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Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2001 Mongolian Mathematical Olympiad
2001 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(6)
Problem 6
2
Hide problems
marking cells on a board, maxmizing # of marked cells
Some cells of a
10
×
10
10\times10
10
×
10
board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the maximum possible number of marked cells.
minimization of # of upset wins in tournament
In a tennis tournament, any two of the
n
n
n
participants played a match (the winner of a match gets
1
1
1
point, the loser gets
0
0
0
). The scores after the tournament were
r
1
≤
r
2
≤
…
≤
r
n
r_1\le r_2\le\ldots\le r_n
r
1
≤
r
2
≤
…
≤
r
n
. A match between two players is called wrong if after it the winner has a score less than or equal to that of the loser. Consider the set
I
=
{
i
∣
r
1
≥
i
}
I=\{i|r_1\ge i\}
I
=
{
i
∣
r
1
≥
i
}
. Prove that the number of wrong matches is not less than
∑
i
∈
I
(
r
i
−
i
+
1
)
\sum_{i\in I}(r_i-i+1)
∑
i
∈
I
(
r
i
−
i
+
1
)
, and show that this value is realizable
Problem 5
2
Hide problems
similar to Pascal's Theorem
Let
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E,F
A
,
B
,
C
,
D
,
E
,
F
be the midpoints of consecutive sides of a hexagon with parallel opposite sides. Prove that the points
A
B
∩
D
E
AB\cap DE
A
B
∩
D
E
,
B
C
∩
E
F
BC\cap EF
BC
∩
EF
,
A
C
∩
D
F
AC\cap DF
A
C
∩
D
F
lie on a line.
geometry config in a circle
Chords
A
C
AC
A
C
and
B
D
BD
B
D
of a circle
w
w
w
intersect at
E
E
E
. A circle that is internally tangent to
w
w
w
at a point
F
F
F
also touches the segments
D
E
DE
D
E
and
E
C
EC
EC
. Prove that the bisector of
∠
A
F
B
\angle AFB
∠
A
FB
passes through the incenter of
△
A
E
B
\triangle AEB
△
A
EB
.
Problem 4
2
Hide problems
coloring of points on a line
On a line are given
n
>
3
n>3
n
>
3
points. Find the number of colorings of these points in red and blue, such that in any set of consequent points the difference between the numbers of red and blue points does not exceed
2
2
2
.
mark cells in 2nx2n board with even # of neighbors
Some cells of a
2
n
×
2
n
2n\times2n
2
n
×
2
n
board are marked so that each cell has an even number of neighboring (i.e. sharing a side) marked cells. Find the number of such markings.
Problem 3
2
Hide problems
m|a^(n-1)-b^(n-1) and n|a^(m-1)-b^(m-1), infinitude of solns.
Let
a
,
b
a,b
a
,
b
be coprime positive integers with
a
a
a
even and
a
>
b
a>b
a
>
b
. Show that there exist infinitely many pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of coprime positive integers such that
m
∣
a
n
−
1
−
b
n
−
1
m\mid a^{n-1}-b^{n-1}
m
∣
a
n
−
1
−
b
n
−
1
and
n
∣
a
m
−
1
−
b
m
−
1
n\mid a^{m-1}-b^{m-1}
n
∣
a
m
−
1
−
b
m
−
1
.
m|n^d+1, find (m^(2^k)-1)/d if integer
Let
k
≥
0
k\ge0
k
≥
0
be a given integer. Suppose there exists positive integer
n
,
d
n,d
n
,
d
and an odd integer
m
>
1
m>1
m
>
1
with
d
∣
m
2
k
−
1
d\mid m^{2^k}-1
d
∣
m
2
k
−
1
and
m
∣
n
d
+
1
m\mid n^d+1
m
∣
n
d
+
1
. Find all possible values of
m
2
k
−
1
d
\frac{m^{2^k}-1}d
d
m
2
k
−
1
.
Problem 2
2
Hide problems
inequality in triangle with medians
In an acute-angled triangle
A
B
C
ABC
A
BC
,
a
,
b
,
c
a,b,c
a
,
b
,
c
are sides,
m
a
,
m
b
,
m
c
m_a,m_b,m_c
m
a
,
m
b
,
m
c
the corresponding medians,
R
R
R
the circumradius and
r
r
r
the inradius. Prove the inequality
a
2
+
b
2
a
+
b
⋅
b
2
+
c
2
b
+
c
⋅
a
2
+
c
2
a
+
c
≥
16
R
2
r
m
a
a
⋅
m
b
b
⋅
m
c
c
.
\frac{a^2+b^2}{a+b}\cdot\frac{b^2+c^2}{b+c}\cdot\frac{a^2+c^2}{a+c}\ge16R^2r\frac{m_a}a\cdot\frac{m_b}b\cdot\frac{m_c}c.
a
+
b
a
2
+
b
2
⋅
b
+
c
b
2
+
c
2
⋅
a
+
c
a
2
+
c
2
≥
16
R
2
r
a
m
a
⋅
b
m
b
⋅
c
m
c
.
inequality in two sequences
For positive real numbers
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots,b_n
b
1
,
b
2
,
…
,
b
n
define
a
1
=
b
1
b
1
+
b
2
+
…
+
b
n
and
a
k
=
b
1
+
…
+
b
k
b
1
+
…
+
b
k
−
1
for
k
>
1.
a_1=\frac{b_1}{b_1+b_2+\ldots+b_n}\enspace\text{ and }\enspace a_k=\frac{b_1+\ldots+b_k}{b_1+\ldots+b_{k-1}}\text{ for }k>1.
a
1
=
b
1
+
b
2
+
…
+
b
n
b
1
and
a
k
=
b
1
+
…
+
b
k
−
1
b
1
+
…
+
b
k
for
k
>
1.
Prove that
a
1
+
a
2
+
…
+
a
n
≤
1
a
1
+
1
a
2
+
…
+
1
a
n
a_1+a_2+\ldots+a_n\le\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_n}
a
1
+
a
2
+
…
+
a
n
≤
a
1
1
+
a
2
1
+
…
+
a
n
1
Problem 1
2
Hide problems
Recurrence relation, maximum of x1
Suppose that a sequence
x
1
,
x
2
,
…
,
x
2001
x_1,x_2,\ldots,x_{2001}
x
1
,
x
2
,
…
,
x
2001
of positive real numbers satisfies
3
x
n
+
1
2
=
7
x
n
x
n
+
1
−
3
x
n
+
1
−
2
x
n
2
+
x
n
and
x
37
=
x
2001
.
3x^2_{n+1}=7x_nx_{n+1}-3x_{n+1}-2x^2_n+x_n\enspace\text{ and }\enspace x_{37}=x_{2001}.
3
x
n
+
1
2
=
7
x
n
x
n
+
1
−
3
x
n
+
1
−
2
x
n
2
+
x
n
and
x
37
=
x
2001
.
Find the maximum possible value of
x
1
x_1
x
1
.
polynomial FE, prove existence in p(x)p(4-x)=p(x(4-x))
Prove that for every positive integer
n
n
n
there exists a polynomial
p
(
x
)
p(x)
p
(
x
)
of degree
n
n
n
with real coefficients, having
n
n
n
distinct real roots and satisfying
p
(
x
)
p
(
4
−
x
)
=
p
(
x
(
4
−
x
)
)
p(x)p(4-x)=p(x(4-x))
p
(
x
)
p
(
4
−
x
)
=
p
(
x
(
4
−
x
))