MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
Taiwan Mathematics Olympiad
2022 Taiwan Mathematics Olympiad
2022 Taiwan Mathematics Olympiad
Part of
Taiwan Mathematics Olympiad
Subcontests
(5)
2
1
Hide problems
2022 black/white balls and 1011 black/white boxes
There are
2022
2022
2022
black balls numbered
1
1
1
to
2022
2022
2022
and
2022
2022
2022
white balls numbered
1
1
1
to
2022
2022
2022
as well. There are also
1011
1011
1011
black boxes and white boxes each. In each box we put two balls that are the same color as the the box. Prove that no matter how the balls are distributed, we can always pick one ball from each box such that the
2022
2022
2022
balls we chose have all the numbers from
1
1
1
to
2022
2022
2022
.
5
1
Hide problems
2022 TMO Geometry
Let
J
J
J
be the
A
A
A
-excenter of an acute triangle
A
B
C
ABC
A
BC
. Let
X
X
X
,
Y
Y
Y
be two points on the circumcircle of the triangle
A
C
J
ACJ
A
C
J
such that
J
X
‾
=
J
Y
‾
<
J
C
‾
\overline{JX} = \overline{JY} < \overline{JC}
J
X
=
J
Y
<
J
C
. Let
P
P
P
be a point lies on
X
Y
XY
X
Y
such that
P
B
PB
PB
is tangent to the circumcircle of the triangle
A
B
C
ABC
A
BC
. Let
Q
Q
Q
be a point lies on the circumcircle of the triangle
B
X
Y
BXY
BX
Y
such that
B
Q
BQ
BQ
is parallel to
A
C
AC
A
C
. Prove that
∠
B
A
P
=
∠
Q
A
C
\angle BAP = \angle QAC
∠
B
A
P
=
∠
Q
A
C
.Proposed by Li4.
4
1
Hide problems
Baby game
Two babies A and B are playing a game with
2022
2022
2022
bottles of milk. Each bottle has a maximum capacity of
200
200
200
ml, and initially each bottle holds
30
30
30
ml of milk. Starting from A, they take turns and do one of the following: (1) Pick a bottle with at least
100
100
100
ml of milk, and drink half of it. (2) Pick two bottles with less than
100
100
100
ml of milk, pour the milk of one bottle into the other one, and toss away the empty bottle. Whoever cannot do any operations loses the game. Who has a winning strategy?Proposed by Chu-Lan Kao and usjl
3
1
Hide problems
L^1 isometry
Find all functions
f
,
g
:
R
2
→
R
f,g:\mathbb{R}^2\to\mathbb{R}
f
,
g
:
R
2
→
R
satisfying that
∣
f
(
a
,
b
)
−
f
(
c
,
d
)
∣
+
∣
g
(
a
,
b
)
−
g
(
c
,
d
)
∣
=
∣
a
−
c
∣
+
∣
b
−
d
∣
|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|
∣
f
(
a
,
b
)
−
f
(
c
,
d
)
∣
+
∣
g
(
a
,
b
)
−
g
(
c
,
d
)
∣
=
∣
a
−
c
∣
+
∣
b
−
d
∣
for all real numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
.Proposed by usjl
1
1
Hide problems
Coprime integers with product being perfect squares
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be three positive integers with
gcd
(
x
,
y
,
z
)
=
1
\gcd(x,y,z)=1
g
cd
(
x
,
y
,
z
)
=
1
. If
x
∣
y
z
(
x
+
y
+
z
)
,
x\mid yz(x+y+z),
x
∣
yz
(
x
+
y
+
z
)
,
y
∣
x
z
(
x
+
y
+
z
)
,
y\mid xz(x+y+z),
y
∣
x
z
(
x
+
y
+
z
)
,
z
∣
x
y
(
x
+
y
+
z
)
,
z\mid xy(x+y+z),
z
∣
x
y
(
x
+
y
+
z
)
,
and
x
+
y
+
z
∣
x
y
z
,
x+y+z\mid xyz,
x
+
y
+
z
∣
x
yz
,
show that
x
y
z
(
x
+
y
+
z
)
xyz(x+y+z)
x
yz
(
x
+
y
+
z
)
is a perfect square.Proposed by usjl