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2014 Stanford Mathematics Tournament
2014 Stanford Mathematics Tournament
Part of
Stanford Mathematics Tournament
Subcontests
(11)
10
1
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SMT RMT 2014 Geometry #10
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
12
AB = 12
A
B
=
12
,
B
C
=
5
BC = 5
BC
=
5
,
A
C
=
13
AC = 13
A
C
=
13
. Let
D
D
D
and
E
E
E
be the feet of the internal and external angle bisectors from
B
B
B
, respectively. (The external angle bisector from
B
B
B
bisects the angle between
B
C
BC
BC
and the extension of
A
B
AB
A
B
.) Let
ω
\omega
ω
be the circumcircle of
△
B
D
E
\vartriangle BDE
△
B
D
E
, extend
A
B
AB
A
B
so that it intersects
ω
\omega
ω
again at
F
F
F
. Extend
F
C
F C
FC
to meet
ω
\omega
ω
again at
X
X
X
, and extend
A
X
AX
A
X
to meet
ω
\omega
ω
again at
G
G
G
. Find
F
G
F G
FG
.
9
1
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SMT RMT 2014 Geometry #9
In cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
,
A
B
=
A
D
AB= AD
A
B
=
A
D
. If
A
C
=
6
AC = 6
A
C
=
6
and
A
B
B
D
=
3
5
\frac{AB}{BD} =\frac35
B
D
A
B
=
5
3
, find the maximum possible area of
A
B
C
D
ABCD
A
BC
D
.
8
1
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SMT RMT 2014 Geometry #8
O
O
O
is a circle with radius
1
1
1
.
A
A
A
and
B
B
B
are fixed points on the circle such that
A
B
=
2
AB =\sqrt2
A
B
=
2
. Let C be any point on the circle, and let
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
B
C
BC
BC
, respectively. As
C
C
C
travels around circle
O
O
O
, find the area of the locus of points on
M
N
MN
MN
.
7
1
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SMT RMT 2014 Geometry #7
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
13
AB = 13
A
B
=
13
,
B
C
=
14
BC = 14
BC
=
14
, and
A
C
=
15
AC = 15
A
C
=
15
. Let
D
D
D
and
E
E
E
be the feet of the altitudes from
A
A
A
and
B
B
B
, respectively. Find the circumference of the circumcircle of
△
C
D
E
\vartriangle CDE
△
C
D
E
6
1
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SMT RMT 2014 Geometry #6
Let
E
E
E
be an ellipse with major axis length
4
4
4
and minor axis length
2
2
2
. Inscribe an equilateral triangle
A
B
C
ABC
A
BC
in
E
E
E
such that
A
A
A
lies on the minor axis and
B
C
BC
BC
is parallel to the major axis. Compute the area of
△
A
B
C
\vartriangle ABC
△
A
BC
.
4
1
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SMT RMT 2014 Geometry #4
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
=
3
AB = 3
A
B
=
3
,
B
C
=
4
BC = 4
BC
=
4
, and
A
C
=
5
AC = 5
A
C
=
5
. Let
X
X
X
be a point in the triangle. Compute the minimal possible value of
A
X
2
+
B
X
2
+
C
X
2
AX^2 + BX^2 + CX^2
A
X
2
+
B
X
2
+
C
X
2
3
2
Hide problems
SMT RMT 2014 Geometry #3
Spencer eats ice cream in a right circular cone with an opening of radius
5
5
5
and a height of
10
10
10
. If Spencer’s ice cream scoops are always perfectly spherical, compute the radius of the largest scoop he can get such that at least half of the scoop is contained within the cone.
SMT RMT 2014 Geometry Tiebreaker #3
Circle
O
O
O
has three chords,
A
D
AD
A
D
,
D
F
DF
D
F
, and
E
F
EF
EF
. Point E lies along the arc
A
D
AD
A
D
. Point
C
C
C
is the intersection of chords
A
D
AD
A
D
and
E
F
EF
EF
. Point
B
B
B
lies on segment
A
C
AC
A
C
such that
E
B
=
E
C
=
8
EB = EC = 8
EB
=
EC
=
8
. Given
A
B
=
6
AB = 6
A
B
=
6
,
B
C
=
10
BC = 10
BC
=
10
, and
C
D
=
9
CD = 9
C
D
=
9
, find
D
F
DF
D
F
. https://cdn.artofproblemsolving.com/attachments/f/c/c36bff9ad04f13f7e227c57bddb53a0bfc0569.png
2
2
Hide problems
SMT RMT 2014 Geometry #2
In a circle, chord
A
B
AB
A
B
has length
5
5
5
and chord
A
C
AC
A
C
has length
7
7
7
. Arc
A
C
AC
A
C
is twice the length of arc
A
B
AB
A
B
, and both arcs have degree less than
180
180
180
. Compute the area of the circle.
