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Subcontests
(15)
15
1
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Mass of Equations
Determine
w
2
+
x
2
+
y
2
+
z
2
w^2+x^2+y^2+z^2
w
2
+
x
2
+
y
2
+
z
2
if
x
2
2
2
−
1
+
y
2
2
2
−
3
2
+
z
2
2
2
−
5
2
+
w
2
2
2
−
7
2
=
1
x
2
4
2
−
1
+
y
2
4
2
−
3
2
+
z
2
4
2
−
5
2
+
w
2
4
2
−
7
2
=
1
x
2
6
2
−
1
+
y
2
6
2
−
3
2
+
z
2
6
2
−
5
2
+
w
2
6
2
−
7
2
=
1
x
2
8
2
−
1
+
y
2
8
2
−
3
2
+
z
2
8
2
−
5
2
+
w
2
8
2
−
7
2
=
1
\begin{array}{l} \displaystyle \frac{x^2}{2^2-1}+\frac{y^2}{2^2-3^2}+\frac{z^2}{2^2-5^2}+\frac{w^2}{2^2-7^2}=1 \\ \displaystyle \frac{x^2}{4^2-1}+\frac{y^2}{4^2-3^2}+\frac{z^2}{4^2-5^2}+\frac{w^2}{4^2-7^2}=1 \\ \displaystyle \frac{x^2}{6^2-1}+\frac{y^2}{6^2-3^2}+\frac{z^2}{6^2-5^2}+\frac{w^2}{6^2-7^2}=1 \\ \displaystyle \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 \\ \end{array}
2
2
−
1
x
2
+
2
2
−
3
2
y
2
+
2
2
−
5
2
z
2
+
2
2
−
7
2
w
2
=
1
4
2
−
1
x
2
+
4
2
−
3
2
y
2
+
4
2
−
5
2
z
2
+
4
2
−
7
2
w
2
=
1
6
2
−
1
x
2
+
6
2
−
3
2
y
2
+
6
2
−
5
2
z
2
+
6
2
−
7
2
w
2
=
1
8
2
−
1
x
2
+
8
2
−
3
2
y
2
+
8
2
−
5
2
z
2
+
8
2
−
7
2
w
2
=
1
14
1
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Hunting for Numbers
What is the largest even integer that cannot be written as the sum of two odd composite numbers?
13
1
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Trig Computation
Find the value of
10
cot
(
cot
−
1
3
+
cot
−
1
7
+
cot
−
1
13
+
cot
−
1
21
)
10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21)
10
cot
(
cot
−
1
3
+
cot
−
1
7
+
cot
−
1
13
+
cot
−
1
21
)
.
12
1
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Yet Another Function
A function
f
f
f
is defined for all real numbers and satisfies
f
(
2
+
x
)
=
f
(
2
−
x
)
and
f
(
7
+
x
)
=
f
(
7
−
x
)
f(2 + x) = f(2 - x)\qquad\text{and}\qquad f(7 + x) = f(7 - x)
f
(
2
+
x
)
=
f
(
2
−
x
)
and
f
(
7
+
x
)
=
f
(
7
−
x
)
for all real
x
x
x
. If
x
=
0
x = 0
x
=
0
is a root of
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
, what is the least number of roots
f
(
x
)
=
0
f(x) = 0
f
(
x
)
=
0
must have in the interval
−
1000
≤
x
≤
1000
-1000 \le x \le 1000
−
1000
≤
x
≤
1000
?
11
1
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Plant Probability
A gardener plants three maple trees, four oak trees, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let
m
n
\frac{m}{n}
n
m
in lowest terms be the probability that no two birch trees are next to one another. Find
m
+
n
m + n
m
+
n
.
10
1
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Playing with ASHME Scores
Mary told John her score on the American High School Mathematics Examination (AHSME), which was over 80. From this, John was able to determine the number of problems Mary solved correctly. If Mary's score had been any lower, but still over 80, John could not have determined this. What was Mary's score? (Recall that the AHSME consists of 30 multiple-choice problems and that one's score,
s
s
s
, is computed by the formula
s
=
30
+
4
c
−
w
s = 30 + 4c - w
s
=
30
+
4
c
−
w
, where
c
c
c
is the number of correct and
w
w
w
is the number of wrong answers; students are not penalized for problems left unanswered.)
