MathDB
Problems
Contests
National and Regional Contests
China Contests
ASDAN Math Tournament
2020 ASDAN Math Tournament
2020 ASDAN Math Tournament
Part of
ASDAN Math Tournament
Subcontests
(15)
15
1
Hide problems
2020 Team #15
For integers
z
z
z
, let
#
(
z
)
\#(z)
#
(
z
)
denote the number of integer ordered pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
that satisfy
x
2
−
x
y
+
y
2
=
z
x^2 - xy + y^2 = z
x
2
−
x
y
+
y
2
=
z
. How many integers
z
z
z
between
0
0
0
and
150
150
150
inclusive satisfy
#
(
z
)
≡
6
\#(z) \equiv 6
#
(
z
)
≡
6
(mod
12
12
12
)?
14
1
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2020 Team #14
If
f
f
f
is a permutation of
S
=
{
0
,
1
,
.
.
.
,
14
}
S = \{0, 1,..., 14\}
S
=
{
0
,
1
,
...
,
14
}
, then for integers
k
≥
1
k \ge 1
k
≥
1
, define
f
k
(
x
)
=
f
(
f
.
.
.
(
f
(
x
)
)
.
.
.
)
)
⏟
k
a
p
p
l
i
c
a
t
i
o
n
s
o
f
f
f^k(x) =\underbrace{f(f...(f(x))... ))}_{k\,\,\, applications \,\,\, of \,\,\, f}
f
k
(
x
)
=
k
a
ppl
i
c
a
t
i
o
n
s
o
f
f
f
(
f
...
(
f
(
x
))
...
))
Compute the number of permutations
f
f
f
of
S
S
S
such that, for some
k
≥
1
k \ge 1
k
≥
1
,
f
k
(
x
)
=
(
x
+
5
)
m
o
d
15
f^k(x) = (x + 5) \mod \,\,\, 15
f
k
(
x
)
=
(
x
+
5
)
mod
15
for all
x
∈
S
x \in S
x
∈
S
.
13
1
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2020 Team #13
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
△
A
B
C
\vartriangle ABC
△
A
BC
is an equilateral triangle. Let
P
P
P
be a point inside the quadrilateral such that
△
A
P
D
\vartriangle APD
△
A
P
D
is an equilateral triangle and
∠
P
C
D
=
3
0
o
\angle PCD = 30^o
∠
PC
D
=
3
0
o
. Suppose
C
P
=
6
CP = 6
CP
=
6
and
C
D
=
8
CD = 8
C
D
=
8
. The area of the triangle formed by
P
P
P
, the midpoint of
B
C
‾
\overline{BC}
BC
, and the midpoint of
A
B
‾
\overline{AB}
A
B
can be expressed in simplest radical form as
a
+
b
c
d
\frac{a+b\sqrt{c}}{d}
d
a
+
b
c
, where
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
are positive integers with
g
c
d
(
a
,
b
,
d
)
=
1
gcd(a, b, d) = 1
g
c
d
(
a
,
b
,
d
)
=
1
and with
c
c
c
not divisible by the square of any prime. Compute
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
.
12
1
Hide problems
2020 Team #12
Let
S
n
S_n
S
n
be the number of subsets of the first
n
n
n
positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer
n
n
n
such that
S
n
S_n
S
n
is a multiple of
2020
2020
2020
.
11
1
Hide problems
2020 Team #11
△
A
B
C
\vartriangle ABC
△
A
BC
is right with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
. The internal angle bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at point
D
D
D
, while the external angle bisectors of
∠
A
\angle A
∠
A
and
∠
B
\angle B
∠
B
meet at point
E
E
E
. Suppose that
A
D
=
1
AD = 1
A
D
=
1
and
B
D
=
2
BD = 2
B
D
=
2
. The value of
D
E
2
DE^2
D
E
2
can be expressed as
x
+
y
z
x+y \sqrt{z}
x
+
y
z
for integers
x
x
x
,
y
y
y
, and
z
z
z
, where
z
z
z
is greater than
1
1
1
and not divisible by the square of any prime. Compute
100
x
+
10
y
+
z
100x + 10y + z
100
x
+
10
y
+
z
. Note: For a generic triangle
△
P
Q
R
\vartriangle PQR
△
PQR
, if we let
Q
′
Q'
Q
′
be the reflection of
Q
Q
Q
over
P
P
P
, then the external angle bisector of
∠
P
\angle P
∠
P
is the line that contains the internal angle bisector of
∠
Q
′
P
R
\angle Q'PR
∠
Q
′
PR
.
10
1
Hide problems
2020 Team #10
Let
r
=
1
−
2
5
+
4
5
−
8
5
+
16
5
r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}
r
=
1
−
5
2
+
5
4
−
5
8
+
5
16
. There exists a unique fifth-degree polynomial
P
P
P
such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is
1
1
1
, and
P
(
r
)
=
0
P(r) = 0
P
(
r
)
=
0
. Evaluate
P
(
10
)
P(10)
P
(
10
)
.
