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Subcontests
(10)
10
2
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2015 Algebra #10: Contrived System of Equations
Find all ordered 4-tuples of integers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
(not necessarily distinct) satisfying the following system of equations: \begin{align*}a^2-b^2-c^2-d^2&=c-b-2\\2ab&=a-d-32\\2ac&=28-a-d\\2ad&=b+c+31.\end{align*}
2015 Geometry #10
Let
G
\mathcal{G}
G
be the set of all points
(
x
,
y
)
(x,y)
(
x
,
y
)
in the Cartesian plane such that
0
≤
y
≤
8
0\le y\le 8
0
≤
y
≤
8
and
(
x
−
3
)
2
+
31
=
(
y
−
4
)
2
+
8
y
(
8
−
y
)
.
(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.
(
x
−
3
)
2
+
31
=
(
y
−
4
)
2
+
8
y
(
8
−
y
)
.
There exists a unique line
ℓ
\ell
ℓ
of negative slope tangent to
G
\mathcal{G}
G
and passing through the point
(
0
,
4
)
(0,4)
(
0
,
4
)
. Suppose
ℓ
\ell
ℓ
is tangent to
G
\mathcal{G}
G
at a unique point
P
P
P
. Find the coordinates
(
α
,
β
)
(\alpha, \beta)
(
α
,
β
)
of
P
P
P
.
9
2
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2015 Algebra #9: Divisible by Large Power
Let
N
=
3
0
2015
N=30^{2015}
N
=
3
0
2015
. Find the number of ordered 4-tuples of integers
(
A
,
B
,
C
,
D
)
∈
{
1
,
2
,
…
,
N
}
4
(A,B,C,D)\in\{1,2,\ldots,N\}^4
(
A
,
B
,
C
,
D
)
∈
{
1
,
2
,
…
,
N
}
4
(not necessarily distinct) such that for every integer
n
n
n
,
A
n
3
+
B
n
2
+
2
C
n
+
D
An^3+Bn^2+2Cn+D
A
n
3
+
B
n
2
+
2
C
n
+
D
is divisible by
N
N
N
.
2015 Geometry #9
Let
A
B
C
D
ABCD
A
BC
D
be a regular tetrahedron with side length
1
1
1
. Let
X
X
X
be the point in the triangle
B
C
D
BCD
BC
D
such that
[
X
B
C
]
=
2
[
X
B
D
]
=
4
[
X
C
D
]
[XBC]=2[XBD]=4[XCD]
[
XBC
]
=
2
[
XB
D
]
=
4
[
XC
D
]
, where
[
ω
‾
]
[\overline{\omega}]
[
ω
]
denotes the area of figure
ω
‾
\overline{\omega}
ω
. Let
Y
Y
Y
lie on segment
A
X
AX
A
X
such that
2
A
Y
=
Y
X
2AY=YX
2
A
Y
=
Y
X
. Let
M
M
M
be the midpoint of
B
D
BD
B
D
. Let
Z
Z
Z
be a point on segment
A
M
AM
A
M
such that the lines
Y
Z
YZ
Y
Z
and
B
C
BC
BC
intersect at some point. Find
A
Z
Z
M
\frac{AZ}{ZM}
ZM
A
Z
.
8
2
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2015 Algebra #8: Double Quadratic Residues
Find the number of ordered pairs of integers
(
a
,
b
)
∈
{
1
,
2
,
…
,
35
}
2
(a,b)\in\{1,2,\ldots,35\}^2
(
a
,
b
)
∈
{
1
,
2
,
…
,
35
}
2
(not necessarily distinct) such that
a
x
+
b
ax+b
a
x
+
b
is a "quadratic residue modulo
x
2
+
1
x^2+1
x
2
+
1
and
35
35
35
", i.e. there exists a polynomial
f
(
x
)
f(x)
f
(
x
)
with integer coefficients such that either of the following
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
e
q
u
i
v
a
l
e
n
t
<
/
s
p
a
n
>
<span class='latex-italic'>equivalent</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
e
q
u
i
v
a
l
e
n
t
<
/
s
p
an
>
conditions holds: [*] there exist polynomials
P
P
P
,
Q
Q
Q
with integer coefficients such that
f
(
x
)
2
−
(
a
x
+
b
)
=
(
x
2
+
1
)
P
(
x
)
+
35
Q
(
x
)
f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)
f
(
x
)
2
−
(
a
x
+
b
)
=
(
x
2
+
1
)
P
(
x
)
+
35
Q
(
x
)
; [*] or more conceptually, the remainder when (the polynomial)
f
(
x
)
2
−
(
a
x
+
b
)
f(x)^2-(ax+b)
f
(
x
)
2
−
(
a
x
+
b
)
is divided by (the polynomial)
x
2
+
1
x^2+1
x
2
+
1
is a polynomial with integer coefficients all divisible by
35
35
35
.
