MathDB
Problems
Contests
National and Regional Contests
China Contests
ASDAN Math Tournament
2021 ASDAN Math Tournament
2021 ASDAN Math Tournament
Part of
ASDAN Math Tournament
Subcontests
(3)
3
1
Hide problems
2021 Individual Tiebreaker #3
Let
V
V
V
be a set of eight points in
3
D
3\text{D}
3
D
space that are the vertices of a cube with side length
1
1
1
. Compute the number of ways we can color the vertices in
V
V
V
yellow or blue such that[*] each vertex receives exactly one color, and [/*] [*] there exists a point in
3
D
3\text{D}
3
D
space whose distance to each yellow vertex is less than
1
1
1
and whose distance to each blue vertex is greater than
1
1
1
. [/*]
2
1
Hide problems
2021 Individual Tiebreaker #2
For a real number
x
,
x,
x
,
let
⌊
x
⌋
\lfloor x\rfloor
⌊
x
⌋
denote the greatest integer less than or equal to
x
,
x,
x
,
and let
{
x
}
=
x
−
⌊
x
⌋
\{x\} = x -\lfloor x\rfloor
{
x
}
=
x
−
⌊
x
⌋
denote the fractional part of
x
.
x.
x
.
The sum of all real numbers
α
\alpha
α
that satisfy the equation
α
2
+
{
α
}
=
21
\alpha^2+\{\alpha\}=21
α
2
+
{
α
}
=
21
can be expressed in the form
a
−
b
c
−
d
\frac{\sqrt{a}-\sqrt{b}}{c}-d
c
a
−
b
−
d
where
a
,
b
,
c
,
a, b, c,
a
,
b
,
c
,
and
d
d
d
are positive integers, and
a
a
a
and
b
b
b
are not divisible by the square of any prime. Compute
a
+
b
+
c
+
d
.
a + b + c + d.
a
+
b
+
c
+
d
.
1
1
Hide problems
2021 Individual Tiebreaker #1
Benny the Bear has
100
100
100
rabbits in his rabbit farm. He observes that
53
53
53
rabbits are spotted, and
73
73
73
rabbits are blue-eyed. Compute the minimum number of rabbits that are both spotted and blue-eyed.