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Problems
Contests
National and Regional Contests
China Contests
XES Mathematics Olympiad
the 9th XMO
the 9th XMO
Part of
XES Mathematics Olympiad
Subcontests
(4)
4
1
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9th XMO 2022 P4: Can you find the maximum profit?
One hundred million cities lie on Planet MO. Initially, there are no air routes between any two cities. Now an airline company comes. It plans to establish
5050
5050
5050
two-way routes, each route connects two different cities, and no two routes connect the same two cities. The "degree" of a city is defined to be the number of routes departing from that city. The "benefit" of a route is the product of the "degrees" of the two cities it connects. Find the maximum possible value of the sum of the benefits of these
5050
5050
5050
routes.
3
1
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9th XMO 2022 P3: Divisibility in a strange sequence
A sequence
{
a
n
}
\{a_n\}
{
a
n
}
satisfies
a
1
a_1
a
1
is a positive integer and
a
n
+
1
a_{n+1}
a
n
+
1
is the largest odd integer that divides
2
n
−
1
+
a
n
2^n-1+a_n
2
n
−
1
+
a
n
for all
n
⩾
1
n\geqslant 1
n
⩾
1
. Given a positive integer
r
r
r
which is greater than
1
1
1
. Is it possible that there exists infinitely many pairs of ordered positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
for which
m
>
n
m>n
m
>
n
and
a
m
=
r
a
n
a_m = ra_n
a
m
=
r
a
n
?In other words, if you successfully find an
a
1
a_1
a
1
that yields infinitely many pairs of
(
m
,
n
)
(m,n)
(
m
,
n
)
which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible
a
1
a_1
a
1
.@below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.
2
1
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9th XMO 2022 P2: Multi-center and a weird computation
Given a
△
A
B
C
\triangle ABC
△
A
BC
with circumcenter
O
O
O
and orthocenter
H
(
O
≠
H
)
H(O\ne H)
H
(
O
=
H
)
. Denote the midpoints of
B
C
,
A
C
BC, AC
BC
,
A
C
as
D
,
E
D, E
D
,
E
and let
D
′
,
E
′
D', E'
D
′
,
E
′
be the reflections of
D
,
E
D, E
D
,
E
w.r.t. point
H
H
H
, respectively. If lines
A
D
′
AD'
A
D
′
and
B
E
′
BE'
B
E
′
meet at
K
K
K
, compute
K
O
K
H
\frac{KO}{KH}
KH
K
O
.
1
1
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9th XMO 2022 P1: On bounding the product of Sum and Harmonic
For any
n
n
n
consecutive integers
a
1
,
⋯
,
a
n
a_1, \cdots, a_n
a
1
,
⋯
,
a
n
, prove that
(
a
1
+
⋯
+
a
n
)
⋅
(
1
a
1
+
⋯
+
1
a
n
)
⩽
n
(
n
+
1
)
ln
(
e
n
)
2
.
(a_1+\cdots+a_n)\cdot\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\leqslant \frac{n(n+1)\ln(\text{e}n)}{2}.
(
a
1
+
⋯
+
a
n
)
⋅
(
a
1
1
+
⋯
+
a
n
1
)
⩽
2
n
(
n
+
1
)
ln
(
e
n
)
.