2015 Geometry #8
Let
S
S
S
be the set of discs
D
D
D
contained completely in the set
{
(
x
,
y
)
:
y
<
0
}
\{ (x,y) : y<0\}
{(
x
,
y
)
:
y
<
0
}
(the region below the
x
x
x
-axis) and centered (at some point) on the curve
y
=
x
2
−
3
4
y=x^2-\frac{3}{4}
y
=
x
2
−
4
3
. What is the area of the union of the elements of
S
S
S
?
7
2
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2015 Algebra #7: Cosine Interpolation
Suppose
(
a
1
,
a
2
,
a
3
,
a
4
)
(a_1,a_2,a_3,a_4)
(
a
1
,
a
2
,
a
3
,
a
4
)
is a 4-term sequence of real numbers satisfying the following two conditions:[*]
a
3
=
a
2
+
a
1
a_3=a_2+a_1
a
3
=
a
2
+
a
1
and
a
4
=
a
3
+
a
2
a_4=a_3+a_2
a
4
=
a
3
+
a
2
; [*] there exist real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
n
2
+
b
n
+
c
=
cos
(
a
n
)
an^2+bn+c=\cos(a_n)
a
n
2
+
bn
+
c
=
cos
(
a
n
)
for all
n
∈
{
1
,
2
,
3
,
4
}
n\in\{1,2,3,4\}
n
∈
{
1
,
2
,
3
,
4
}
.Compute the maximum possible value of
cos
(
a
1
)
−
cos
(
a
4
)
\cos(a_1)-\cos(a_4)
cos
(
a
1
)
−
cos
(
a
4
)
over all such sequences
(
a
1
,
a
2
,
a
3
,
a
4
)
(a_1,a_2,a_3,a_4)
(
a
1
,
a
2
,
a
3
,
a
4
)
.
2015 Geometry #7
Let
A
B
C
D
ABCD
A
BC
D
be a square pyramid of height
1
2
\frac{1}{2}
2
1
with square base
A
B
C
D
ABCD
A
BC
D
of side length
A
B
=
12
AB=12
A
B
=
12
(so
E
E
E
is the vertex of the pyramid, and the foot of the altitude from
E
E
E
to
A
B
C
D
ABCD
A
BC
D
is the center of square
A
B
C
D
ABCD
A
BC
D
). The faces
A
D
E
ADE
A
D
E
and
C
D
E
CDE
C
D
E
meet at an acute angle of measure
α
\alpha
α
(so that
0
∘
<
α
<
9
0
∘
0^{\circ}<\alpha<90^{\circ}
0
∘
<
α
<
9
0
∘
). Find
tan
α
\tan \alpha
tan
α
.
6
2
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2015 Algebra #6: Another Maximization Problem
Let
a
,
b
,
c
,
d
,
e
a,b,c,d,e
a
,
b
,
c
,
d
,
e
be nonnegative integers such that
625
a
+
250
b
+
100
c
+
40
d
+
16
e
=
1
5
3
625a+250b+100c+40d+16e=15^3
625
a
+
250
b
+
100
c
+
40
d
+
16
e
=
1
5
3
. What is the maximum possible value of
a
+
b
+
c
+
d
+
e
a+b+c+d+e
a
+
b
+
c
+
d
+
e
?
