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Problems
Contests
National and Regional Contests
Israel Contests
Israel Team Selection Test
2025 Israel TST
2025 Israel TST
Part of
Israel Team Selection Test
Subcontests
(2)
P2
1
Hide problems
Special line through antipodal
Triangle
△
A
B
C
\triangle ABC
△
A
BC
is inscribed in circle
Ω
\Omega
Ω
. Let
I
I
I
denote its incenter and
I
A
I_A
I
A
its
A
A
A
-excenter. Let
N
N
N
denote the midpoint of arc
B
A
C
BAC
B
A
C
. Line
N
I
A
NI_A
N
I
A
meets
Ω
\Omega
Ω
a second time at
T
T
T
. The perpendicular to
A
I
AI
A
I
at
I
I
I
meets sides
A
C
AC
A
C
and
A
B
AB
A
B
at
E
E
E
and
F
F
F
respectively. The circumcircle of
△
B
F
T
\triangle BFT
△
BFT
meets
B
I
A
BI_A
B
I
A
a second time at
P
P
P
, and the circumcircle of
△
C
E
T
\triangle CET
△
CET
meets
C
I
A
CI_A
C
I
A
a second time at
Q
Q
Q
. Prove that
P
Q
PQ
PQ
passes through the antipodal to
A
A
A
on
Ω
\Omega
Ω
.
P1
1
Hide problems
Rational sequence reaches 1
A sequence starts at some rational number
x
1
>
1
x_1>1
x
1
>
1
, and is subsequently defined using the recurrence relation
x
n
+
1
=
x
n
⋅
n
⌊
x
n
⋅
n
⌋
x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }
x
n
+
1
=
⌊
x
n
⋅
n
⌋
x
n
⋅
n
Show that
k
>
0
k>0
k
>
0
exists with
x
k
=
1
x_k=1
x
k
=
1
.