MathDB
Problems
Contests
National and Regional Contests
China Contests
ASDAN Math Tournament
2015 ASDAN Math Tournament
2015 ASDAN Math Tournament
Part of
ASDAN Math Tournament
Subcontests
(36)
36
1
Hide problems
2015 Guts #36
A blue square of side length
10
10
10
is laid on top of a coordinate grid with corners at
(
0
,
0
)
(0,0)
(
0
,
0
)
,
(
0
,
10
)
(0,10)
(
0
,
10
)
,
(
10
,
0
)
(10,0)
(
10
,
0
)
, and
(
10
,
10
)
(10,10)
(
10
,
10
)
. Red squares of side length
2
2
2
are randomly placed on top of the grid, changing the color of a
2
×
2
2\times2
2
×
2
square section red. Each red square when placed lies completely within the blue square, and each square's four corners take on integral coordinates. In addition, randomly placed red squares may overlap, keeping overlapped regions red. Compute the expected value of the number of red squares necessary to turn the entire blue square red, rounded to the nearest integer. Your score will be given by
⌊
25
min
{
(
A
C
)
2
,
(
C
A
)
2
}
⌋
\lfloor25\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor
⌊
25
min
{(
C
A
)
2
,
(
A
C
)
2
}⌋
, where
A
A
A
is your answer and
C
C
C
is the actual answer.
35
1
Hide problems
2015 Guts #35
Let
S
S
S
be the set of positive integers less than
1
0
6
10^6
1
0
6
that can be written as the sum of two perfect squares. Compute the number of elements in
S
S
S
. Your score will be given by
max
{
⌊
75
(
min
{
(
A
C
)
2
,
(
C
A
)
2
}
−
2
3
)
⌋
,
0
}
\max\{\lfloor75(\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}-\tfrac{2}{3})\rfloor,0\}
max
{⌊
75
(
min
{(
C
A
)
2
,
(
A
C
)
2
}
−
3
2
)⌋
,
0
}
, where
A
A
A
is your answer and
C
C
C
is the actual answer.
34
1
Hide problems
2015 Guts #34
Compute the number of natural numbers
1
≤
n
≤
1
0
6
1\leq n\leq10^6
1
≤
n
≤
1
0
6
such that the least prime divisor of
n
n
n
is
17
17
17
. Your score will be given by
⌊
26
min
{
(
A
C
)
2
,
(
C
A
)
2
}
⌋
\lfloor26\min\{(\tfrac{A}{C})^2,(\tfrac{C}{A})^2\}\rfloor
⌊
26
min
{(
C
A
)
2
,
(
A
C
)
2
}⌋
, where
A
A
A
is your answer and
C
C
C
is the actual answer.
33
1
Hide problems
2015 Guts #33
Compute the number of digits is
2015
!
2015!
2015
!
. Your score will be given by
max
{
⌊
125
(
min
{
A
C
,
C
A
}
−
1
5
)
⌋
,
0
}
\max\{\lfloor125(\min\{\tfrac{A}{C},\tfrac{C}{A}\}-\tfrac{1}{5})\rfloor,0\}
max
{⌊
125
(
min
{
C
A
,
A
C
}
−
5
1
)⌋
,
0
}
, where
A
A
A
is your answer and
C
C
C
is the actual answer.
32
1
Hide problems
2015 Guts #32
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
=
8
AB=8
A
B
=
8
,
B
C
=
7
BC=7
BC
=
7
, and
A
C
=
11
AC=11
A
C
=
11
. Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be the two possible circles that are tangent to
A
B
AB
A
B
,
A
C
AC
A
C
, and
B
C
BC
BC
when
A
C
AC
A
C
and
B
C
BC
BC
are extended, with
Γ
1
\Gamma_1
Γ
1
having the smaller radius.
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
are tangent to
A
B
AB
A
B
to
D
D
D
and
E
E
E
, respectively, and
C
E
CE
CE
intersects the perpendicular bisector of
A
B
AB
A
B
at a point
F
F
F
. What is
C
F
F
D
\tfrac{CF}{FD}
F
D
CF
?
31
1
Hide problems
2015 Guts #31
Compute the sum of the irrational solutions of the equation
x
2
+
16
x
+
54
x
2
+
11
x
+
35
=
x
2
+
13
x
+
35
x
2
+
14
x
+
54
.
\frac{x^2+16x+54}{x^2+11x+35}=\frac{x^2+13x+35}{x^2+14x+54}.
x
2
+
11
x
+
35
x
2
+
16
x
+
54
=
x
2
+
14
x
+
54
x
2
+
13
x
+
35
.
