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Problems
Contests
National and Regional Contests
Mongolia Contests
Mongolian Mathematical Olympiad
2002 Mongolian Mathematical Olympiad
2002 Mongolian Mathematical Olympiad
Part of
Mongolian Mathematical Olympiad
Subcontests
(6)
Problem 6
2
Hide problems
prove angles equal given side ratios
Let
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
be the midpoints of the sides
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively of a triangle
A
B
C
ABC
A
BC
. Points
K
K
K
on segment
C
1
A
1
C_1A_1
C
1
A
1
and
L
L
L
on segment
A
1
B
1
A_1B_1
A
1
B
1
are taken such that
C
1
K
K
A
1
=
B
C
+
A
C
A
C
+
A
B
and
A
1
L
L
B
1
=
A
C
+
A
B
B
C
+
A
B
.
\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.
K
A
1
C
1
K
=
A
C
+
A
B
BC
+
A
C
and
L
B
1
A
1
L
=
BC
+
A
B
A
C
+
A
B
.
If
B
K
BK
B
K
and
C
L
CL
C
L
meet at
S
S
S
, prove that
∠
C
1
A
1
S
=
∠
B
1
A
1
S
\angle C_1A_1S=\angle B_1A_1S
∠
C
1
A
1
S
=
∠
B
1
A
1
S
.
stingin two squares connected by a curve
Two squares of area
38
38
38
are given. Each of the squares is divided into
38
38
38
connected pieces of unit area by simple curves. Then the two squares are patched together. Show that one can sting the patched squares with
38
38
38
needles so that every piece of each square is stung exactly once.
Problem 5
2
Hide problems
inequality in sequences
Let
a
0
,
a
1
,
…
a_0,a_1,\ldots
a
0
,
a
1
,
…
be an infinite sequence of positive numbers. Prove that the inequality
1
+
a
n
>
2
n
a
n
−
1
1+a_n>\sqrt[n]2a_{n-1}
1
+
a
n
>
n
2
a
n
−
1
holds for infinitely many positive integers
n
n
n
.
cyclic pentagon, ratio of product of sides to diagonals
Let
A
A
A
be the ratio of the product of sides to the product of diagonals in a circumscribed pentagon. Find the maximum possible value of
A
A
A
.
Problem 4
2
Hide problems
choose two natural numbers whose product is a square
Let there be
131
131
131
given distinct natural numbers, each having prime divisors not exceeding
42
42
42
. Prove that one can choose four of them whose product is a perfect square.
Fermat-like nondivisibility
Let
p
≥
5
p\ge5
p
≥
5
be a prime number. Prove that there exists
a
∈
{
1
,
2
,
…
,
p
−
2
}
a\in\{1,2,\ldots,p-2\}
a
∈
{
1
,
2
,
…
,
p
−
2
}
satisfying
p
2
∤
a
p
−
1
−
1
p^2\nmid a^{p-1}-1
p
2
∤
a
p
−
1
−
1
and
p
2
∤
(
a
+
1
)
p
−
1
−
1
p^2\nmid(a+1)^{p-1}-1
p
2
∤
(
a
+
1
)
p
−
1
−
1
.
Problem 3
2
Hide problems
prove that lines are concurrent with incircle intouch points
The incircle of a triangle
A
B
C
ABC
A
BC
with
A
B
≠
B
C
AB\ne BC
A
B
=
BC
touches
B
C
BC
BC
at
A
1
A_1
A
1
and
A
C
AC
A
C
at
B
1
B_1
B
1
. The segments
A
A
1
AA_1
A
A
1
and
B
B
1
BB_1
B
B
1
meet the incircle at
A
2
A_2
A
2
and
B
2
B_2
B
2
, respectively. Prove that the lines
A
B
,
A
1
B
1
,
A
2
B
2
AB,A_1B_1,A_2B_2
A
B
,
A
1
B
1
,
A
2
B
2
are concurrent.
sets equal in terms of integral parameter
Find all positive integer
n
n
n
for which there exist real number
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
such that
{
a
j
−
a
i
∣
1
≤
i
<
j
≤
n
}
=
{
1
,
2
,
…
,
n
(
n
−
1
)
2
}
.
\{a_j-a_i|1\le i<j\le n\}=\left\{1,2,\ldots,\frac{n(n-1)}2\right\}.
{
a
j
−
a
i
∣1
≤
i
<
j
≤
n
}
=
{
1
,
2
,
…
,
2
n
(
n
−
1
)
}
.
Problem 2
2
Hide problems
moving between lattice point if distance=p
For a natural number
p
p
p
, one can move between two points with integer coordinates if the distance between them equals
p
p
p
. Find all prime numbers
p
p
p
for which it is possible to reach the point
(
2002
,
38
)
(2002,38)
(
2002
,
38
)
starting from the origin
(
0
,
0
)
(0,0)
(
0
,
0
)
.
irreducible: (x^2+x)^(2^n)+1
Prove that for each
n
∈
N
n\in\mathbb N
n
∈
N
the polynomial
(
x
2
+
x
)
2
n
+
1
(x^2+x)^{2^n}+1
(
x
2
+
x
)
2
n
+
1
is irreducible over the polynomials with integer coefficients.
Problem 1
1
Hide problems
minimal cardinality of a set such that there exist subsets
Let
n
,
k
n,k
n
,
k
be given natural numbers. Find the smallest possible cardinality of a set
A
A
A
with the following property: There exist subsets
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\ldots,A_n
A
1
,
A
2
,
…
,
A
n
of
A
A
A
such that the union of any
k
k
k
of them is
A
A
A
, but the union of any
k
−
1
k-1
k
−
1
of them is never
A
A
A
.