MathDB
Problems
Contests
National and Regional Contests
India Contests
India IMO Training Camp
2023 India IMO Training Camp
2023 India IMO Training Camp
Part of
India IMO Training Camp
Subcontests
(3)
3
3
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Hard triangle geo (impossible?)
In triangle
A
B
C
ABC
A
BC
, with orthocenter
H
H
H
and circumcircle
Γ
\Gamma
Γ
, the bisector of angle
B
A
C
BAC
B
A
C
meets
B
C
‾
\overline{BC}
BC
at
K
K
K
. Point
Q
Q
Q
lies on
Γ
\Gamma
Γ
such that
A
Q
‾
⊥
Q
K
‾
\overline{AQ} \perp \overline{QK}
A
Q
⊥
Q
K
. Circumcircle of
△
A
Q
H
\triangle AQH
△
A
Q
H
meets
A
C
‾
\overline{AC}
A
C
at
Y
Y
Y
and
A
B
‾
\overline{AB}
A
B
at
Z
Z
Z
. Let
B
Y
‾
\overline{BY}
B
Y
and
C
Z
‾
\overline{CZ}
CZ
meet at
T
T
T
. Prove that
T
H
‾
⊥
K
A
‾
\overline{TH} \perp \overline{KA}
T
H
⊥
K
A
Sum of reciprocals of product of pairs is 1
Prove that for all integers
k
>
2
k>2
k
>
2
, there exists
k
k
k
distinct positive integers
a
1
,
…
,
a
k
a_1, \dots, a_k
a
1
,
…
,
a
k
such that
∑
1
≤
i
<
j
≤
k
1
a
i
a
j
=
1.
\sum_{1 \le i<j \le k} \frac{1}{a_ia_j} =1.
1
≤
i
<
j
≤
k
∑
a
i
a
j
1
=
1.
Proposed by Anant Mudgal
Sum of fourth powers is prime
Let
n
n
n
be any positive integer, and let
S
(
n
)
S(n)
S
(
n
)
denote the number of permutations
τ
\tau
τ
of
{
1
,
…
,
n
}
\{1,\dots,n\}
{
1
,
…
,
n
}
such that
k
4
+
(
τ
(
k
)
)
4
k^4+(\tau(k))^4
k
4
+
(
τ
(
k
)
)
4
is prime for all
k
=
1
,
…
,
n
k=1,\dots,n
k
=
1
,
…
,
n
. Show that
S
(
n
)
S(n)
S
(
n
)
is always a square.
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5
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