2015 Geometry #6
In triangle
A
B
C
ABC
A
BC
,
A
B
=
2
AB=2
A
B
=
2
,
A
C
=
1
+
5
AC=1+\sqrt{5}
A
C
=
1
+
5
, and
∠
C
A
B
=
5
4
∘
\angle CAB=54^{\circ}
∠
C
A
B
=
5
4
∘
. Suppose
D
D
D
lies on the extension of
A
C
AC
A
C
through
C
C
C
such that
C
D
=
5
−
1
CD=\sqrt{5}-1
C
D
=
5
−
1
. If
M
M
M
is the midpoint of
B
D
BD
B
D
, determine the measure of
∠
A
C
M
\angle ACM
∠
A
CM
, in degrees.
5
2
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2015 Algebra #5: Maximum of Minimum of Expression
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that
a
+
b
+
c
=
10
a+b+c=10
a
+
b
+
c
=
10
and
a
b
+
b
c
+
c
a
=
25
ab+bc+ca=25
ab
+
b
c
+
c
a
=
25
. Let
m
=
min
{
a
b
,
b
c
,
c
a
}
m=\min\{ab,bc,ca\}
m
=
min
{
ab
,
b
c
,
c
a
}
. Find the largest possible value of
m
m
m
.
2015 Geometry #5
Let
I
I
I
be the set of points
(
x
,
y
)
(x,y)
(
x
,
y
)
in the Cartesian plane such that
x
>
(
y
4
9
+
2015
)
1
/
4
x>\left(\frac{y^4}{9}+2015\right)^{1/4}
x
>
(
9
y
4
+
2015
)
1/4
Let
f
(
r
)
f(r)
f
(
r
)
denote the area of the intersection of
I
I
I
and the disk
x
2
+
y
2
≤
r
2
x^2+y^2\le r^2
x
2
+
y
2
≤
r
2
of radius
r
>
0
r>0
r
>
0
centered at the origin
(
0
,
0
)
(0,0)
(
0
,
0
)
. Determine the minimum possible real number
L
L
L
such that
f
(
r
)
<
L
r
2
f(r)<Lr^2
f
(
r
)
<
L
r
2
for all
r
>
0
r>0
r
>
0
.
4
2
Hide problems
2015 Algebra #4: Sums Divisible by 203
Compute the number of sequences of integers
(
a
1
,
…
,
a
200
)
(a_1,\ldots,a_{200})
(
a
1
,
…
,
a
200
)
such that the following conditions hold. [*]
0
≤
a
1
<
a
2
<
⋯
<
a
200
≤
202.
0\leq a_1<a_2<\cdots<a_{200}\leq 202.
0
≤
a
1
<
a
2
<
⋯
<
a
200
≤
202.
[*] There exists a positive integer
N
N
N
with the following property: for every index
i
∈
{
1
,
…
,
200
}
i\in\{1,\ldots,200\}
i
∈
{
1
,
…
,
200
}
there exists an index
j
∈
{
1
,
…
,
200
}
j\in\{1,\ldots,200\}
j
∈
{
1
,
…
,
200
}
such that
a
i
+
a
j
−
N
a_i+a_j-N
a
i
+
a
j
−
N
is divisible by
203
203
203
.
2015 Geometry #4
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
A
B
=
3
AB=3
A
B
=
3
,
B
C
=
2
BC=2
BC
=
2
,
C
D
=
2
CD=2
C
D
=
2
,
D
A
=
4
DA=4
D
A
=
4
. Let lines perpendicular to
B
C
‾
\overline{BC}
BC
from
B
B
B
and
C
C
C
meet
A
D
‾
\overline{AD}
A
D
at
B
′
B'
B
′
and
C
′
C'
C
′
, respectively. Let lines perpendicular to
B
C
‾
\overline{BC}
BC
from
A
A
A
and
D
D
D
meet
A
D
‾
\overline{AD}
A
D
at
A
′
A'
A
′
and
D
′
D'
D
′
, respectively. Compute the ratio
[
B
C
C
′
B
′
]
[
D
A
A
′
D
′
]
\frac{[BCC'B']}{[DAA'D']}
[
D
A
A
′
D
′
]
[
BC
C
′
B
′
]
, where
[
ω
‾
]
[\overline{\omega}]
[
ω
]
denotes the area of figure
ω
‾
\overline{\omega}
ω
.
