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Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2023 Mongolian Mathematical Olympiad
2023 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(3)
3
2
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Directed graph
Five girls and five boys took part in a competition. Suppose that we can number the boys and girls
1
,
2
,
3
,
4
,
5
1, 2, 3, 4, 5
1
,
2
,
3
,
4
,
5
such that for each
1
≤
i
,
j
≤
5
1 \leq i,j \leq 5
1
≤
i
,
j
≤
5
, there are exactly
∣
i
−
j
∣
|i-j|
∣
i
−
j
∣
contestants that the girl numbered
i
i
i
and the boy numbered
j
j
j
both know. Let
a
i
a_i
a
i
and
b
i
b_i
b
i
be the number of contestants that the girl numbered
i
i
i
knows and the number of contestants that the boy numbered
i
i
i
knows respectively. Find the minimum value of
max
(
∑
i
=
1
5
a
i
,
∑
i
=
1
5
b
i
)
\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)
max
(
i
=
1
∑
5
a
i
,
i
=
1
∑
5
b
i
)
. (Note that for a pair of contestants
A
A
A
and
B
B
B
,
A
A
A
knowing
B
B
B
doesn't mean that
B
B
B
knows
A
A
A
and a contestant cannot know themself.)
An absolutely breathtaking number theory
Let
m
m
m
be a positive integer. We say that a sequence of positive integers written on a circle is good , if the sum of any
m
m
m
consecutive numbers on this circle is a power of
m
m
m
.1. Let
n
≥
2
n \geq 2
n
≥
2
be a positive integer. Prove that for any good sequence with
m
n
mn
mn
numbers, we can remove
m
m
m
numbers such that the remaining
m
n
−
m
mn-m
mn
−
m
numbers form a good sequence.2. Prove that in any good sequence with
m
2
m^2
m
2
numbers, we can always find a number that was repeated at least
m
m
m
times in the sequence.
2
1
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Easy geo
In an acute triangle
A
B
C
ABC
A
BC
the points
D
,
E
D, E
D
,
E
are the feet of the altitudes through
B
,
C
B, C
B
,
C
respectively. Let
L
L
L
be the point on segment
B
D
BD
B
D
such that
A
D
=
D
L
AD=DL
A
D
=
D
L
. Similarly, let
K
K
K
be the point on segment
C
E
CE
CE
such that
A
E
=
E
K
AE=EK
A
E
=
E
K
. Let
M
M
M
be the midpoint of
K
L
KL
K
L
. The circumcircle of
A
B
C
ABC
A
BC
intersect the lines
A
L
AL
A
L
and
A
K
AK
A
K
for a second time at
T
,
S
T, S
T
,
S
respectively. Prove that the lines
B
S
,
C
T
,
A
M
BS, CT, AM
BS
,
CT
,
A
M
are concurrent.
1
2
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FE with two functions
Find all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
and
h
:
R
2
→
R
h : \mathbb{R}^2 \to \mathbb{R}
h
:
R
2
→
R
such that
f
(
x
+
y
−
z
)
2
=
f
(
x
y
)
+
h
(
x
+
y
+
z
,
x
y
+
y
z
+
z
x
)
f(x+y-z)^2=f(xy)+h(x+y+z, xy+yz+zx)
f
(
x
+
y
−
z
)
2
=
f
(
x
y
)
+
h
(
x
+
y
+
z
,
x
y
+
yz
+
z
x
)
for all real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
.
Free inequality
Let
u
,
v
u, v
u
,
v
be arbitrary positive real numbers. Prove that
min
(
u
,
100
v
,
v
+
2023
u
)
≤
2123
.
\min{(u, \frac{100}{v}, v+\frac{2023}{u})} \leq \sqrt{2123}.
min
(
u
,
v
100
,
v
+
u
2023
)
≤
2123
.