30
1
Hide problems
2015 Guts #30
Suppose that
10
10
10
mathematics teachers gather at a circular table with
25
25
25
seats to discuss the upcoming mathematics competition. Each teacher is assigned a unique integer ID number between
1
1
1
and
10
10
10
, and the teachers arrange themselves in such a way that teachers with consecutive ID numbers are not separated by any other teacher (IDs
1
1
1
and
10
10
10
are considered consecutive). In addition, each pair of teachers is separated by at least one empty seat. Given that seating arrangements obtained by rotation are considered identical, how many ways are there for the teachers to sit at the table?
29
1
Hide problems
2015 Guts #29
Suppose that the following equations hold for positive integers
x
x
x
,
y
y
y
, and
n
n
n
, where
n
>
18
n>18
n
>
18
: \begin{align*} x+3y&\equiv7\pmod{n}\\ 2x+2y&\equiv18\pmod{n}\\ 3x+y&\equiv7\pmod{n} \end{align*} Compute the smallest nonnegative integer
a
a
a
such that
2
x
≡
a
(
m
o
d
n
)
2x\equiv a\pmod{n}
2
x
≡
a
(
mod
n
)
.
28
1
Hide problems
2015 Guts #28
Consider
13
13
13
marbles that are labeled with positive integers such that the product of all
13
13
13
integers is
360
360
360
. Moor randomly picks up
5
5
5
marbles and multiplies the integers on top of them together, obtaining a single number. What is the maximum number of different products that Moor can obtain?
27
1
Hide problems
2015 Guts #27
In triangle
A
B
C
ABC
A
BC
,
D
D
D
is a point on
A
B
AB
A
B
between
A
A
A
and
B
B
B
,
E
E
E
is a point on
A
C
AC
A
C
between
A
A
A
and
C
C
C
, and
F
F
F
is a point on
B
C
BC
BC
between
B
B
B
and
C
C
C
such that
A
F
AF
A
F
,
B
E
BE
BE
, and
C
D
CD
C
D
all meet inside
△
A
B
C
\triangle ABC
△
A
BC
at a point
G
G
G
. Given that the area of
△
A
B
C
\triangle ABC
△
A
BC
is
15
15
15
, the area of
△
A
B
E
\triangle ABE
△
A
BE
is
5
5
5
, and the area of
△
A
C
D
\triangle ACD
△
A
C
D
is
10
10
10
, compute the area of
△
A
B
F
\triangle ABF
△
A
BF
.
26
1
Hide problems
2015 Guts #26
Lennart and Eddy are playing a betting game. Lennart starts with
7
7
7
dollars and Eddy starts with
3
3
3
dollars. Each round, both Lennart and Eddy bet an amount equal to the amount of the player with the least money. For example, on the first round, both players bet
3
3
3
dollars. A fair coin is then tossed. If it lands heads, Lennart wins all the money bet; if it lands tails, Eddy wins all the money bet. They continue playing this game until one person has no money. What is the probability that Eddy ends with
10
10
10
dollars?
25
1
Hide problems
2015 Guts #25
Let
a
n
a_n
a
n
be a sequence with
a
0
=
1
a_0=1
a
0
=
1
and defined recursively by
a
n
+
1
=
{
a
n
+
2
if
n
is even
,
2
a
n
if
n
is odd.
a_{n+1}=\begin{cases}a_n+2&\text{if }n\text{ is even},\\2a_n&\text{if }n\text{ is odd.}\end{cases}
a
n
+
1
=
{
a
n
+
2
2
a
n
if
n
is even
,
if
n
is odd.
What are the last two digits of
a
2015
a_{2015}
a
2015
?
24
1
Hide problems
2015 Guts #24
Trains
A
A
A
and
B
B
B
are on the same track a distance
100
100
100
miles apart heading towards one another, each at a speed of
50
50
50
miles per hour. A fly starting out at the front of train
A
A
A
flies towards train
B
B
B
at a speed of
75
75
75
miles per hour. Upon reaching train
B
B
B
, the fly turns around and flies towards train
A
A
A
, again at
75
75
75
miles per hour. The fly continues flying back and forth between the two trains at
75
75
75
miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train
A
A
A
than to train
B
B
B
before getting squashed?
23
1
Hide problems
2015 Guts #23
Two regular hexagons of side length
2
2
2
are laid on top of each other such that they share the same center point and one hexagon is rotated
3
0
∘
30^\circ
3
0
∘
about the center from the other. Compute the area of the union of the two hexagons.
