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Problems
Contests
National and Regional Contests
Albania Contests
Kosovo & Albania Mathematical Olympiad
2022 Kosovo & Albania Mathematical Olympiad
2022 Kosovo & Albania Mathematical Olympiad
Part of
Kosovo & Albania Mathematical Olympiad
Subcontests
(5)
0
1
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Positive integer solutions to inequality
Let
a
>
0
a>0
a
>
0
. If the inequality
22
<
a
x
<
222
22<ax<222
22
<
a
x
<
222
holds for precisely
10
10
10
positive integers
x
x
x
, find how many positive integers satisfy the inequality
222
<
a
x
<
2022
222<ax<2022
222
<
a
x
<
2022
? Note: The first 8 problems of the competition are questions which the contestants are expected to solve quickly and only write the answer of. This problem turned out to be a lot more difficult than anticipated for an answer-only question.
4
2
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Diophantine equation over a restricted set of integers
Let
A
A
A
be the set of natural numbers
n
n
n
such that the distance of the real number
n
2022
−
1
3
n\sqrt{2022} - \frac13
n
2022
−
3
1
from the nearest integer is at most
1
2022
\frac1{2022}
2022
1
. Show that the equation
20
x
+
21
y
=
22
z
20x + 21y = 22z
20
x
+
21
y
=
22
z
has no solutions over the set
A
A
A
.
Counting lines forming a small angle
Consider
n
>
9
n>9
n
>
9
lines on the plane such that no two lines are parallel. Show that there exist at least
n
/
9
n/9
n
/9
lines such that the angle between any two of the lines is at most
2
0
∘
20^\circ
2
0
∘
.
3
2
Hide problems
Equivalence between a right angle and sum of two segments in a square
Let
A
B
C
D
ABCD
A
BC
D
be a square and let
M
M
M
be the midpoint of
B
C
BC
BC
. Let
X
X
X
and
Y
Y
Y
be points on the segments
A
B
AB
A
B
and
C
D
CD
C
D
, respectively. Prove that
∠
X
M
Y
=
9
0
∘
\angle XMY = 90^\circ
∠
XM
Y
=
9
0
∘
if and only if
B
X
+
C
Y
=
X
Y
BX + CY = XY
BX
+
C
Y
=
X
Y
. Note: In the competition, students were only asked to prove the 'only if' direction.
Difference of sum of squares of parts of a bipartition
Is it possible to partition
{
1
,
2
,
3
,
…
,
28
}
\{1, 2, 3, \ldots, 28\}
{
1
,
2
,
3
,
…
,
28
}
into two sets
A
A
A
and
B
B
B
such that both of the following conditions hold simultaneously: (i) the number of odd integers in
A
A
A
is equal to the number of odd integers in
B
B
B
;(ii) the difference between the sum of squares of the integers in
A
A
A
and the sum of squares of the integers in
B
B
B
is
16
16
16
?
2
2
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Coloring a grid
Consider a
5
×
5
5\times 5
5
×
5
grid with
25
25
25
cells. What is the least number of cells that should be colored, such that every
2
×
3
2\times 3
2
×
3
or
3
×
2
3\times 2
3
×
2
rectangle in the grid has at least two colored cells?
Relation in a convex quadrilateral formed from an acute triangle
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
D
D
D
be a point on the line parallel to
A
C
AC
A
C
that passes through
B
B
B
, such that
∠
B
D
C
=
2
∠
B
A
C
\angle BDC = 2\angle BAC
∠
B
D
C
=
2∠
B
A
C
as well as such that
A
B
D
C
ABDC
A
B
D
C
is a convex quadrilateral. Show that
B
D
+
D
C
=
A
C
BD + DC = AC
B
D
+
D
C
=
A
C
.
1
2
Hide problems
Easy algebra question
If
(
2
x
−
4
x
)
+
(
2
−
x
−
4
−
x
)
=
3
(2^x - 4^x) + (2^{-x} - 4^{-x}) = 3
(
2
x
−
4
x
)
+
(
2
−
x
−
4
−
x
)
=
3
, find the numerical value of the expression
(
8
x
+
3
⋅
2
x
)
+
(
8
−
x
+
3
⋅
2
−
x
)
.
(8^x + 3\cdot 2^x) + (8^{-x} + 3\cdot 2^{-x}).
(
8
x
+
3
⋅
2
x
)
+
(
8
−
x
+
3
⋅
2
−
x
)
.
Easy diophantine equation
Find all pairs of integers
(
m
,
n
)
(m, n)
(
m
,
n
)
such that
m
+
n
=
3
(
m
n
+
10
)
.
m+n = 3(mn+10).
m
+
n
=
3
(
mn
+
10
)
.