MathDB
Problems
Contests
National and Regional Contests
Latvia Contests
Latvia TST
2021 Latvia TST
2021 Latvia TST
Part of
Latvia TST
Subcontests
(5)
1.4
1
Hide problems
Game on the board
Initially, on the board, all integers from
1
1
1
to
400
400
400
are written. Two players play a game alternating their moves. In one move it is allowed to erase from the board any 3 integers, which form a triangle. The player, who can not perform a move loses. Who has a winning strategy?
1.3
1
Hide problems
Isosceles triangle and strange point and angle
Given isosceles
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
B
A
C
=
2
2
∘
\angle BAC = 22^{\circ}
∠
B
A
C
=
2
2
∘
. On the side
B
C
BC
BC
point
D
D
D
is chosen such that
B
D
=
2
C
D
BD = 2CD
B
D
=
2
C
D
. The foots of perpendiculars from
B
B
B
to lines
A
D
AD
A
D
and
A
C
AC
A
C
are points
E
E
E
,
F
F
F
respectively. Find with the proof value of the angle
∠
C
E
F
\angle CEF
∠
CEF
.
1.2
1
Hide problems
Cute construction problem
Prove it is possible to find
2
2021
2^{2021}
2
2021
different pairs of positive integers
(
a
i
,
b
i
)
(a_i,b_i)
(
a
i
,
b
i
)
such that:
1
a
i
b
i
+
1
a
2
b
2
+
…
+
1
a
2
2021
b
2
2021
=
1
\frac{1}{a_ib_i}+\frac{1}{a_2b_2} + \ldots + \frac{1}{a_{2^{2021}}b_{2^{2021}}} = 1
a
i
b
i
1
+
a
2
b
2
1
+
…
+
a
2
2021
b
2
2021
1
=
1
a
1
+
a
2
+
…
a
2
2021
+
b
1
+
b
2
+
…
+
b
2
2021
=
3
2022
a_1+a_2 +\ldots a_{2^{2021}} +b_1+b_2 + \ldots +b_{2^{2021}} = 3^{2022}
a
1
+
a
2
+
…
a
2
2021
+
b
1
+
b
2
+
…
+
b
2
2021
=
3
2022
Note: Pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
and
(
c
,
d
)
(c,d)
(
c
,
d
)
are different if
a
≠
c
a \neq c
a
=
c
or
b
≠
d
b \neq d
b
=
d
1.1
1
Hide problems
Simple algebra warm up
Given real numbers
x
,
y
,
z
,
a
x,y,z,a
x
,
y
,
z
,
a
satisfying:
x
+
y
+
z
=
a
x+y+z = a
x
+
y
+
z
=
a
1
x
+
1
y
+
1
z
=
1
a
\frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{a}
x
1
+
y
1
+
z
1
=
a
1
Prove that at least one of the numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
is equal to
a
a
a
.
1.5
1
Hide problems
Exponential and cubic NT equation
Find all positive integers
n
,
k
n,k
n
,
k
satisfying:
n
3
−
5
n
+
10
=
2
k
n^3 -5n+10 =2^k
n
3
−
5
n
+
10
=
2
k