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Contests
International Contests
Caucasus Mathematical Olympiad
2021 Caucasus Mathematical Olympiad
2021 Caucasus Mathematical Olympiad
Part of
Caucasus Mathematical Olympiad
Subcontests
(7)
3
1
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An estimate for the number of triangles with vertices in given points
Let
n
≥
3
n\ge 3
n
≥
3
be a positive integer. In the plane
n
n
n
points which are not all collinear are marked. Find the least possible number of triangles whose vertices are all marked. (Recall that the vertices of a triangle are not collinear.)
8
1
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Number of elements is equal to the average of all its elements
Let us call a set of positive integers nice, if its number of elements is equal to the average of all its elements. Call a number
n
n
n
amazing, if one can partition the set
{
1
,
2
,
…
,
n
}
\{1,2,\ldots,n\}
{
1
,
2
,
…
,
n
}
into nice subsets. a) Prove that any perfect square is amazing.b) Prove that there exist infinitely many positive integers which are not amazing.
7
1
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One more construction involving H and O
An acute triangle
A
B
C
ABC
A
BC
is given. Let
A
D
AD
A
D
be its altitude, let
H
H
H
and
O
O
O
be its orthocenter and its circumcenter, respectively. Let
K
K
K
be the point on the segment
A
H
AH
A
H
with
A
K
=
H
D
AK=HD
A
K
=
HD
; let
L
L
L
be the point on the segment
C
D
CD
C
D
with
C
L
=
D
B
CL=DB
C
L
=
D
B
. Prove that line
K
L
KL
K
L
passes through
O
O
O
.
5
2
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The product of pairwise gcd-s is a perfect square
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive integers such that the product
gcd
(
a
,
b
)
⋅
gcd
(
b
,
c
)
⋅
gcd
(
c
,
a
)
\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a)
g
cd
(
a
,
b
)
⋅
g
cd
(
b
,
c
)
⋅
g
cd
(
c
,
a
)
is a perfect square. Prove that the product
lcm
(
a
,
b
)
⋅
lcm
(
b
,
c
)
⋅
lcm
(
c
,
a
)
\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a)
lcm
(
a
,
b
)
⋅
lcm
(
b
,
c
)
⋅
lcm
(
c
,
a
)
is also a perfect square.
No triangle with heights as sides
A triangle
Δ
\Delta
Δ
with sidelengths
a
≤
b
≤
c
a\leq b\leq c
a
≤
b
≤
c
is given. It appears that it is impossible to construct a triangle from three segments whose lengths are equal to the altitudes of
Δ
\Delta
Δ
. Prove that
b
2
>
a
c
b^2>ac
b
2
>
a
c
.
4
2
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Another problem on a square grid
A square grid
2
n
×
2
n
2n \times 2n
2
n
×
2
n
is constructed of matches (each match is a segment of length 1). By one move Peter can choose a vertex which (at this moment) is the endpoint of 3 or 4 matches and delete two matches whose union is a segment of length 2. Find the least possible number of matches that could remain after a number of Peter's moves.
The reflections of the vertices and the intersections of circumcircle with HaHb
In an acute triangle
A
B
C
ABC
A
BC
let
A
H
a
AH_a
A
H
a
and
B
H
b
BH_b
B
H
b
be altitudes. Let
H
a
H
b
H_aH_b
H
a
H
b
intersect the circumcircle of
A
B
C
ABC
A
BC
at
P
P
P
and
Q
Q
Q
. Let
A
′
A'
A
′
be the reflection of
A
A
A
in
B
C
BC
BC
, and let
B
′
B'
B
′
be the reflection of
B
B
B
in
C
A
CA
C
A
. Prove that
A
′
,
B
′
A', B'
A
′
,
B
′
,
P
P
P
,
Q
Q
Q
are concyclic.
2
1
Hide problems
A point on the median
In a triangle
A
B
C
ABC
A
BC
let
K
K
K
be a point on the median
B
M
BM
BM
such that
C
K
=
C
M
CK=CM
C
K
=
CM
. It appears that
∠
C
B
M
=
2
∠
A
B
M
\angle CBM = 2 \angle ABM
∠
CBM
=
2∠
A
BM
. Prove that
B
C
=
M
K
BC=MK
BC
=
M
K
.
1
2
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On a^2+b=c^2
Let
a
a
a
,
b
b
b
,
c
c
c
be real numbers such that
a
2
+
b
=
c
2
a^2+b=c^2
a
2
+
b
=
c
2
,
b
2
+
c
=
a
2
b^2+c=a^2
b
2
+
c
=
a
2
,
c
2
+
a
=
b
2
c^2+a=b^2
c
2
+
a
=
b
2
. Find all possible values of
a
b
c
abc
ab
c
.
Numbers that are greater than all its right\left neighbours
Integers from 1 to 100 are placed in a row in some order. Let us call a number large-right, if it is greater than each number to the right of it; let us call a number large-left, is it is greater than each number to the left of it. It appears that in the row there are exactly
k
k
k
large-right numbers and exactly
k
k
k
large-left numbers. Find the maximal possible value of
k
k
k
.