22
1
Hide problems
2015 Guts #22
You flip a fair coin which results in heads (
H
\text{H}
H
) or tails (
T
\text{T}
T
) with equal probability. What is the probability that you see the consecutive sequence
THH
\text{THH}
THH
before the sequence
HHH
\text{HHH}
HHH
?
21
1
Hide problems
2015 Guts #21
Parallelogram
A
B
C
D
ABCD
A
BC
D
has
A
B
=
C
D
=
6
AB=CD=6
A
B
=
C
D
=
6
and
B
C
=
A
D
=
10
BC=AD=10
BC
=
A
D
=
10
, where
∠
A
B
C
\angle ABC
∠
A
BC
is obtuse. The circumcircle of
△
A
B
D
\triangle ABD
△
A
B
D
intersects
B
C
BC
BC
at
E
E
E
such that
C
E
=
4
CE=4
CE
=
4
. Compute
B
D
BD
B
D
.
20
1
Hide problems
2015 Guts #20
The sequence
a
1
,
a
2
,
…
,
a
13
a_1,a_2,\dots,a_{13}
a
1
,
a
2
,
…
,
a
13
is a geometric sequence with
a
1
=
a
a_1=a
a
1
=
a
and common ratio
r
r
r
, where
a
a
a
and
r
r
r
are positive integers. Given that
log
2015
a
1
+
log
2015
a
2
+
⋯
+
log
2015
a
13
=
2015
,
\log_{2015}a_1+\log_{2015}a_2+\dots+\log_{2015}a_{13}=2015,
lo
g
2015
a
1
+
lo
g
2015
a
2
+
⋯
+
lo
g
2015
a
13
=
2015
,
find the number of possible ordered pairs
(
a
,
r
)
(a,r)
(
a
,
r
)
.
19
1
Hide problems
2015 Guts #19
Compute the number of
0
≤
n
≤
2015
0\leq n\leq2015
0
≤
n
≤
2015
such that
6
n
+
8
n
6^n+8^n
6
n
+
8
n
is divisible by
7
7
7
.
18
1
Hide problems
2015 Guts #18
Andrew takes a square sheet of paper
A
B
C
D
ABCD
A
BC
D
of side length
1
1
1
and folds a kite shape. To do this, he takes the corners at
B
B
B
and
D
D
D
and folds the paper such that both corners now rest at a point
E
E
E
on
A
C
AC
A
C
. This fold results in two creases
C
F
CF
CF
and
C
G
CG
CG
, respectively, where
F
F
F
lies on
A
B
AB
A
B
and
G
G
G
lies on
A
D
AD
A
D
. Compute the length of
F
G
FG
FG
.
17
1
Hide problems
2015 Guts #17
How many ways are there to write
91
91
91
as the sum of at least
2
2
2
consecutive positive integers?
16
1
Hide problems
2015 Guts #16
Find the maximum value of
c
c
c
such that \begin{align*} 1&=-cx+y\\ -7&=x^2+y^2+8y \end{align*} has a unique real solution
(
x
,
y
)
(x,y)
(
x
,
y
)
.
15
2
Hide problems
2015 Guts #15
Let
A
B
C
D
ABCD
A
BC
D
be a regular tetrahedron of side length
12
12
12
. Select points
E
,
F
,
G
,
H
E,F,G,H
E
,
F
,
G
,
H
on
A
C
,
B
C
,
B
D
,
A
D
AC,BC,BD,AD
A
C
,
BC
,
B
D
,
A
D
, respectively, such that
A
E
=
B
F
=
B
G
=
A
H
=
3
AE=BF=BG=AH=3
A
E
=
BF
=
BG
=
A
H
=
3
. Compute the area of quadrilateral
E
F
G
H
EFGH
EFG
H
.
2015 Team #15
In a given acute triangle
△
A
B
C
\triangle ABC
△
A
BC
with the values of angles given (known as
a
a
a
,
b
b
b
, and
c
c
c
), the inscribed circle has points of tangency
D
,
E
,
F
D,E,F
D
,
E
,
F
where
D
D
D
is on
B
C
BC
BC
,
E
E
E
is on
A
B
AB
A
B
, and
F
F
F
is on
A
C
AC
A
C
. Circle
γ
\gamma
γ
has diameter
B
C
BC
BC
, and intersects
E
F
‾
\overline{EF}
EF
at points
X
X
X
and
Y
Y
Y
. Find
X
Y
B
C
\tfrac{XY}{BC}
BC
X
Y
in terms of the angles
a
a
a
,
b
b
b
, and
c
c
c
.
