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Problems
Contests
International Contests
Caucasus Mathematical Olympiad
2024 Caucasus Mathematical Olympiad
2024 Caucasus Mathematical Olympiad
Part of
Caucasus Mathematical Olympiad
Subcontests
(7)
6
1
Hide problems
Permuting numbers in a table
The integers from
1
1
1
to
320000
320000
320000
are placed in the cells of a
8
×
40000
8 \times 40000
8
×
40000
board. Prove that it is possible to permute the rows of the table so that the numbers in each column will not be sorted from the top to the bottom in increasing order.
1
2
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Boxes and colored balls
Balls of
3
3
3
colours — red, blue and white — are placed in two boxes. If you take out
3
3
3
balls from the first box, there would definitely be a blue one among them. If you take out
4
4
4
balls from the second box, there would definitely be a red one among them. If you take out any
5
5
5
balls (only from the first, only from the second, or from two boxes at the same time), then there would definitely be a white ball among them. Find the greatest possible total number of balls in two boxes.
Easy algebra on 4 positive reals
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive real numbers. It is given that at least one of the following two conditions holds:
a
b
>
min
(
c
d
,
d
c
)
,
c
d
>
min
(
a
b
,
b
a
)
.
ab >\min(\frac{c}{d}, \frac{d}{c}), cd >\min(\frac{a}{b}, \frac{b}{a}).
ab
>
min
(
d
c
,
c
d
)
,
c
d
>
min
(
b
a
,
a
b
)
.
Show that at least one of the following two conditions holds:
b
d
>
min
(
c
a
,
a
c
)
,
c
a
>
min
(
d
b
,
b
d
)
.
bd>\min(\frac{c}{a}, \frac{a}{c}), ca >\min(\frac{d}{b}, \frac{b}{d}).
b
d
>
min
(
a
c
,
c
a
)
,
c
a
>
min
(
b
d
,
d
b
)
.
2
2
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Easy perpendicularity
The rhombuses
A
B
D
K
ABDK
A
B
DK
and
C
B
E
L
CBEL
CBE
L
are arranged so that
B
B
B
lies on the segment
A
C
AC
A
C
and
E
E
E
lies on the segment
B
D
BD
B
D
. Point
M
M
M
is the midpoint of
K
L
KL
K
L
. Prove that
∠
D
M
E
=
9
0
∘
\angle DME=90^{\circ}
∠
D
ME
=
9
0
∘
.
Easy conditional geometry
In an acute-angled triangle
A
B
C
ABC
A
BC
let
B
L
BL
B
L
be the bisector, and let
B
K
BK
B
K
be the altitude. Let the lines
B
L
BL
B
L
and
B
K
BK
B
K
meet the circumcircle of
A
B
C
ABC
A
BC
again at
W
W
W
and
T
T
T
, respectively. Given that
B
C
=
B
W
BC = BW
BC
=
B
W
, prove that
T
L
⊥
B
C
TL \perp BC
T
L
⊥
BC
.
3
2
Hide problems
Product of 10 factorials is 10-th power
Given
10
10
10
positive integers with a sum equal to
1000
1000
1000
. The product of their factorials is a
10
10
10
-th power of an integer. Prove that all these numbers are equal.
Product of n and its reverse equals 888...8
Let
n
n
n
be a
d
d
d
-digit (i.e., having
d
d
d
digits in its decimal representation) positive integer not divisible by
10
10
10
. Writing all the digits of
n
n
n
in reverse order, we obtain the number
n
′
n'
n
′
. Determine if it is possible that the decimal representation of the product
n
⋅
n
′
n\cdot n'
n
⋅
n
′
consists of digits
8
8
8
only, if (a)
d
=
9998
d = 9998
d
=
9998
; (b)
d
=
9999
?
d = 9999?
d
=
9999
?
4
2
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Existence of a line intersecting many segments
Given a set
P
P
P
of
n
>
100
n>100
n
>
100
points on the plane such that no three of them are collinear, and a set
S
S
S
of
20
n
20n
20
n
distinct segments, each joining a pair of points from
P
P
P
. Prove that there exists a line not passing through a point from
P
P
P
and intersecting at least
200
200
200
segments from
S
S
S
.
Maximal guaranteed sum of numbers in non-adjacent cells
Yasha writes in the cells of the table
99
×
99
99 \times 99
99
×
99
all positive integers from 1 to
9
9
2
99^2
9
9
2
(each number once). Grisha looks at the table and selects several cells, among which there are no two cells sharing a common side, and then sums up the numbers in all selected cells. Find the largest sum Grisha can guarantee to achieve.
5
2
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Ratio of two products
Alex calculated the value of function
f
(
n
)
=
n
2
+
n
+
1
f(n) = n^2 + n + 1
f
(
n
)
=
n
2
+
n
+
1
for each integer from
1
1
1
to
100
100
100
. Marina calculated the value of function
g
(
n
)
=
n
2
−
n
+
1
g(n) = n^2-n+1
g
(
n
)
=
n
2
−
n
+
1
for the same numbers. Who of them has greater product of values and what is their ratio?
Easy algebra
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be reals and consider three lines
y
=
a
x
+
b
,
y
=
b
x
+
c
,
y
=
c
x
+
a
y=ax+b, y=bx+c, y=cx+a
y
=
a
x
+
b
,
y
=
b
x
+
c
,
y
=
c
x
+
a
. Two of these lines meet at a point with
x
x
x
-coordinate
1
1
1
. Show that the third one passes through a point with two integer coordinates.
8
1
Hide problems
Incircle geometry with locus
Let
A
B
C
ABC
A
BC
be an acute triangle and let
X
X
X
be a variable point on
A
C
AC
A
C
. The incircle of
△
A
B
X
\triangle ABX
△
A
BX
touches
A
X
,
B
X
AX, BX
A
X
,
BX
at
K
,
P
K, P
K
,
P
, respectively. The incircle of
△
B
C
X
\triangle BCX
△
BCX
touches
C
X
,
B
X
CX, BX
CX
,
BX
at
L
,
Q
L, Q
L
,
Q
, respectively. Find the locus of
K
P
∩
L
Q
KP \cap LQ
K
P
∩
L
Q
.