9
1
Hide problems
2020 Team #9
A positive integer
n
n
n
has the property that, for any
2
2
2
integers
a
a
a
and
b
b
b
, if
a
b
+
1
ab + 1
ab
+
1
is divisible by
n
n
n
, then
a
+
b
a + b
a
+
b
is also divisible by
n
n
n
. What is the largest possible value of
n
n
n
?
8
1
Hide problems
2020 Team #8
For nonzero integers
n
n
n
, let
f
(
n
)
f(n)
f
(
n
)
be the sum of all positive integers
b
b
b
for which all solutions
x
x
x
to
x
2
+
b
x
+
n
=
0
x^2 +bx+n = 0
x
2
+
b
x
+
n
=
0
are integers, and let
g
(
n
)
g(n)
g
(
n
)
be the sum of all positive integers
c
c
c
for which all solutions
x
x
x
to
c
x
+
n
=
0
cx + n = 0
c
x
+
n
=
0
are integers. Compute
∑
n
=
1
2020
(
f
(
n
)
−
g
(
n
)
)
\sum^{2020}_{n=1} (f(n) - g(n))
∑
n
=
1
2020
(
f
(
n
)
−
g
(
n
))
.
7
1
Hide problems
2020 Team #7
Alex scans the list of integers between
1
1
1
and
2020
2020
2020
inclusive using the following algorithm. First, he reads off perfect squares between
1
1
1
and
2020
2020
2020
in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are
2
2
2
,
6
6
6
,
12
12
12
,
.
.
.
...
...
, and removes these numbers from the list. He repeats this algorithm until he reads off
2020
2020
2020
, which is the nth number he has read o so far. Compute
n
n
n
.
6
1
Hide problems
2020 Team #6
Triangle
△
A
B
C
\vartriangle ABC
△
A
BC
has side lengths
A
B
=
26
AB = 26
A
B
=
26
,
B
C
=
34
BC = 34
BC
=
34
, and
C
A
=
24
2
CA = 24\sqrt2
C
A
=
24
2
. A fourth point
D
D
D
makes a right angle
∠
B
D
C
\angle BDC
∠
B
D
C
. What is the smallest possible length of
A
D
‾
\overline{AD}
A
D
?
5
1
Hide problems
2020 Team #5
Two quadratic polynomials
A
(
x
)
A(x)
A
(
x
)
and
B
(
x
)
B(x)
B
(
x
)
have a leading term of
x
2
x^2
x
2
. For some real numbers
a
a
a
and
b
b
b
, the roots of
A
(
x
)
A(x)
A
(
x
)
are
1
1
1
and
a
a
a
, and the roots of
B
(
x
)
B(x)
B
(
x
)
are
6
6
6
and
b
b
b
. If the roots of
A
(
x
)
+
B
(
x
)
A(x) + B(x)
A
(
x
)
+
B
(
x
)
are
a
+
3
a + 3
a
+
3
and
b
+
1
2
b + \frac1 2
b
+
2
1
, then compute
a
2
+
b
2
a^2 + b^2
a
2
+
b
2
.
4
1
Hide problems
2020 Team #4
There are
2
2
2
ways to write
2020
2020
2020
as a sum of
2
2
2
squares:
2020
=
a
2
+
b
2
2020 = a^2 + b^2
2020
=
a
2
+
b
2
and
2020
=
c
2
+
d
2
2020 = c^2 + d^2
2020
=
c
2
+
d
2
, where
a
a
a
,
b
b
b
,
c
c
c
, and
d
d
d
are distinct positive integers with
a
<
b
a < b
a
<
b
and
c
<
d
c < d
c
<
d
. Compute
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
.
3
1
Hide problems
2020 Team #3
A fair coin is flipped
6
6
6
times. The probability that the coin lands on the same side
3
3
3
flips in a row at some point can be expressed as a common fraction
m
n
\frac{m}{n}
n
m
, where
m
m
m
and
n
n
n
are relatively prime positive integers. Compute
100
m
+
n
100m + n
100
m
+
n
.
2
1
Hide problems
2020 Team #2
Sam's cup has a
400
400
400
mL mixture of coffee and milk tea. He pours
200
200
200
mL into Ben's empty cup. Ben then adds 100mL of co ee to his cup and stirs well. Finally, Ben pours
200
200
200
mL out of his cup back into Sam's cup. If the mixture in Sam's cup is now
50
%
50\%
50%
milk tea, then how many milliliters of milk tea were in it originally?
1
1
Hide problems
2020 Team #1
Consider triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with
∠
C
=
9
0
o
\angle C = 90^o
∠
C
=
9
0
o
. Let
P
P
P
be the midpoint of
A
C
‾
\overline{AC}
A
C
so that
A
P
=
P
C
=
1
AP = PC = 1
A
P
=
PC
=
1
, and suppose
∠
B
A
C
=
∠
C
B
P
\angle BAC = \angle CBP
∠
B
A
C
=
∠
CBP
. Compute
A
B
2
AB^2
A
B
2
.