14
2
Hide problems
2015 Guts #14
A standard deck of
52
52
52
cards is shuffled and randomly arranged in a queue, with each card having a suit
(
♢
,
♣
,
♡
,
♠
)
(\diamondsuit,\clubsuit,\heartsuit,\spadesuit)
(
♢
,
♣
,
♡
,
♠
)
and a rank
(
Ace
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
Jack
,
Queen
,
King
)
(\text{Ace},2,3,4,5,6,7,8,9,10,\text{Jack},\text{Queen},\text{ King})
(
Ace
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
,
Jack
,
Queen
,
King
)
. For example, a card with the
♢
\diamondsuit
♢
suit and the
7
7
7
rank would be denoted as
♢
7
\diamondsuit7
♢7
, and a card with the
♠
\spadesuit
♠
and the
Ace
\text{Ace}
Ace
rank would be denoted as
♠
Ace
\spadesuit\text{Ace}
♠
Ace
. In the queue, there exists a card with a rank of
Ace
\text{Ace}
Ace
that appears for the first time in the queue. Let the card immediately following the above card be denoted as card
C
C
C
. Is the probability that
C
C
C
is a
♠
A
\spadesuit\text{A}
♠
A
higher than, equal to, or lower than the probability that
C
C
C
is a
♣
2
\clubsuit2
♣2
?
2015 Team #14
For a given positive integer
m
m
m
, the series
∑
k
=
1
,
k
≠
m
∞
1
(
k
+
m
)
(
k
−
m
)
\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}
k
=
1
,
k
=
m
∑
∞
(
k
+
m
)
(
k
−
m
)
1
evaluates to
a
b
m
2
\frac{a}{bm^2}
b
m
2
a
, where
a
a
a
and
b
b
b
are positive integers. Compute
a
+
b
a+b
a
+
b
.
13
2
Hide problems
2015 Guts #13
A three-digit number
x
x
x
in base
10
10
10
has a units-digit of
6
6
6
. When
x
x
x
is written is base
9
9
9
, the second digit of the number is
4
4
4
, and the first and third digit are equal in value. Compute
x
x
x
in base
10
10
10
.
2015 Team #13
The incircle of triangle
△
A
B
C
\triangle ABC
△
A
BC
is the unique inscribed circle that is internally tangent to the sides
A
B
‾
\overline{AB}
A
B
,
B
C
‾
\overline{BC}
BC
, and
C
A
‾
\overline{CA}
C
A
. How many non-congruent right triangles with integer side lengths have incircles of radius
2015
2015
2015
?
12
2
Hide problems
2015 Guts #12
The rectangular faces of rectangular prism
A
A
A
have perimeters
12
12
12
,
16
16
16
, and
24
24
24
. The rectangular faces of rectangular prism
B
B
B
have perimeters
12
12
12
,
16
16
16
, and
20
20
20
. Let
V
A
V_A
V
A
denote the volume of
A
A
A
and
V
B
V_B
V
B
denote the volume of
B
B
B
. Find
V
A
−
V
B
V_A-V_B
V
A
−
V
B
.
2015 Team #12
Find the smallest positive integer solution to the equation
2
2
k
≡
k
(
m
o
d
29
)
2^{2^k}\equiv k\pmod{29}
2
2
k
≡
k
(
mod
29
)
.
11
2
Hide problems
2015 Guts #11
In the following diagram, each circle has radius
6
6
6
and each circle passes through the center of the other two circles. Compute the area of the white center region and express your answer in terms of
π
\pi
π
.
2015 Team #11
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a regular hexagon, and let
G
G
G
,
H
H
H
,
I
I
I
,
J
J
J
,
K
K
K
, and
L
L
L
be the midpoints of sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
E
DE
D
E
,
E
F
EF
EF
, and
F
A
FA
F
A
, respectively. The intersection of lines
A
H
‾
\overline{AH}
A
H
,
B
I
‾
\overline{BI}
B
I
,
C
J
‾
\overline{CJ}
C
J
,
D
K
‾
\overline{DK}
DK
,
E
L
‾
\overline{EL}
E
L
, and
F
G
‾
\overline{FG}
FG
bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
.
10
5
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9
5
Show problems
8
5
Show problems
7
5
Show problems
6
5
Show problems
5
5
Show problems
4
5
Show problems
3
8
Show problems
2
8
Show problems
1
8
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