MathDB
Problems
Contests
National and Regional Contests
China Contests
China Northern MO
2022 China Northern MO
2022 China Northern MO
Part of
China Northern MO
Subcontests
(4)
4
1
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friendships among 22 mathematicians
22
22
22
mathematicians are meeting together. Each mathematician has at least
3
3
3
friends (friends are mutual). And each mathematician can pass his or her information to any mathematician through the transfer between friends. Is it possible to divide these
22
22
22
mathematicians into
2
2
2
-person groups (that is, two people in each group, a total of
11
11
11
groups), so that the mathematicians in each group are friends?[hide=original wording in Chinese]仃22位数学家一起开会.每位数学家都至少有3个朋友(朋友是相互的).而且每 位数学家都可以通过朋友之间的传递.将门已的资料传给任意一位数学家.问：是否一定可 以将这22位数学家两两分组(即每组两人,共11组),使得每组的数学家都是朋友？
3
1
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a_{n+1}=a_n+ \frac{n^2}{a_n}, b_n =a_n-n
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence of positive terms such that
a
n
+
1
=
a
n
+
n
2
a
n
a_{n+1}=a_n+ \frac{n^2}{a_n}
a
n
+
1
=
a
n
+
a
n
n
2
. Let
b
n
=
a
n
−
n
b_n =a_n-n
b
n
=
a
n
−
n
. (1) Are there infinitely many
n
n
n
such that
b
n
≥
0
b_n \ge 0
b
n
≥
0
? (2) Prove that there is a positive number
M
M
M
such that
∑
n
=
3
∞
b
n
n
+
1
<
M
\sum^{\infty}_{n=3} \frac{b_n}{n+1}<M
∑
n
=
3
∞
n
+
1
b
n
<
M
.
1
1
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<AFD= <CEF if AB _l_ AC, AB=BC, AD=//DB, DE=_I_DF, BE _|_ EC
As shown in the figure, given
△
A
B
C
\vartriangle ABC
△
A
BC
with
A
B
⊥
A
C
AB \perp AC
A
B
⊥
A
C
,
A
B
=
B
C
AB=BC
A
B
=
BC
,
D
D
D
is the midpoint of the side
A
B
AB
A
B
,
D
F
⊥
D
E
DF\perp DE
D
F
⊥
D
E
,
D
E
=
D
F
DE=DF
D
E
=
D
F
and
B
E
⊥
E
C
BE \perp EC
BE
⊥
EC
. Prove that
∠
A
F
D
=
∠
C
E
F
\angle AFD= \angle CEF
∠
A
F
D
=
∠
CEF
. https://cdn.artofproblemsolving.com/attachments/9/2/f16a8c8c463874f3ccb333d91cdef913c34189.png
2
1
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221|3^a -2^a, n and n+1 lie in {n\in N^*: 211|1+2^n+3^n+4^n}
(1) Find the smallest positive integer
a
a
a
such that
221
∣
3
a
−
2
a
221|3^a -2^a
221∣
3
a
−
2
a
,(2) Let
A
=
{
n
∈
N
∗
:
211
∣
1
+
2
n
+
3
n
+
4
n
}
A=\{n\in N^*: 211|1+2^n+3^n+4^n\}
A
=
{
n
∈
N
∗
:
211∣1
+
2
n
+
3
n
+
4
n
}
. Are there infinitely many numbers
n
n
n
such that both
n
n
n
and
n
+
1
n+1
n
+
1
belong to set
A
A
A
?