MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2018 Turkey EGMO TST
2018 Turkey EGMO TST
Part of
Turkey EGMO TST
Subcontests
(6)
4
1
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Maximum possible value of x
There are
n
n
n
stone piles each consisting of
2018
2018
2018
stones. The weight of each stone is equal to one of the numbers
1
,
2
,
3
,
.
.
.
25
1, 2, 3, ...25
1
,
2
,
3
,
...25
and the total weights of any two piles are different. It is given that if we choose any two piles and remove the heaviest and lightest stones from each of these piles then the pile which has the heavier one becomes the lighter one. Determine the maximal possible value of
n
n
n
.
3
1
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A counting problem about coloring
In how many ways every unit square of a
2018
2018
2018
x
2018
2018
2018
board can be colored in red or white such that number of red unit squares in any two rows are distinct and number of red squares in any two columns are distinct.
1
1
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Turkey Egmo Tst 2018 p1
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral and
w
w
w
be its circumcircle. For a given point
E
E
E
inside
w
w
w
,
D
E
DE
D
E
intersects
A
B
AB
A
B
at
F
F
F
inside
w
w
w
. Let
l
l
l
be a line passes through
E
E
E
and tangent to circle
A
E
F
AEF
A
EF
. Let
G
G
G
be any point on
l
l
l
and inside the quadrilateral
A
B
C
D
ABCD
A
BC
D
. Show that if
∠
G
A
D
=
∠
B
A
E
\angle GAD =\angle BAE
∠
G
A
D
=
∠
B
A
E
and
∠
G
C
B
+
∠
G
A
B
=
∠
E
A
D
+
∠
A
G
D
+
∠
A
B
E
\angle GCB + \angle GAB = \angle EAD + \angle AGD + \angle ABE
∠
GCB
+
∠
G
A
B
=
∠
E
A
D
+
∠
A
G
D
+
∠
A
BE
then
B
C
BC
BC
,
A
D
AD
A
D
and
E
G
EG
EG
are concurrent.
5
1
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Turkey Egmo tst 2018 p5
Prove that
x
2
+
1
(
x
+
y
)
2
+
4
(
z
+
1
)
+
y
2
+
1
(
y
+
z
)
2
+
4
(
x
+
1
)
+
z
2
+
1
(
z
+
x
)
2
+
4
(
y
+
1
)
≥
1
2
\dfrac {x^2+1}{(x+y)^2+4 (z+1)}+\dfrac {y^2+1}{(y+z)^2+4 (x+1)}+\dfrac {z^2+1}{(z+x)^2+4 (y+1)} \ge \dfrac{1}{2}
(
x
+
y
)
2
+
4
(
z
+
1
)
x
2
+
1
+
(
y
+
z
)
2
+
4
(
x
+
1
)
y
2
+
1
+
(
z
+
x
)
2
+
4
(
y
+
1
)
z
2
+
1
≥
2
1
for all positive reals
x
,
y
,
z
x,y,z
x
,
y
,
z
6
1
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Turkey Egmo Tst 2018 p6
Let
f
:
Z
+
→
Z
+
f:\mathbb{Z}_{+}\rightarrow\mathbb{Z}_{+}
f
:
Z
+
→
Z
+
is one to one and bijective function. Prove that
f
(
m
n
)
=
f
(
m
)
f
(
n
)
f(mn)=f (m)f (n)
f
(
mn
)
=
f
(
m
)
f
(
n
)
if and only if
l
c
m
(
f
(
m
)
,
f
(
n
)
)
=
f
(
l
c
m
(
m
,
n
)
)
lcm (f (m),f (n))=f(lcm(m,n))
l
c
m
(
f
(
m
)
,
f
(
n
))
=
f
(
l
c
m
(
m
,
n
))
2
1
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Turkey Egmo Tst 2018 p2
Determine all pairs
(
m
,
n
)
(m,n)
(
m
,
n
)
of positive integers such that
m
2
+
n
2
=
2018
(
m
−
n
)
m^2+n^2=2018(m-n)
m
2
+
n
2
=
2018
(
m
−
n
)