MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2023 Turkey Junior National Olympiad
2023 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
Hide problems
Playing with floor and ceiling
Let
x
1
,
x
2
,
…
,
x
31
x_1,x_2,\dots,x_{31}
x
1
,
x
2
,
…
,
x
31
be real numbers. Then find the maximum value can
∑
i
,
j
=
1
,
2
,
…
,
31
,
i
≠
j
⌈
x
i
x
j
⌉
−
30
(
∑
i
=
1
,
2
,
…
,
31
⌊
x
i
2
⌋
)
\sum_{i,j=1,2,\dots,31, \; i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1,2,\dots,31}{\lfloor x_i^2 \rfloor } \right)
i
,
j
=
1
,
2
,
…
,
31
,
i
=
j
∑
⌈
x
i
x
j
⌉
−
30
(
i
=
1
,
2
,
…
,
31
∑
⌊
x
i
2
⌋
)
achieve. P.S.: For a real number
x
x
x
we denote the smallest integer that does not subseed
x
x
x
by
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
and the biggest integer that does not exceed
x
x
x
by
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
. For example
⌈
2.7
⌉
=
3
\lceil 2.7 \rceil=3
⌈
2.7
⌉
=
3
,
⌊
2.7
⌋
=
2
\lfloor 2.7 \rfloor=2
⌊
2.7
⌋
=
2
and
⌊
4
⌋
=
⌈
4
⌉
=
4
\lfloor 4 \rfloor=\lceil 4 \rceil=4
⌊
4
⌋
=
⌈
4
⌉
=
4
3
1
Hide problems
Either \frac{n^4+m}{m^2+n^2} or \frac{n^4-m}{m^2-n^2} is integer
Let
m
,
n
m,n
m
,
n
be relatively prime positive integers. Prove that the numbers
n
4
+
m
m
2
+
n
2
n
4
−
m
m
2
−
n
2
\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}
m
2
+
n
2
n
4
+
m
m
2
−
n
2
n
4
−
m
cannot be integer at the same time.
2
1
Hide problems
Incenters on an inscribed quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be an inscribed quadrilateral. Let the incenters of
B
A
D
BAD
B
A
D
and
C
A
D
CAD
C
A
D
be
I
I
I
and
J
J
J
respectively. Let the intersection point of the line that passes through
I
I
I
and perpendicular to
B
D
BD
B
D
and the line that passes through
J
J
J
and perpendicular to
A
C
AC
A
C
be
K
K
K
. Prove that
K
I
=
K
J
KI=KJ
K
I
=
K
J
1
1
Hide problems
Boxes and Balls
Initially, there are
n
n
n
red boxes numbered with the numbers
1
,
2
,
…
,
n
1,2,\dots ,n
1
,
2
,
…
,
n
and
n
n
n
white boxes numbered with the numbers
1
,
2
,
…
,
n
1,2,\dots ,n
1
,
2
,
…
,
n
on the table. At every move, we choose
2
2
2
different colored boxes and put a ball on each of them. After some moves, every pair of the same numbered boxes has the property of either the number of balls from the red one is
6
6
6
more than the number of balls from the white one or the number of balls from the white one is
16
16
16
more than the number of balls from the red one. With that given information find all possible values of
n
n
n