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Contests
National and Regional Contests
Turkey Contests
Turkey Junior National Olympiad
2011 Turkey Junior National Olympiad
2011 Turkey Junior National Olympiad
Part of
Turkey Junior National Olympiad
Subcontests
(4)
4
1
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20 math and 11 physics problems
Each student chooses
1
1
1
math problem and
1
1
1
physics problem among
20
20
20
math problems and
11
11
11
physics problems. No same pair of problem is selected by two students. And at least one of the problems selected by any student is selected by at most one other student. At most how many students are there?
3
1
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Primes in the form of 8k+1
m
<
n
m < n
m
<
n
are positive integers. Let
p
=
n
2
+
m
2
n
2
−
m
2
p=\frac{n^2+m^2}{\sqrt{n^2-m^2}}
p
=
n
2
−
m
2
n
2
+
m
2
.(a) Find three pairs of positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
that make
p
p
p
prime.(b) If
p
p
p
is prime, then show that
p
≡
1
(
m
o
d
8
)
p \equiv 1 \pmod 8
p
≡
1
(
mod
8
)
.
2
1
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Circumcircle of half of an isosceles triangle
Let
A
B
C
ABC
A
BC
be a triangle with
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
.
D
D
D
is the midpoint of
[
B
C
]
[BC]
[
BC
]
.
E
E
E
is the foot of the altitude from
D
D
D
to
A
C
AC
A
C
.
B
E
BE
BE
cuts the circumcircle of triangle
A
B
D
ABD
A
B
D
at
B
B
B
and
F
F
F
.
D
E
DE
D
E
and
A
F
AF
A
F
meet at
G
G
G
. Prove that
∣
D
G
∣
=
∣
G
E
∣
|DG|=|GE|
∣
D
G
∣
=
∣
GE
∣
1
1
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(x+y)(x^3+y^3)/(x^2+y^2)^2
Show that
1
≤
(
x
+
y
)
(
x
3
+
y
3
)
(
x
2
+
y
2
)
2
≤
9
8
1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98
1
≤
(
x
2
+
y
2
)
2
(
x
+
y
)
(
x
3
+
y
3
)
≤
8
9
holds for all positive real numbers
x
,
y
x,y
x
,
y
.