SMT RMT 2014 Geometry Tiebreaker #2
Let
A
B
C
ABC
A
BC
be a triangle with sides
A
B
=
19
AB = 19
A
B
=
19
,
B
C
=
21
BC = 21
BC
=
21
and
A
C
=
20
AC = 20
A
C
=
20
. Let
ω
\omega
ω
be the incircle of
A
B
C
ABC
A
BC
with center
I
I
I
. Extend
B
I
BI
B
I
so that it intersects
A
C
AC
A
C
at
E
E
E
. If
ω
\omega
ω
is tangent to
A
C
AC
A
C
at the point
D
D
D
, then find the length of
D
E
DE
D
E
.
1
2
Hide problems
SMT RMT 2014 Geometry #1
The coordinates of three vertices of a parallelogram are
A
(
1
,
1
)
A(1, 1)
A
(
1
,
1
)
,
B
(
2
,
4
)
B(2, 4)
B
(
2
,
4
)
, and
C
(
−
5
,
1
)
C(-5, 1)
C
(
−
5
,
1
)
. Compute the area of the parallelogram.
SMT RMT 2014 Geometry Tiebreaker #1
A square
A
B
C
D
ABCD
A
BC
D
with side length
1
1
1
is inscribed in a circle. A smaller square lies in the circle with two vertices lying on segment
A
B
AB
A
B
and the other two vertices lying on minor arc
A
B
AB
A
B
. Compute the area of the smaller square.
1
Hide problems
2014 RMT + SMT Team Round - Rice + Stanford Math Tournament
p1. Given that the three points where the parabola
y
=
b
x
2
−
2
y = bx^2 -2
y
=
b
x
2
−
2
intersects the
x
x
x
-axis and
y
y
y
-axis form an equilateral triangle, compute
b
b
b
. p2. Compute the last digit of
2
(
3
(
4
.
.
.
2014
)
)
)
2^{(3^{(4^{...2014)})})}
2
(
3
(
4
...2014
)
)
)
p3. A math tournament has a test which contains
10
10
10
questions, each of which come from one of three different subjects. The subject of each question is chosen uniformly at random from the three subjects, and independently of the subjects of all the other questions. The test is unfair if any one subject appears at least 5 times. Compute the probability that the test is unfair. p4. Let
S
n
S_n
S
n
be the sum
S
n
=
1
+
11
+
111
+
1111
+
.
.
.
+
111...11
S_n = 1 + 11 + 111 + 1111 + ... + 111... 11
S
n
=
1
+
11
+
111
+
1111
+
...
+
111...11
where the last number
111...11
111...11
111...11
has exactly
n
n
n
1
1
1
’s. Find
⌊
1
0
2017
/
S
2014
⌋
\lfloor 10^{2017}/S_{2014} \rfloor
⌊
1
0
2017
/
S
2014
⌋
. p5.
A
B
C
ABC
A
BC
is an equilateral triangle with side length
12
12
12
. Let
O
A
O_A
O
A
be the point inside
A
B
C
ABC
A
BC
that is equidistant from
B
B
B
and
C
C
C
and is
3
\sqrt3
3
units from
A
A
A
. Define
O
B
O_B
O
B
and
O
C
O_C
O
C
symmetrically. Find the area of the intersection of triangles
O
A
B
C
O_ABC
O
A
BC
,
A
O
B
C
AO_BC
A
O
B
C
, and
A
B
O
C
ABO_C
A
B
O
C
. p6. A composition of a natural number
n
n
n
is a way of writing it as a sum of natural numbers, such as
3
=
1
+
2
3 = 1 + 2
3
=
1
+
2
. Let
P
(
n
)
P(n)
P
(
n
)
denote the sum over all compositions of
n
n
n
of the number of terms in the composition. For example, the compositions of
3
3
3
are
3
3
3
,
1
+
2
1+2
1
+
2
,
2
+
1
2+1
2
+
1
, and
1
+
1
+
1
1+1+1
1
+
1
+
1
; the first has one term, the second and third have two each, and the last has
3
3
3
terms, so
P
(
3
)
=
1
+
2
+
2
+
3
=
8
P(3) = 1 + 2 + 2 + 3 = 8
P
(
3
)
=
1
+
2
+
2
+
3
=
8
. Compute
P
(
9
)
P(9)
P
(
9
)
. p7. Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
7
AB = 7
A
B
=
7
,
A
C
=
8
AC = 8
A
C
=
8
, and
B
C
=
9
BC = 9
BC
=
9
. Let the angle bisector of
A
A
A
intersect
B
C
BC
BC
at
D
D
D
. Let
E
E
E
be the foot of the perpendicular from
C
C
C
to line
A
D
AD
A
D
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Find
M
E
ME
ME
. p8. Call a function
g
g
g
lower-approximating for
f
f
f
on the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
