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National and Regional Contests
Nepal Contests
Nepal National Olympiad
2024 Nepal Mathematics Olympiad (Pre-TST)
2024 Nepal Mathematics Olympiad (Pre-TST)
Part of
Nepal National Olympiad
Subcontests
(4)
Problem 4
1
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NMO (Nepal) Problem 4
Find all integer/s
n
n
n
such that
5
n
−
1
3
\displaystyle{\frac{5^n-1}{3}}
3
5
n
−
1
is a prime or a perfect square of an integer.Proposed by Prajit Adhikari, Nepal
Problem 3
1
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NMO (Nepal) Problem 3
Let
A
B
C
ABC
A
BC
be an acute triangle and
H
H
H
be its orthocenter. Let
E
E
E
be the foot of the altitude from
C
C
C
to
A
B
AB
A
B
,
F
F
F
be the foot of the altitude from
B
B
B
to
A
C
AC
A
C
. Let
G
≠
H
G \neq H
G
=
H
be the intersection of the circles
(
A
E
F
)
(AEF)
(
A
EF
)
and
(
B
H
C
)
(BHC)
(
B
H
C
)
. Prove that
A
G
AG
A
G
bisects
B
C
BC
BC
.Proposed by Kang Taeyoung, South Korea
Problem 2
1
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NMO(Nepal) Problem 2
Let,
S
=
∑
i
=
1
k
n
i
2
\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}
S
=
i
=
1
∑
k
n
i
2
. Prove that for
n
i
∈
R
+
n_i \in \mathbb{R}^+
n
i
∈
R
+
∑
i
=
1
k
n
i
S
−
n
i
2
≥
4
n
1
+
n
2
+
⋯
+
n
k
\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}
i
=
1
∑
k
S
−
n
i
2
n
i
≥
n
1
+
n
2
+
⋯
+
n
k
4
Proposed by Kang Taeyoung, South Korea
Problem 1
1
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NMO (Nepal) Problem 1
Nirajan is trapped in a magical dungeon. He has infinitely many magical cards with arbitrary MPs(Mana Points) which is always an integer
Z
\mathbb{Z}
Z
. To escape, he must give the dungeon keeper some magical cards whose MPs add up to an integer with at least
2024
2024
2024
divisors. Can Nirajan always escape?( Proposed by Vlad Spǎtaru, Romania)