MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
Turkey EGMO TST
2017 Turkey EGMO TST
2017 Turkey EGMO TST
Part of
Turkey EGMO TST
Subcontests
(6)
6
1
Hide problems
Turkey EGMO TST 2017 P6
Find all pairs of prime numbers
(
p
,
q
)
(p,q)
(
p
,
q
)
, such that
(
2
p
2
−
1
)
q
+
1
p
+
q
\frac{(2p^2-1)^q+1}{p+q}
p
+
q
(
2
p
2
−
1
)
q
+
1
and
(
2
q
2
−
1
)
p
+
1
p
+
q
\frac{(2q^2-1)^p+1}{p+q}
p
+
q
(
2
q
2
−
1
)
p
+
1
are both integers.
5
1
Hide problems
Turkey EGMO TST 2017 P5
In a
12
×
12
12\times 12
12
×
12
square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?Note. Each diagonal is parallel to one of two main diagonals of the table and consists of
1
,
2
…
,
11
1,2\ldots,11
1
,
2
…
,
11
or
12
12
12
cells.
4
1
Hide problems
Turkey EGMO TST 2017 P4
On the inside of the triangle
A
B
C
ABC
A
BC
a point
P
P
P
is chosen with
∠
B
A
P
=
∠
C
A
P
\angle BAP = \angle CAP
∠
B
A
P
=
∠
C
A
P
. If
∣
A
B
∣
⋅
∣
C
P
∣
=
∣
A
C
∣
⋅
∣
B
P
∣
=
∣
B
C
∣
⋅
∣
A
P
∣
\left | AB \right |\cdot \left | CP \right |= \left | AC \right |\cdot \left | BP \right |= \left | BC \right |\cdot \left | AP \right |
∣
A
B
∣
⋅
∣
CP
∣
=
∣
A
C
∣
⋅
∣
BP
∣
=
∣
BC
∣
⋅
∣
A
P
∣
, find all possible values of the angle
∠
A
B
P
\angle ABP
∠
A
BP
.
3
1
Hide problems
Turkey EGMO TST 2017 P3
For all positive real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying the inequality
x
y
z
+
y
z
x
+
z
x
y
≤
3
,
\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}\leq 3,
z
x
y
+
x
yz
+
y
z
x
≤
3
,
prove that
x
2
y
3
+
y
2
z
3
+
z
2
x
3
≥
x
y
+
y
z
+
z
x
.
\frac{x^2}{y^3}+\frac{y^2}{z^3}+\frac{z^2}{x^3}\geq \frac{x}{y}+\frac{y}{z}+\frac{z}{x}.
y
3
x
2
+
z
3
y
2
+
x
3
z
2
≥
y
x
+
z
y
+
x
z
.
2
1
Hide problems
Turkey EGMO TST 2017 P2
At the beginning there are
2017
2017
2017
marbles in each of
1000
1000
1000
boxes. On each move Aybike chooses a box, grabs some of the marbles from that box and delivers them one for each to the boxes she wishes. At least how many moves does Aybike have to make to have different number of marbles in each box?
1
1
Hide problems
Turkey EGMO TST 2017 P1
Let
m
,
k
,
n
m,k,n
m
,
k
,
n
be positive integers. Determine all triples
(
m
,
k
,
n
)
(m,k,n)
(
m
,
k
,
n
)
satisfying the following equation:
3
m
5
k
=
n
3
+
125
3^m5^k=n^3+125
3
m
5
k
=
n
3
+
125