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Problems
Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2004 Iran Team Selection Test
2004 Iran Team Selection Test
Part of
Iran Team Selection Test
Subcontests
(5)
4
1
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Conjugate Points
Let
M
,
M
′
M,M'
M
,
M
′
be two conjugates point in triangle
A
B
C
ABC
A
BC
(in the sense that \angle MAB\equal{}\angle M'AC,\dots). Let
P
,
Q
,
R
,
P
′
,
Q
′
,
R
′
P,Q,R,P',Q',R'
P
,
Q
,
R
,
P
′
,
Q
′
,
R
′
be foots of perpendiculars from
M
M
M
and
M
′
M'
M
′
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
. Let E\equal{}QR\cap Q'R', F\equal{}RP\cap R'P' and G\equal{}PQ\cap P'Q'. Prove that the lines
A
G
,
B
F
,
C
E
AG, BF, CE
A
G
,
BF
,
CE
are parallel.
3
1
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Convex quadrilateral
Suppose that
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral. Let F \equal{} AB\cap CD, E \equal{} AD\cap BC and T \equal{} AC\cap BD. Suppose that
A
,
B
,
T
,
E
A,B,T,E
A
,
B
,
T
,
E
lie on a circle which intersects with
E
F
EF
EF
at
P
P
P
. Prove that if
M
M
M
is midpoint of
A
B
AB
A
B
, then \angle APM \equal{} \angle BPT.
2
1
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Pell equation
Suppose that
p
p
p
is a prime number. Prove that the equation x^2\minus{}py^2\equal{}\minus{}1 has a solution if and only if
p
≡
1
(
m
o
d
4
)
p\equiv1\pmod 4
p
≡
1
(
mod
4
)
.
1
1
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Legendre Symbol
Suppose that
p
p
p
is a prime number. Prove that for each
k
k
k
, there exists an
n
n
n
such that: \left(\begin{array}{c}n\\ \hline p\end{array}\right)\equal{}\left(\begin{array}{c}n\plus{}k\\ \hline p\end{array}\right)
5
1
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Graph
This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one. Suppose
G
G
G
is a graph and
S
⊂
V
(
G
)
S\subset V(G)
S
⊂
V
(
G
)
. Suppose we have arbitrarily assign real numbers to each element of
S
S
S
. Prove that we can assign numbers to each vertex in
G
\
S
G\backslash S
G
\
S
that for each
v
∈
G
\
S
v\in G\backslash S
v
∈
G
\
S
number assigned to
v
v
v
is average of its neighbors.