MathDB
Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2007 Mongolian Mathematical Olympiad
2007 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(6)
Problem 6
2
Hide problems
bicentric quadrilateral
Given a quadrilateral
A
B
C
D
ABCD
A
BC
D
simultaneously inscribed and circumscribed, assume that none of its diagonals or sides is a diameter of the circumscribed circle. Let
P
P
P
be the intersection point of the external bisectors of the angles near
A
A
A
and
B
B
B
. Similarly, let
Q
Q
Q
be the intersection point of the external bisectors of the angles
C
C
C
and
D
D
D
. If
J
J
J
and
O
O
O
respectively are the incenter and circumcenter of
A
B
C
D
ABCD
A
BC
D
prove that
O
J
⊥
P
Q
OJ\perp PQ
O
J
⊥
PQ
.
prove number divides sum of a^a
Let
n
=
p
1
α
1
⋯
p
s
α
s
≥
2
n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2
n
=
p
1
α
1
⋯
p
s
α
s
≥
2
. If for any
α
∈
N
\alpha\in\mathbb N
α
∈
N
,
p
i
−
1
∤
α
p_i-1\nmid\alpha
p
i
−
1
∤
α
, where
i
=
1
,
2
,
…
,
s
i=1,2,\ldots,s
i
=
1
,
2
,
…
,
s
, prove that
n
∣
∑
α
∈
Z
n
∗
α
α
n\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha}
n
∣
∑
α
∈
Z
n
∗
α
α
where
Z
n
∗
=
{
a
∈
Z
n
:
gcd
(
a
,
n
)
=
1
}
\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}
Z
n
∗
=
{
a
∈
Z
n
:
g
cd
(
a
,
n
)
=
1
}
.
Problem 5
2
Hide problems
single-player game in nxn table
Given a
n
×
n
n\times n
n
×
n
table with non-negative real entries such that the sums of entries in each column and row are equal, a player plays the following game: The step of the game consists of choosing
n
n
n
cells, no two of which share a column or a row, and subtracting the same number from each of the entries of the
n
n
n
cells, provided that the resulting table has all non-negative entries. Prove that the player can change all entries to zeros.
find area of triangle formed by segments of circumcircle
Given a point
P
P
P
in the circumcircle
ω
\omega
ω
of an equilateral triangle
A
B
C
ABC
A
BC
, prove that the segments
P
A
PA
P
A
,
P
B
PB
PB
, and
P
C
PC
PC
form a triangle
T
T
T
. Let
R
R
R
be the radius of the circumcircle
ω
\omega
ω
and let
d
d
d
be the distance between
P
P
P
and the circumcenter. Find the area of
T
T
T
.
Problem 3
2
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reflections to get point inside triangle
Let
P
P
P
be a point outside of the triangle
A
B
C
ABC
A
BC
in the plane of
A
B
C
ABC
A
BC
. Prove that by using reflections
S
A
B
S_{AB}
S
A
B
,
S
A
C
S_{AC}
S
A
C
, and
S
B
C
S_{BC}
S
BC
across the lines
A
B
AB
A
B
,
A
C
AC
A
C
, and
B
C
BC
BC
one can shift point
P
P
P
inside the triangle
A
B
C
ABC
A
BC
.
equal sets mod p
Let
p
p
p
be an odd prime number. Let
g
g
g
be a primitive root of unity modulo
p
p
p
. Find all the values of
p
p
p
such that the sets
A
=
{
k
2
+
1
:
1
≤
k
≤
p
−
1
2
}
A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}
A
=
{
k
2
+
1
:
1
≤
k
≤
2
p
−
1
}
and
B
=
{
g
m
:
1
≤
m
≤
p
−
1
2
}
B=\left\{g^m:1\le m\le\frac{p-1}2\right\}
B
=
{
g
m
:
1
≤
m
≤
2
p
−
1
}
are equal modulo
p
p
p
.
Problem 2
2
Hide problems
product of k<n such that gcd(k,n)=1
For all
n
≥
2
n\ge2
n
≥
2
, let
a
n
a_n
a
n
be the product of all coprime natural numbers less than
n
n
n
. Prove that(a)
n
∣
a
n
+
1
⇔
n
=
2
,
4
,
p
α
,
2
p
α
n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha
n
∣
a
n
+
1
⇔
n
=
2
,
4
,
p
α
,
2
p
α
(b)
n
∣
a
n
−
1
⇔
n
≠
2
,
4
,
p
α
,
2
p
α
n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha
n
∣
a
n
−
1
⇔
n
=
2
,
4
,
p
α
,
2
p
α
Here
p
p
p
is an odd prime number and
α
∈
N
\alpha\in\mathbb N
α
∈
N
.
101 segments on a line
Given
101
101
101
segments in a line, prove that there exists
11
11
11
segments meeting in
1
1
1
point or
11
11
11
segments such that every two of them are disjoint.
Problem 1
2
Hide problems
Prove that lines parallel in triangle
Let
M
M
M
be the midpoint of the side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. The bisector of the exterior angle of point
A
A
A
intersects the side
B
C
BC
BC
in
D
D
D
. Let the circumcircle of triangle
A
D
M
ADM
A
D
M
intersect the lines
A
B
AB
A
B
and
A
C
AC
A
C
in
E
E
E
and
F
F
F
respectively. If the midpoint of
E
F
EF
EF
is
N
N
N
, prove that
M
N
∥
A
D
MN\parallel AD
MN
∥
A
D
.
# of subsets of 1,2,3,...,5n with 5|sum
Find the number of subsets of the set
{
1
,
2
,
3
,
.
.
.
,
5
n
}
\{1,2,3,...,5n\}
{
1
,
2
,
3
,
...
,
5
n
}
such that the sum of the elements in each subset are divisible by
5
5
5
.
Problem 4
2
Hide problems
simple three variable
Let
a
,
b
,
c
>
0
a,b,c>0
a
,
b
,
c
>
0
. Prove that \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}
syseq, sums of two squares
If
x
,
y
,
z
∈
N
x,y,z\in\mathbb N
x
,
y
,
z
∈
N
and
x
y
=
z
2
+
1
xy=z^2+1
x
y
=
z
2
+
1
prove that there exists integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
x
=
a
2
+
b
2
x=a^2+b^2
x
=
a
2
+
b
2
,
y
=
c
2
+
d
2
y=c^2+d^2
y
=
c
2
+
d
2
,
z
=
a
c
+
b
d
z=ac+bd
z
=
a
c
+
b
d
.