MathDB
Problems
Contests
National and Regional Contests
China Contests
ASDAN Math Tournament
2017 ASDAN Math Tournament
2017 ASDAN Math Tournament
Part of
ASDAN Math Tournament
Subcontests
(28)
27
1
Hide problems
2017 Guts #27
How many primes between
2
2
2
and
2
30
2^{30}
2
30
are
1
1
1
more than a multiple of
2017
2017
2017
? If
C
C
C
is the correct answer and
A
A
A
is your answer, then your score will be rounded up from
max
(
0
,
25
−
15
∣
ln
A
C
∣
)
\max(0,25-15|\ln\tfrac{A}{C}|)
max
(
0
,
25
−
15∣
ln
C
A
∣
)
.
26
1
Hide problems
2017 Guts #26
A lattice point is a coordinate pair
(
a
,
b
)
(a,b)
(
a
,
b
)
where both
a
,
b
a,b
a
,
b
are integers. What is the number of lattice points
(
x
,
y
)
(x,y)
(
x
,
y
)
that satisfy
x
2
2017
+
2
y
2
2017
<
1
\tfrac{x^2}{2017}+\tfrac{2y^2}{2017}<1
2017
x
2
+
2017
2
y
2
<
1
and
y
≡
2
x
(
m
o
d
7
)
y\equiv2x\pmod{7}
y
≡
2
x
(
mod
7
)
?Let
C
C
C
be the actual answer,
A
A
A
be the answer you submit, and
D
=
∣
A
−
C
∣
D=|A-C|
D
=
∣
A
−
C
∣
. Your score will be rounded up from
max
(
0
,
25
−
e
D
/
100
)
\max(0,25-e^{D/100})
max
(
0
,
25
−
e
D
/100
)
.
25
1
Hide problems
2017 Guts #25
Consider the sequence
{
a
n
}
\{a_n\}
{
a
n
}
defined so that
a
n
a_n
a
n
is the leftmost digit of
2
n
2^n
2
n
. The first few terms of this sequence are
1
,
2
,
4
,
8
,
1
,
3
,
6
,
…
1,2,4,8,1,3,6,\dots
1
,
2
,
4
,
8
,
1
,
3
,
6
,
…
. For how many
0
≤
n
≤
100000
0\le n\le100000
0
≤
n
≤
100000
is
a
n
=
1
a_n=1
a
n
=
1
? If
C
C
C
is the correct answer and
A
A
A
is your answer, then your score will be rounded up from
max
(
0
,
25
−
1
6
∣
A
−
C
∣
)
\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)
max
(
0
,
25
−
6
1
∣
A
−
C
∣
)
.
24
1
Hide problems
2017 Guts #24
Consider all rational numbers of the form
p
q
\tfrac{p}{q}
q
p
where
p
,
q
p,q
p
,
q
are relatively prime positive integers less than or equal to
8
8
8
, and plot them on the
x
y
xy
x
y
-plane, where
p
q
\tfrac{p}{q}
q
p
corresponds to point
(
p
,
q
)
(p,q)
(
p
,
q
)
. Arrange the rationals in increasing order
{
P
1
,
P
2
,
…
,
P
n
}
\{P_1,P_2,\dots,P_n\}
{
P
1
,
P
2
,
…
,
P
n
}
and form a polygon by connecting points
P
i
P_i
P
i
and
P
i
+
1
P_{i+1}
P
i
+
1
for
1
≤
i
<
n
1\le i<n
1
≤
i
<
n
and connecting both
P
1
P_1
P
1
and
P
n
P_n
P
n
to the origin. What is the area of the polygon?
23
1
Hide problems
2017 Guts #23
Ben creates an
8
×
8
8\times8
8
×
8
grid of coins, where each coin faces heads with probability
1
2
\tfrac{1}{2}
2
1
, and tails with probability
1
2
\tfrac{1}{2}
2
1
. Ben then makes a series of moves; each move consists of selecting a coin in the grid and flipping over all coins in the same row and column as the selected coin. Suppose that in Ben’s current grid of coins, it is possible to make a series of moves so that all coins in the grid are heads, and that Ben will make the fewest number of moves to do so. What is the expected number of moves that Ben makes?
