point
L
L
L
, and the angle bisector of angle
B
B
B
intersects
A
C
‾
\overline{AC}
A
C
at point
K
K
K
. Let
M
M
M
and
N
N
N
be the feet of the perpendiculars from
C
C
C
to
B
K
‾
\overline{BK}
B
K
and
A
L
‾
\overline{AL}
A
L
, respectively. Find
M
N
MN
MN
.
Angle bisectors [2011.II.4]
In triangle
A
B
C
ABC
A
BC
,
A
B
=
20
11
A
C
AB=\frac{20}{11} AC
A
B
=
11
20
A
C
. The angle bisector of
∠
A
\angle A
∠
A
intersects
B
C
BC
BC
at point
D
D
D
, and point
M
M
M
is the midpoint of
A
D
AD
A
D
. Let
P
P
P
be the point of the intersection of
A
C
AC
A
C
and
B
M
BM
BM
. The ratio of
C
P
CP
CP
to
P
A
PA
P
A
can be expressed in the form
m
n
\dfrac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Find
m
+
n
m+n
m
+
n
.
3
2
Hide problems
Changed axes in coordinate system
Let
L
L
L
be the line with slope
5
12
\tfrac{5}{12}
12
5
that contains the point
A
=
(
24
,
−
1
)
A=(24,-1)
A
=
(
24
,
−
1
)
, and let
M
M
M
be the line perpendicular to line
L
L
L
that contains the point
B
=
(
5
,
6
)
B=(5,6)
B
=
(
5
,
6
)
. The original coordinate axes are erased, and line
L
L
L
is made the
x
x
x
-axis, and line
M
M
M
the
y
y
y
-axis. In the new coordinate system, point
A
A
A
is on the positive
x
x
x
-axis, and point
B
B
B
is on the positive
y
y
y
-axis. The point
P
P
P
with coordinates
(
−
14
,
27
)
(-14,27)
(
−
14
,
27
)
in the original system has coordinates
(
α
,
β
)
(\alpha,\beta)
(
α
,
β
)
in the new coordinate system. Find
α
+
β
\alpha+\beta
α
+
β
.
Angles in an arithmetic sequence [2011.II.3]
The degree measures of the angles of a convex 18-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.
2
2
Hide problems
Rectangle and parallel lines
In rectangle
A
B
C
D
ABCD
A
BC
D
,
A
B
=
12
AB=12
A
B
=
12
and
B
C
=
10
BC=10
BC
=
10
. Points
E
E
E
and
F
F
F
lie inside rectangle
A
B
C
D
ABCD
A
BC
D
so that
B
E
=
9
BE=9
BE
=
9
,
D
F
=
8
DF=8
D
F
=
8
,
B
E
‾
∥
D
F
‾
\overline{BE} \parallel \overline{DF}
BE
∥
D
F
,
E
F
‾
∥
A
B
‾
\overline{EF} \parallel \overline{AB}
EF
∥
A
B
, and line
B
E
BE
BE
intersects segment
A
D
‾
\overline{AD}
A
D
. The length
E
F
EF
EF
can be expressed in the form
m
n
−
p
m\sqrt{n}-p
m
n
−
p
, where
m
,
n
,
m,n,
m
,
n
,
and
p
p
p
are positive integers and
n
n
n
is not divisible by the square of any prime. Find
m
+
n
+
p
m+n+p
m
+
n
+
p
.
A square. [2011.II.2]
On square
A
B
C
D
ABCD
A
BC
D
, point
E
E
E
lies on side
A
D
‾
\overline{AD}
A
D
and point
F
F
F
lies on side
B
C
‾
\overline{BC}
BC
, so that
B
E
=
E
F
=
F
D
=
30
BE=EF=FD=30
BE
=
EF
=
F
D
=
30
. Find the area of square
A
B
C
D
ABCD
A
BC
D
.
1
2
Hide problems
Large beverage [2011.II.1]
Gary purchased a large beverage, but drank only
m
/
n
m/n
m
/
n
of this beverage, where
m
m
m
and
n
n
n
are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only
2
9
\frac{2}{9}
9
2
as much beverage. Find
m
+
n
m+n
m
+
n
.
Jars of Acid Solution
Jar A contains four liters of a solution that is
45
%
45\%
45%
acid. Jar B contains five liters of a solution that is
48
%
48\%
48%
acid. Jar C contains one liter of a solution that is
k
%
k\%
k
%
acid. From jar C,
m
n
\tfrac{m}{n}
n
m
liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end, both jar A and jar B contain solutions that are
50
%
50\%
50%
acid. Given that
m
m
m
and
n
n
n
are relatively prime positive integers, find
k
+
m
+
n
k+m+n
k
+
m
+
n
.