MathDB
Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2024 Mongolian Mathematical Olympiad
2024 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(3)
2
2
Hide problems
Unexpectedly difficult combo geo
We call a triangle consisting of three vertices of a pentagon big if it's area is larger than half of the pentagon's area. Find the maximum number of big triangles that can be in a convex pentagon.Proposed by Gonchigdorj Sandag
Insimilicenter collinearity
Let
A
B
C
ABC
A
BC
be an acute-angled triangle and let
E
E
E
and
F
F
F
be the feet of the altitudes from
B
B
B
and
C
C
C
to the sides
A
C
AC
A
C
and
A
B
AB
A
B
respectively. Suppose
A
D
AD
A
D
is the diameter of the circle
A
B
C
ABC
A
BC
. Let
M
M
M
be the midpoint of
B
C
BC
BC
. Let
K
K
K
be the imsimilicenter of the incircles of the triangles
B
M
F
BMF
BMF
and
C
M
E
CME
CME
. Prove that the points
K
,
M
,
D
K, M, D
K
,
M
,
D
are collinear.Proposed by Bilegdembrel Bat-Amgalan.
1
2
Hide problems
Diophantine equation involving factorials
Find all triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
of positive integers such that
a
≤
b
a \leq b
a
≤
b
and
a
!
+
b
!
=
c
4
+
2024
a!+b!=c^4+2024
a
!
+
b
!
=
c
4
+
2024
Proposed by Otgonbayar Uuye.
Derivative in nationals?
Let
P
(
x
)
P(x)
P
(
x
)
and
Q
(
x
)
Q(x)
Q
(
x
)
be polynomials with nonnegative coefficients. We denote by
P
′
(
x
)
P'(x)
P
′
(
x
)
the derivative of
P
(
x
)
P(x)
P
(
x
)
. Suppose that
P
(
0
)
=
Q
(
0
)
=
0
P(0)=Q(0)=0
P
(
0
)
=
Q
(
0
)
=
0
and
Q
(
1
)
≤
1
≤
P
′
(
0
)
Q(1) \leq 1 \leq P'(0)
Q
(
1
)
≤
1
≤
P
′
(
0
)
.
(
1
)
(1)
(
1
)
Prove that
0
≤
Q
(
x
)
≤
x
≤
P
(
x
)
0 \leq Q(x) \leq x \leq P(x)
0
≤
Q
(
x
)
≤
x
≤
P
(
x
)
for all
0
≤
x
≤
1
0 \leq x \leq 1
0
≤
x
≤
1
.
(
2
)
(2)
(
2
)
Prove that
P
(
Q
(
x
)
)
≤
Q
(
P
(
x
)
)
P(Q(x)) \leq Q(P(x))
P
(
Q
(
x
))
≤
Q
(
P
(
x
))
for all
0
≤
x
≤
1
0 \leq x \leq 1
0
≤
x
≤
1
.Proposed by Otgonbayar Uuye.
3
2
Hide problems
Unexpectedly easy functional equation
Let
R
+
\mathbb{R}^+
R
+
denote the set of positive real numbers. Determine all functions
f
:
R
+
→
R
+
f: \mathbb{R}^+ \to \mathbb{R}^+
f
:
R
+
→
R
+
such that for all positive real numbers
x
x
x
and
y
y
y
:
f
(
x
)
f
(
y
+
f
(
x
)
)
=
f
(
1
+
x
y
)
f(x)f(y+f(x))=f(1+xy)
f
(
x
)
f
(
y
+
f
(
x
))
=
f
(
1
+
x
y
)
Proposed by Otgonbayar Uuye.
Subsets of a set form a complete residue system mod 2^n
A set
X
X
X
consisting of
n
n
n
positive integers is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
o
o
d
<
/
s
p
a
n
>
<span class='latex-italic'>good</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
g
oo
d
<
/
s
p
an
>
if the following condition holds: For any two different subsets of
X
X
X
, say
A
A
A
and
B
B
B
, the number
s
(
A
)
−
s
(
B
)
s(A) - s(B)
s
(
A
)
−
s
(
B
)
is not divisible by
2
n
2^n
2
n
. (Here, for a set
A
A
A
,
s
(
A
)
s(A)
s
(
A
)
denotes the sum of the elements of
A
A
A
) Given
n
n
n
, find the number of good sets of size
n
n
n
, all of whose elements is strictly less than
2
n
2^n
2
n
.