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Contests
National and Regional Contests
Turkey Contests
Turkey MO (2nd round)
1999 Turkey MO (2nd round)
1999 Turkey MO (2nd round)
Part of
Turkey MO (2nd round)
Subcontests
(6)
6
1
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40 Processors, adding 40 numbers
We wish to find the sum of
40
40
40
given numbers utilizing
40
40
40
processors. Initially, we have the number
0
0
0
on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to find the desired sum.
1
1
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Turkey NMO 1999, P-1, find quadruples (x,y,z,w) in (mod 37)
Find the number of ordered quadruples
(
x
,
y
,
z
,
w
)
(x,y,z,w)
(
x
,
y
,
z
,
w
)
of integers with
0
≤
x
,
y
,
z
,
w
≤
36
0\le x,y,z,w\le 36
0
≤
x
,
y
,
z
,
w
≤
36
such that
x
2
+
y
2
≡
z
3
+
w
3
(mod 37)
{{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}
x
2
+
y
2
≡
z
3
+
w
3
(mod 37)
.
4
1
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Turkey NMO 1999, P-4, find all sequences of real numbers
Find all sequences
a
1
,
a
2
,
.
.
.
,
a
2000
{{a}_{1}},{{a}_{2}},...,{{a}_{2000}}
a
1
,
a
2
,
...
,
a
2000
of real numbers such that
∑
n
=
1
2000
a
n
=
1999
\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}
n
=
1
∑
2000
a
n
=
1999
and such that
1
2
<
a
n
<
1
\frac{1}{2}<{{a}_{n}}<1
2
1
<
a
n
<
1
and
a
n
+
1
=
a
n
(
2
−
a
n
)
{{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})
a
n
+
1
=
a
n
(
2
−
a
n
)
for all
n
≥
1
n\ge 1
n
≥
1
.
3
1
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Turkey NMO 1999, P-3, the number of the functions
For any two positive integers
n
n
n
and
p
p
p
, prove that there are exactly
(
p
+
1
)
n
+
1
−
p
n
+
1
{{(p+1)}^{n+1}}-{{p}^{n+1}}
(
p
+
1
)
n
+
1
−
p
n
+
1
functions
f
:
{
1
,
2
,
.
.
.
,
n
}
→
{
−
p
,
−
p
+
1
,
−
p
+
2
,
.
.
.
.
,
p
−
1
,
p
}
f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}
f
:
{
1
,
2
,
...
,
n
}
→
{
−
p
,
−
p
+
1
,
−
p
+
2
,
....
,
p
−
1
,
p
}
such that
∣
f
(
i
)
−
f
(
j
)
∣
≤
p
\left| f(i)-f(j) \right|\le p
∣
f
(
i
)
−
f
(
j
)
∣
≤
p
for all
i
,
j
∈
{
1
,
2
,
.
.
.
,
n
}
i,j\in \left\{ 1,2,...,n \right\}
i
,
j
∈
{
1
,
2
,
...
,
n
}
.
2
1
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Turkey NMO 1999, P-2,a relation on a circle
Problem-2: Given a circle with center
O
O
O
, the two tangent lines from a point
S
S
S
outside the circle touch the circle at points
P
P
P
and
Q
Q
Q
. Line
S
O
SO
SO
intersects the circle at
A
A
A
and
B
B
B
, with
B
B
B
closer to
S
S
S
. Let
X
X
X
be an interior point of minor arc
P
B
PB
PB
, and let line
O
S
OS
OS
intersect lines
Q
X
QX
QX
and
P
X
PX
PX
at
C
C
C
and
D
D
D
, respectively. Prove that
1
∣
A
C
∣
+
1
∣
A
D
∣
=
2
∣
A
B
∣
\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}
∣
A
C
∣
1
+
∣
A
D
∣
1
=
∣
A
B
∣
2
.
5
1
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Turkey NMO 1999, P-5, nice geometric inequality on altitudes
In an acute triangle
△
A
B
C
\vartriangle ABC
△
A
BC
with circumradius
R
R
R
, altitudes
A
D
‾
,
B
E
‾
,
C
F
‾
\overline{AD},\overline{BE},\overline{CF}
A
D
,
BE
,
CF
have lengths
h
1
,
h
2
,
h
3
{{h}_{1}},{{h}_{2}},{{h}_{3}}
h
1
,
h
2
,
h
3
, respectively. If
t
1
,
t
2
,
t
3
{{t}_{1}},{{t}_{2}},{{t}_{3}}
t
1
,
t
2
,
t
3
are lengths of the tangents from
A
,
B
,
C
A,B,C
A
,
B
,
C
, respectively, to the circumcircle of triangle
△
D
E
F
\vartriangle DEF
△
D
EF
, prove that
∑
i
=
1
3
(
t
i
h
i
)
2
≤
3
2
R
\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R
i
=
1
∑
3
(
h
i
t
i
)
2
≤
2
3
R
.