if for all
x
∈
[
a
,
b
]
x \in [a, b]
x
∈
[
a
,
b
]
,
f
(
x
)
≥
g
(
x
)
f(x) \ge g(x)
f
(
x
)
≥
g
(
x
)
. Find the maximum possible value of
∫
1
2
g
(
x
)
d
x
\int_1^2 g(x)dx
∫
1
2
g
(
x
)
d
x
where
g
(
x
)
g(x)
g
(
x
)
is a linear lower-approximating function for
f
(
x
)
=
x
x
f(x) = x^x
f
(
x
)
=
x
x
on
[
1
,
2
]
[1, 2]
[
1
,
2
]
. p9. Determine the smallest positive integer
x
x
x
such that
1.24
x
1.24x
1.24
x
is the same number as the number obtained by taking the first (leftmost) digit of
x
x
x
and moving it to be the last (rightmost) digit of
x
x
x
. p10. Let
a
a
a
and
b
b
b
be real numbers chosen uniformly and independently at random from the interval
[
−
10
,
10
]
[-10, 10]
[
−
10
,
10
]
. Find the probability that the polynomial
x
5
+
a
x
+
b
x^5+ax+b
x
5
+
a
x
+
b
has exactly one real root (ignoring multiplicity). p11. Let
b
b
b
be a positive real number, and let
a
n
a_n
a
n
be the sequence of real numbers defined by
a
1
=
a
2
=
a
3
=
1
a_1 = a_2 = a_3 = 1
a
1
=
a
2
=
a
3
=
1
, and
a
n
=
a
n
−
1
+
a
n
−
2
+
b
a
n
−
3
a_n = a_{n-1} + a_{n-2} + ba_{n-3}
a
n
=
a
n
−
1
+
a
n
−
2
+
b
a
n
−
3
for all
n
>
3
n > 3
n
>
3
. Find the smallest value of
b
b
b
such that
∑
n
=
1
∞
a
n
2
n
\sum_{n=1}^{\infty} \frac{\sqrt{a_n}}{2^n}
∑
n
=
1
∞
2
n
a
n
diverges. p12. Find the smallest
L
L
L
such that
(
1
−
1
a
)
b
(
1
−
1
2
b
)
c
(
1
−
1
3
c
)
a
≤
L
\left( 1 - \frac{1}{a} \right)^b \left( 1 - \frac{1}{2b} \right)^c \left( 1 - \frac{1}{3c} \right)^a \le L
(
1
−
a
1
)
b
(
1
−
2
b
1
)
c
(
1
−
3
c
1
)
a
≤
L
for all real numbers
a
,
b
a, b
a
,
b
, and
c
c
c
greater than
1
1
1
. p13. Find the number of distinct ways in which
3
0
(
3
0
30
)
30^{(30^{30})}
3
0
(
3
0
30
)
can be written in the form
a
(
b
c
)
a^{(b^c)}
a
(
b
c
)
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are integers greater than
1
1
1
. p14. Convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has sidelengths
A
B
=
7
AB = 7
A
B
=
7
,
B
C
=
9
BC = 9
BC
=
9
,
C
D
=
15
CD = 15
C
D
=
15
. A circle with center
I
I
I
lies inside the quadrilateral, and is tangent to all four of its sides. Let
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
B
D
BD
B
D
, respectively. It can be proven that
I
I
I
always lies on segment
M
N
MN
MN
. If
I
I
I
is in fact the midpoint of
M
N
MN
MN
, find the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
. p15. Marc has a bag containing
10
10
10
balls, each with a different color. He draws out two balls uniformly at random and then paints the first ball he drew to match the color of the second ball. Then he places both balls back in the bag. He repeats until all the balls are the same color. Compute the expected number of times Marc has to perform this procedure before all the balls are the same color. PS. You had better use hide for answers.
5
1
Hide problems
2014 SMT Geometry #5
Let
A
B
C
ABC
A
BC
be a triangle where
∠
B
A
C
=
3
0
∘
\angle BAC = 30^\circ
∠
B
A
C
=
3
0
∘
. Construct
D
D
D
in
△
A
B
C
\triangle ABC
△
A
BC
such that
∠
A
B
D
=
∠
A
C
D
=
3
0
∘
\angle ABD = \angle ACD = 30^\circ
∠
A
B
D
=
∠
A
C
D
=
3
0
∘
. Let the circumcircle of
△
A
B
D
\triangle ABD
△
A
B
D
intersect
A
C
AC
A
C
at
X
X
X
. Let the circumcircle of
△
A
C
D
\triangle ACD
△
A
C
D
intersect
A
B
AB
A
B
at
Y
Y
Y
. Given that
D
B
−
D
C
=
10
DB - DC = 10
D
B
−
D
C
=
10
and
B
C
=
20
BC = 20
BC
=
20
, find
A
X
⋅
A
Y
AX \cdot AY
A
X
⋅
A
Y
.