3
2
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2015 Algebra #3: Basically Limits
Let
p
p
p
be a real number and
c
≠
0
c\neq 0
c
=
0
such that
c
−
0.1
<
x
p
(
1
−
(
1
+
x
)
10
1
+
(
1
+
x
)
10
)
<
c
+
0.1
c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1
c
−
0.1
<
x
p
(
1
+
(
1
+
x
)
10
1
−
(
1
+
x
)
10
)
<
c
+
0.1
for all (positive) real numbers
x
x
x
with
0
<
x
<
1
0
−
100
0<x<10^{-100}
0
<
x
<
1
0
−
100
. (The exact value
1
0
−
100
10^{-100}
1
0
−
100
is not important. You could replace it with any "sufficiently small number".)Find the ordered pair
(
p
,
c
)
(p,c)
(
p
,
c
)
.
2015 Geometry #3
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
∠
B
A
D
=
∠
A
B
C
=
9
0
∘
\angle BAD = \angle ABC = 90^{\circ}
∠
B
A
D
=
∠
A
BC
=
9
0
∘
, and suppose
A
B
=
B
C
=
1
AB=BC=1
A
B
=
BC
=
1
,
A
D
=
2
AD=2
A
D
=
2
. The circumcircle of
A
B
C
ABC
A
BC
meets
A
D
‾
\overline{AD}
A
D
and
B
D
‾
\overline{BD}
B
D
at point
E
E
E
and
F
F
F
, respectively. If lines
A
F
AF
A
F
and
C
D
CD
C
D
meet at
K
K
K
, compute
E
K
EK
E
K
.
2
2
Hide problems
2015 Algebra #2: Partial Fraction Decomposition
The fraction
1
2015
\tfrac1{2015}
2015
1
has a unique "(restricted) partial fraction decomposition'' of the form
1
2015
=
a
5
+
b
13
+
c
31
,
\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},
2015
1
=
5
a
+
13
b
+
31
c
,
where
a
a
a
,
b
b
b
, and
c
c
c
are integers with
0
≤
a
<
5
0\leq a<5
0
≤
a
<
5
and
0
≤
b
<
13
0\leq b<13
0
≤
b
<
13
. Find
a
+
b
a+b
a
+
b
.
2015 Geometry #2
Let
A
B
C
ABC
A
BC
be a triangle with orthocenter
H
H
H
; suppose
A
B
=
13
AB=13
A
B
=
13
,
B
C
=
14
BC=14
BC
=
14
,
C
A
=
15
CA=15
C
A
=
15
. Let
G
A
G_A
G
A
be the centroid of triangle
H
B
C
HBC
H
BC
, and define
G
B
G_B
G
B
,
G
C
G_C
G
C
similarly. Determine the area of triangle
G
A
G
B
G
C
G_AG_BG_C
G
A
G
B
G
C
.
1
2
Hide problems
2015 Algebra #1: Unique Polynomial
Let
Q
Q
Q
be a polynomial
Q
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
,
Q(x)=a_0+a_1x+\cdots+a_nx^n,
Q
(
x
)
=
a
0
+
a
1
x
+
⋯
+
a
n
x
n
,
where
a
0
,
…
,
a
n
a_0,\ldots,a_n
a
0
,
…
,
a
n
are nonnegative integers. Given that
Q
(
1
)
=
4
Q(1)=4
Q
(
1
)
=
4
and
Q
(
5
)
=
152
Q(5)=152
Q
(
5
)
=
152
, find
Q
(
6
)
Q(6)
Q
(
6
)
.
2015 Geometry #1
Let
R
R
R
be the rectangle in the Cartesian plane with vertices at
(
0
,
0
)
(0,0)
(
0
,
0
)
,
(
2
,
0
)
(2,0)
(
2
,
0
)
,
(
2
,
1
)
(2,1)
(
2
,
1
)
, and
(
0
,
1
)
(0,1)
(
0
,
1
)
.
R
R
R
can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7)); draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point
P
P
P
at random in the interior of
R
R
R
. Find the probability that the line through
P
P
P
with slope
1
2
\frac{1}{2}
2
1
will pass through both unit squares.