MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
JBMO TST - Turkey
2022 JBMO TST - Turkey
2022 JBMO TST - Turkey
Part of
JBMO TST - Turkey
Subcontests
(6)
6
1
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f(c)f(-c)>=f(a) where f(x)=x^2-2ax+b
Let
c
c
c
be a real number. If the inequality
f
(
c
)
⋅
f
(
−
c
)
≥
f
(
a
)
f(c)\cdot f(-c)\ge f(a)
f
(
c
)
⋅
f
(
−
c
)
≥
f
(
a
)
holds for all
f
(
x
)
=
x
2
−
2
a
x
+
b
f(x)=x^2-2ax+b
f
(
x
)
=
x
2
−
2
a
x
+
b
where
a
a
a
and
b
b
b
are arbitrary real numbers, find all possible values of
c
c
c
.
5
1
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29 students write numbers on boards
Each of the
n
n
n
students writes one of the numbers
1
,
2
1,2
1
,
2
or
3
3
3
on each of the
29
29
29
boards. If any two students wrote different numbers on at least one of the boards and any three students wrote the same number on at least one of the boards, what is the maximum possible value of
n
n
n
?
4
1
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Parallelity and equal angles given, wanted an angle equality
Given a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
such that
m
(
A
B
C
^
)
=
m
(
B
C
D
^
)
m(\widehat{ABC})=m(\widehat{BCD})
m
(
A
BC
)
=
m
(
BC
D
)
. The lines
A
D
AD
A
D
and
B
C
BC
BC
intersect at a point
P
P
P
and the line passing through
P
P
P
which is parallel to
A
B
AB
A
B
, intersects
B
D
BD
B
D
at
T
T
T
. Prove that
m
(
A
C
B
^
)
=
m
(
P
C
T
^
)
m(\widehat{ACB})=m(\widehat{PCT})
m
(
A
CB
)
=
m
(
PCT
)
3
1
Hide problems
29 peoples attending a party, replacing a hat of a person
Each of the
29
29
29
people attending a party wears one of three different types of hats. Call a person lucky if at least two of his friends wear different types of hats. Show that it is always possible to replace the hat of a person at this party with a hat of one of the other two types, in a way that the total number of lucky people is not reduced.
2
1
Hide problems
Solve x*[x]+2022=[x^2]
For a real number
a
a
a
,
[
a
]
[a]
[
a
]
denotes the largest integer not exceeding
a
a
a
. Find all positive real numbers
x
x
x
satisfying the equation
x
⋅
[
x
]
+
2022
=
[
x
2
]
x\cdot [x]+2022=[x^2]
x
⋅
[
x
]
+
2022
=
[
x
2
]
1
1
Hide problems
(a^2+b^2)/(a-b)^2 is an integer => (a^3+b^3)/(a-b)^3 is an integer
For positive integers
a
a
a
and
b
b
b
, if the expression
a
2
+
b
2
(
a
−
b
)
2
\frac{a^2+b^2}{(a-b)^2}
(
a
−
b
)
2
a
2
+
b
2
is an integer, prove that the expression
a
3
+
b
3
(
a
−
b
)
3
\frac{a^3+b^3}{(a-b)^3}
(
a
−
b
)
3
a
3
+
b
3
is an integer as well.