22
1
Hide problems
2017 Guts #22
Let
x
=
2
sin
8
∘
+
2
sin
1
6
∘
+
⋯
+
2
sin
17
6
∘
x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ
x
=
2
sin
8
∘
+
2
sin
1
6
∘
+
⋯
+
2
sin
17
6
∘
. What is
arctan
(
x
)
\arctan(x)
arctan
(
x
)
?
21
1
Hide problems
2017 Guts #21
In trapezoid
A
B
C
D
ABCD
A
BC
D
, we have
A
D
‾
∥
B
C
‾
\overline{AD}\parallel\overline{BC}
A
D
∥
BC
,
B
C
=
3
BC=3
BC
=
3
, and
C
D
=
4
CD=4
C
D
=
4
. In addition,
cos
∠
A
D
C
=
1
3
\cos\angle ADC=\tfrac{1}{3}
cos
∠
A
D
C
=
3
1
and
∠
A
B
C
=
2
∠
A
D
C
\angle ABC=2\angle ADC
∠
A
BC
=
2∠
A
D
C
. Compute
A
C
AC
A
C
.
20
1
Hide problems
2017 Guts #20
Let
α
\alpha
α
and
β
\beta
β
be positive rational numbers so that
α
+
β
5
\alpha+\beta\sqrt{5}
α
+
β
5
is a root of some polynomial
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
where
a
a
a
and
b
b
b
are integers. What is the smallest possible value of
α
β
\alpha\beta
α
β
?
19
1
Hide problems
2017 Guts #19
How many ways can you tile a
2
×
5
2\times5
2
×
5
rectangle with
2
×
1
2\times1
2
×
1
dominoes of
4
4
4
different colors if no two dominoes of the same color may be adjacent?
18
1
Hide problems
2017 Guts #18
Find the sum of all integers
0
≤
a
≤
124
0\le a \le124
0
≤
a
≤
124
so that
a
3
−
2
a^3-2
a
3
−
2
is a multiple of
125
125
125
.
17
1
Hide problems
2017 Guts #17
For
△
A
B
C
\triangle ABC
△
A
BC
,
A
B
=
B
C
=
5
AB=BC=5
A
B
=
BC
=
5
, and
A
C
=
6
AC=6
A
C
=
6
. Circle
O
O
O
is inscribed in
△
A
B
C
\triangle ABC
△
A
BC
, and circle
P
P
P
is tangent to circle
O
O
O
,
A
B
AB
A
B
, and
A
C
AC
A
C
. Compute the area of
△
A
B
C
\triangle ABC
△
A
BC
not covered by circles
O
O
O
and
P
P
P
.
16
1
Hide problems
2017 Guts #16
Let
x
x
x
and
y
y
y
be real numbers satisfying
9
x
2
+
16
y
2
=
144
9x^2+16y^2=144
9
x
2
+
16
y
2
=
144
. What is the maximum possible value of
x
y
xy
x
y
?
15
1
Hide problems
2017 Guts #15
Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.
14
1
Hide problems
2017 Guts #14
What are the last two digits of
201
7
2017
2017^{2017}
201
7
2017
?
13
1
Hide problems
2017 Guts #13
Let
S
1
S_1
S
1
be a square of side length
3
3
3
. For
i
=
2
,
3
,
4
,
…
i=2,3,4,\dots
i
=
2
,
3
,
4
,
…
, inscribe a square
S
i
S_i
S
i
inside
S
i
−
1
S_{i-1}
S
i
−
1
such that the sides of the inner square form four
3
0
∘
−
6
0
∘
−
9
0
∘
30^\circ-60^\circ-90^\circ
3
0
∘
−
6
0
∘
−
9
0
∘
triangles with the outer square. Compute the total sum
∑
i
=
1
∞
area
(
S
i
)
.
\sum_{i=1}^\infty\text{area}(S_i).
i
=
1
∑
∞
area
(
S
i
)
.
12
1
Hide problems
2017 Guts #12
Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after
6
6
6
minutes?
11
1
Hide problems
2017 Guts #11
If
a
+
b
+
c
=
12
a+b+c=12
a
+
b
+
c
=
12
and
a
2
+
b
2
+
c
2
=
62
a^2+b^2+c^2=62
a
2
+
b
2
+
c
2
=
62
, what is
a
b
+
b
c
+
a
c
ab+bc+ac
ab
+
b
c
+
a
c
?