9
1
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A Tetrahedron ABCD
In tetrahedron
A
B
C
D
ABCD
A
BC
D
, edge
A
B
AB
A
B
has length 3 cm. The area of face
A
B
C
ABC
A
BC
is 15
cm
2
\text{cm}^2
cm
2
and the area of face
A
B
D
ABD
A
B
D
is 12
cm
2
\text{cm}^2
cm
2
. These two faces meet each other at a
3
0
∘
30^\circ
3
0
∘
angle. Find the volume of the tetrahedron in
cm
3
\text{cm}^3
cm
3
.
8
1
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Complex Roots
The equation
z
6
+
z
3
+
1
z^6 + z^3 + 1
z
6
+
z
3
+
1
has one complex root with argument
θ
\theta
θ
between
9
0
∘
90^\circ
9
0
∘
and
18
0
∘
180^\circ
18
0
∘
in the complex plane. Determine the degree measure of
θ
\theta
θ
.
7
1
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Another Function
The function
f
f
f
is defined on the set of integers and satisfies
f
(
n
)
=
{
n
−
3
if
n
≥
1000
f
(
f
(
n
+
5
)
)
if
n
<
1000
f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases}
f
(
n
)
=
{
n
−
3
f
(
f
(
n
+
5
))
if
n
≥
1000
if
n
<
1000
Find
f
(
84
)
f(84)
f
(
84
)
.
6
1
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Splitting Circle Areas
Three circles, each of radius 3, are drawn with centers at
(
14
,
92
)
(14,92)
(
14
,
92
)
,
(
17
,
76
)
(17,76)
(
17
,
76
)
, and
(
19
,
84
)
(19,84)
(
19
,
84
)
. A line passing through
(
17
,
76
)
(17,76)
(
17
,
76
)
is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?
5
1
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Easy Log
Determine the value of
a
b
ab
ab
if
log
8
a
+
log
4
b
2
=
5
\log_8 a + \log_4 b^2 = 5
lo
g
8
a
+
lo
g
4
b
2
=
5
and
log
8
b
+
log
4
a
2
=
7
\log_8 b + \log_4 a^2 = 7
lo
g
8
b
+
lo
g
4
a
2
=
7
.
4
1
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Arithmetic Means
Let
S
S
S
be a list of positive integers - not necessarily distinct - in which the number 68 appears. The average (arithmetic mean) of the numbers in
S
S
S
is 56. However, if 68 is removed, the average of the remaining numbers drops to 55. What is the largest number that can appear in
S
S
S
?
3
1
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Fun Area Probem
A point
P
P
P
is chosen in the interior of
△
A
B
C
\triangle ABC
△
A
BC
so that when lines are drawn through
P
P
P
parallel to the sides of
△
A
B
C
\triangle ABC
△
A
BC
, the resulting smaller triangles,
t
1
t_1
t
1
,
t
2
t_2
t
2
, and
t
3
t_3
t
3
in the figure, have areas 4, 9, and 49, respectively. Find the area of
△
A
B
C
\triangle ABC
△
A
BC
. [asy] size(200); pathpen=black+linewidth(0.65);pointpen=black; pair A=(0,0),B=(12,0),C=(4,5); D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12); MP("A",C,N);MP("B",A,SW);MP("C",B,SE); /* sorry mixed up points according to resources diagram. */ MP("t_3",(A+B+(B-A)*3/4+(A-B)*5/6)/2+(-1,0.8),N); MP("t_2",(B+C+(B-C)*5/12+(C-B)*5/6)/2+(-0.3,0.1),WSW); MP("t_1",(A+C+(C-A)*3/4+(A-C)*5/12)/2+(0,0.15),ESE);[/asy]
2
1
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Special Multiple of 15
The integer
n
n
n
is the smallest positive multiple of 15 such that every digit of
n
n
n
is either 8 or 0. Compute
n
15
\frac{n}{15}
15
n
.
1
1
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Arithmetic Progression
Find the value of
a
2
+
a
4
+
a
6
+
⋯
+
a
98
a_2 + a_4 + a_6 + \dots + a_{98}
a
2
+
a
4
+
a
6
+
⋯
+
a
98
if
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
,
…
\dots
…
is an arithmetic progression with common difference 1, and
a
1
+
a
2
+
a
3
+
⋯
+
a
98
=
137
a_1 + a_2 + a_3 + \dots + a_{98} = 137
a
1
+
a
2
+
a
3
+
⋯
+
a
98
=
137
.