MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia TST
2020 Latvia TST
2020 Latvia TST
Part of
Latvia TST
Subcontests
(5)
1.5
1
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Combi with primes
Given a
6
×
6
6\times 6
6
×
6
square consisting of unit squares, denote its rows and columns from
1
1
1
to
6
6
6
. Figure p-horse can move from square
(
x
;
y
)
(x; y)
(
x
;
y
)
to
(
x
’
;
y
’
)
(x’; y’)
(
x
’
;
y
’
)
if and only if both
x
+
x
’
x + x’
x
+
x
’
and
y
+
y
’
y + y’
y
+
y
’
are primes. At the start the p-horse is located in one of the unit squares.
a
)
a)
a
)
Can the p-horse visit every unit square exactly once?
b
b
b
) Can the p-horse visit every unit square exactly once and with the last move return to the initial starting position?
1.1
1
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Parallelograms
It is given parallelogram
A
B
C
D
ABCD
A
BC
D
. On it's sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
are chosen points
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
such that area of
E
F
G
H
EFGH
EFG
H
is half of the area of
A
B
C
D
ABCD
A
BC
D
. Show that at least one of the quadrilaterals
A
B
F
H
ABFH
A
BF
H
and
A
E
G
D
AEGD
A
EG
D
is parallelogram.
1.4
1
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Perpendicular lines
It is given isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB = AC
A
B
=
A
C
.
A
D
AD
A
D
is diameter of circumcircle of triangle
A
B
C
ABC
A
BC
. On the side
B
C
BC
BC
is chosen point
E
E
E
. On the sides
A
C
,
A
B
AC, AB
A
C
,
A
B
there are points
F
,
G
F, G
F
,
G
respectively such that
A
F
E
G
AFEG
A
FEG
is parallelogram. Prove that
D
E
DE
D
E
is perpendicular to
F
G
FG
FG
.
1.2
1
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Sequence
Sequences
(
a
n
)
,
(
b
n
)
(a_n), (b_n)
(
a
n
)
,
(
b
n
)
are defined by
a
1
=
1
,
b
1
=
2
a_1 = 1, b_1 = 2
a
1
=
1
,
b
1
=
2
,
a
n
+
1
=
1
+
a
n
+
a
n
b
n
b
n
a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}
a
n
+
1
=
b
n
1
+
a
n
+
a
n
b
n
,
b
n
+
1
=
1
+
b
n
+
a
n
b
n
a
n
b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}
b
n
+
1
=
a
n
1
+
b
n
+
a
n
b
n
for all positive integers
n
n
n
. Prove that
a
2020
<
5
a_{2020} < 5
a
2020
<
5
.
1.3
1
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Infinitely many solutions
Prove that equation
a
2
−
b
2
=
a
b
−
1
a^2 - b^2=ab - 1
a
2
−
b
2
=
ab
−
1
has infinitely many solutions, if
a
,
b
a,b
a
,
b
are positive integers