MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2020 Turkey Junior National Olympiad
2020 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
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Maximum number of dwarves in a forest
There are dwarves in a forest and each one of them owns exactly 3 hats which are numbered with numbers
1
,
2
,
…
28
1, 2, \dots 28
1
,
2
,
…
28
. Three hats of a dwarf are numbered with different numbers and there are 3 festivals in this forest in a day. In the first festival, each dwarf wears the hat which has the smallest value, in the second festival, each dwarf wears the hat which has the second smallest value and in the final festival each dwarf wears the hat which has the biggest value. After that, it is realized that there is no dwarf pair such that both of two dwarves wear the same value in at least two festivals. Find the maximum value of number of dwarves.
3
1
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good problem for p3 jr
The circumcenter of an acute-triangle
A
B
C
ABC
A
BC
with
∣
A
B
∣
<
∣
B
C
∣
|AB|<|BC|
∣
A
B
∣
<
∣
BC
∣
is
O
O
O
,
D
D
D
and
E
E
E
are midpoints of
∣
A
B
∣
|AB|
∣
A
B
∣
and
∣
A
C
∣
|AC|
∣
A
C
∣
, respectively.
O
E
OE
OE
intersects
B
C
BC
BC
at
K
K
K
, the circumcircle of
O
K
B
OKB
O
K
B
intersects
O
D
OD
O
D
second time at
L
L
L
.
F
F
F
is the foot of altitude from
A
A
A
to line
K
L
KL
K
L
. Show that the point
F
F
F
lies on the line
D
E
DE
D
E
2
1
Hide problems
a nice problem for juniors
If the ratio
17
m
+
43
n
m
−
n
\frac{17m+43n}{m-n}
m
−
n
17
m
+
43
n
is an integer where
m
m
m
and
n
n
n
positive integers, let's call
(
m
,
n
)
(m,n)
(
m
,
n
)
is a special pair. How many numbers can be selected from
1
,
2
,
.
.
.
,
2021
1,2,..., 2021
1
,
2
,
...
,
2021
, any two of which do not form a special pair?
1
1
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a very easy quadratic equation problem
Determine all real number
(
x
,
y
)
(x,y)
(
x
,
y
)
pairs that satisfy the equation.
2
x
2
+
y
2
+
7
=
2
(
x
+
1
)
(
y
+
1
)
2x^2+y^2+7=2(x+1)(y+1)
2
x
2
+
y
2
+
7
=
2
(
x
+
1
)
(
y
+
1
)