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Russian Team Selection Tests
Russian TST 2015
Russian TST 2015
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Russian Team Selection Tests
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P3
3
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Duck swims to water lily :)
Let
0
<
α
<
1
0<\alpha<1
0
<
α
<
1
be a fixed number. On a lake shaped like a convex polygon, at some point there is a duck and at another point a water lily grows. If the duck is at point
X
X{}
X
, then in one move it can swim towards one of the vertices
Y
Y
Y
of the polygon a distance equal to a
α
⋅
X
Y
\alpha\cdot XY
α
⋅
X
Y
. Find all
α
\alpha{}
α
for which, regardless of the shape of the lake and the initial positions of the duck and the lily, after a sequence of adequate moves, the distance between the duck and the lily will be at most one meter.
NT with perfect squares
Find all integers
k
k{}
k
for which there are infinitely many triples of integers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
such that
(
a
2
−
k
)
(
b
2
−
k
)
=
c
2
−
k
.
(a^2-k)(b^2-k)=c^2-k.
(
a
2
−
k
)
(
b
2
−
k
)
=
c
2
−
k
.
Incenter coincides with orthocenter
The triangle
A
B
C
ABC
A
BC
is given. Let
A
′
A'
A
′
be the midpoint of the side
B
C
BC
BC
,
B
c
B_c{}
B
c
be the projection of
B
B{}
B
onto the bisector of the angle
A
C
B
ACB{}
A
CB
and
C
b
C_b
C
b
be the projection of the point
C
C{}
C
onto the bisector of the angle
A
B
C
ABC
A
BC
. Let
A
0
A_0
A
0
be the center of the circle passing through
A
′
,
B
c
,
C
b
A', B_c, C_b
A
′
,
B
c
,
C
b
. The points
B
0
B_0
B
0
and
C
0
C_0
C
0
are defined similarly. Prove that the incenter of the triangle
A
B
C
ABC
A
BC
coincides with the orthocenter of the triangle
A
0
B
0
C
0
A_0B_0C_0
A
0
B
0
C
0
.
P1
7
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P2
5
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