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Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2021 JBMO TST - Turkey
2021 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(8)
8
1
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Tangency, intersections and linearity
w
1
w_1
w
1
and
w
2
w_2
w
2
circles have different diameters and externally tangent to each other at
X
X
X
. Points
A
A
A
and
B
B
B
are on
w
1
w_1
w
1
, points
C
C
C
and
D
D
D
are on
w
2
w_2
w
2
such that
A
C
AC
A
C
and
B
D
BD
B
D
are common tangent lines of these two circles.
C
X
CX
CX
intersects
A
B
AB
A
B
at
E
E
E
and
w
1
w_1
w
1
at
F
F
F
second time.
(
E
F
B
)
(EFB)
(
EFB
)
intersects
A
F
AF
A
F
at
G
G
G
second time. If
A
X
∩
C
D
=
H
AX \cap CD =H
A
X
∩
C
D
=
H
, show that points
E
,
G
,
H
E, G, H
E
,
G
,
H
are collinear.
7
1
Hide problems
Finite algebraic moves on a blackboard
Initially on a blackboard, the equation
a
1
x
2
+
b
1
x
+
c
=
0
a_1x^2+b_1x+c=0
a
1
x
2
+
b
1
x
+
c
=
0
is written where
a
1
,
b
1
,
c
1
a_1, b_1, c_1
a
1
,
b
1
,
c
1
are integers and
(
a
1
+
c
1
)
b
1
>
0
(a_1+c_1)b_1 > 0
(
a
1
+
c
1
)
b
1
>
0
. At each move, if the equation
a
x
2
+
b
x
+
c
=
0
ax^2+bx+c=0
a
x
2
+
b
x
+
c
=
0
is written on the board and there is a
x
∈
R
x \in \mathbb{R}
x
∈
R
satisfying the equation, Alice turns this equation into
(
b
+
c
)
x
2
+
(
c
+
a
)
x
+
(
a
+
b
)
=
0
(b+c)x^2+(c+a)x+(a+b)=0
(
b
+
c
)
x
2
+
(
c
+
a
)
x
+
(
a
+
b
)
=
0
. Prove that Alice will stop after a finite number of moves.
6
1
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Modular arithmetic at mod n
Integers
a
1
,
a
2
,
…
a
n
a_1, a_2, \dots a_n
a
1
,
a
2
,
…
a
n
are different at
mod n
\text{mod n}
mod n
. If
a
1
,
a
2
−
a
1
,
a
3
−
a
2
,
…
a
n
−
a
n
−
1
a_1, a_2-a_1, a_3-a_2, \dots a_n-a_{n-1}
a
1
,
a
2
−
a
1
,
a
3
−
a
2
,
…
a
n
−
a
n
−
1
are also different at
mod n
\text{mod n}
mod n
, we call the ordered
n
n
n
-tuple
(
a
1
,
a
2
,
…
a
n
)
(a_1, a_2, \dots a_n)
(
a
1
,
a
2
,
…
a
n
)
lucky. For which positive integers
n
n
n
, one can find a lucky
n
n
n
-tuple?
5
1
Hide problems
Reaching a perfect square with d(n) function
d
(
n
)
d(n)
d
(
n
)
shows the number of positive integer divisors of positive integer
n
n
n
. For which positive integers
n
n
n
one cannot find a positive integer
k
k
k
such that
d
(
…
d
(
d
⏟
k
times
(
n
)
…
)
\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )
k
times
d
(
…
d
(
d
(
n
)
…
)
is a perfect square.
3
1
Hide problems
Maximum number of flights in a country
In a country, there are
28
28
28
cities and between some cities there are two-way flights. In every city there is exactly one airport and this airport is either small or medium or big. For every route which contains more than two cities, doesn't contain a city twice and ends where it begins; has all types of airports. What is the maximum number of flights in this country?
4
1
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Find the maximum value of x^3+2y
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that
∣
y
z
−
x
z
∣
≤
1
and
∣
y
z
+
x
z
∣
≤
1
\left|\dfrac yz-xz\right|\leq 1\text{ and }\left|yz+\dfrac xz\right|\leq 1
z
y
−
x
z
≤
1
and
yz
+
z
x
≤
1
Find the maximum value of the expression
x
3
+
2
y
x^3+2y
x
3
+
2
y
2
1
Hide problems
x is a non-integer rational number, x^n+(x+1)^n is an integer
For which positive integers
n
n
n
, one can find a non-integer rational number
x
x
x
such that
x
n
+
(
x
+
1
)
n
x^n+(x+1)^n
x
n
+
(
x
+
1
)
n
is an integer?
1
1
Hide problems
AF is perpendicular to DE wanted
In an acute-angled triangle
A
B
C
ABC
A
BC
, the circle with diameter
[
A
B
]
[AB]
[
A
B
]
intersects the altitude drawn from vertex
C
C
C
at a point
D
D
D
and the circle with diameter
[
A
C
]
[AC]
[
A
C
]
intersects the altitude drawn from vertex
B
B
B
at a point
E
E
E
. Let the lines
B
D
BD
B
D
and
C
E
CE
CE
intersect at
F
F
F
. Prove that
A
F
⊥
D
E
AF\perp DE
A
F
⊥
D
E