1
Hide problems
2017 Estimation Round
DescriptionThe Estimation Round is a 60-minute team competition with a set of 8 short answer questions. In these questions, students must approximate a diverse set of real-world quantities as accurately as possible using their mathematical skills and worldly knowledge.Answer Format and ScoringEach problem’s answer is a positive integer as close to the true answer as our staff can produce. Teams will submit a positive integer answer for each problem in either fully simplified form (for example,
123456789
123456789
123456789
) or scientific notation (for example,
1.23456789
×
1
0
8
1.23456789\times10^8
1.23456789
×
1
0
8
). For problem
k
k
k
, let the staff answer be
z
k
z_k
z
k
. If a team leaves the answer space for problem
k
k
k
blank, their score on that problem will be
S
k
=
0
S_k=0
S
k
=
0
. Otherwise, let the team’s submission be
x
k
x_k
x
k
. Then their score on problem
k
k
k
is
S
k
=
max
(
0
,
1
−
∣
log
10
∣
x
k
/
z
k
∣
5
∣
)
S_k=\max\left(0,1-\left|\frac{\log_{10}|x_k/z_k|}{5}\right|\right)
S
k
=
max
(
0
,
1
−
5
lo
g
10
∣
x
k
/
z
k
∣
)
This is constructed such that the score is in the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
, and so that an estimate that is at most an order of magnitude from the staff answer earns a high score (at least
0.8
0.8
0.8
). A team's total score
S
S
S
on the exam, which will be used for rankings, is the sum of the scores for each problem:
S
=
∑
k
=
1
8
S
k
S=\sum_{k=1}^8S_k
S
=
k
=
1
∑
8
S
k
VerificationA team may ask, up to three times during the duration of the exam, whether, for a problem
k
k
k
, a closed interval of integers contains the staff answer
z
k
z_k
z
k
. (This does not mean three times per problem, but rather three attempts for the entire exam.) ——-- Problems1. How many pet cats were there in the United States in
2012
2012
2012
?2. How many passengers passed through Beijing Capital International Airport in
2015
2015
2015
?3. What was the total worldwide revenue of all films in
2015
2015
2015
, in US dollars?4. How many candies are in the container displayed in the front of the room?5. If you stacked them from head to toe, how many male giraffes, on average, would it take to reach from the surface of the Earth to that of the moon, when the Earth and the moon are closest to each other?6. How many total millimeters of rain does Singapore receive in an average year?7. What is the mass of the Empire State Building in kilograms?8. How many minutes before the scheduled start of this tournament’s opening ceremony was this problem written?AppendixHere are some facts that may or may not be useful for these problems.[*] Population of the United States on December
31
31
31
,
2012
2012
2012
:
315219776
315219776
315219776
[/*] [*] Aircraft movements at Beijing Capital International Airport in
2015
2015
2015
:
590169
590169
590169
[/*] [*] Exchange rate for US dollars to CNY (Chinese Yuan/Renminbi) on December
31
31
31
,
2015
2015
2015
:
1
USD
=
6.5138
CNY
1\text{ USD}=6.5138\text{ CNY}
1
USD
=
6.5138
CNY
[/*] [*] Total sales for the
10
10
10
th highest selling film in
2015
2015
2015
(The Martian):
630161890
USD
630161890\text{ USD}
630161890
USD
[/*] [*] Average height of the International Space Station from Earth:
400
km
400\text{ km}
400
km
[/*] [*] Land area of Singapore at the end of
2016
2016
2016
:
719.7
km
2
719.7\text{ km}^2
719.7
km
2
[/*] [*] Density of iron:
7.874
g/cm
3
7.874\text{ g/cm}^3
7.874
g/cm
3
[/*] [*] The Empire State Building is a skyscraper in New York City and is
443.2
m
443.2\text{ m}
443.2
m
high. Refer to the projector screen for a picture of the building; it’s the largest building in the middle with the antenna. [/*]
10
5
Show problems
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
9
Show problems
2
9
Show problems
1
9
Show problems