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Contests
National and Regional Contests
China Contests
China Team Selection Test
2015 China Team Selection Test
2015 China Team Selection Test
Part of
China Team Selection Test
Subcontests
(6)
6
2
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Ping Pong and directed graph
There are some players in a Ping Pong tournament, where every
2
2
2
players play with each other at most once. Given: \$$1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$) \$$2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$). \\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
Squarefree Numbers
Prove that there exist infinitely many integers
n
n
n
such that
n
2
+
1
n^2+1
n
2
+
1
is squarefree.
5
2
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Adding weights to numbers such that they sum to 0
FIx positive integer
n
n
n
. Prove: For any positive integers
a
,
b
,
c
a,b,c
a
,
b
,
c
not exceeding
3
n
2
+
4
n
3n^2+4n
3
n
2
+
4
n
, there exist integers
x
,
y
,
z
x,y,z
x
,
y
,
z
with absolute value not exceeding
2
n
2n
2
n
and not all
0
0
0
, such that
a
x
+
b
y
+
c
z
=
0
ax+by+cz=0
a
x
+
b
y
+
cz
=
0
subsets of subset has same sum
Set
S
S
S
to be a subset of size
68
68
68
of
{
1
,
2
,
.
.
.
,
2015
}
\{1,2,...,2015\}
{
1
,
2
,
...
,
2015
}
. Prove that there exist
3
3
3
pairwise disjoint, non-empty subsets
A
,
B
,
C
A,B,C
A
,
B
,
C
such that
∣
A
∣
=
∣
B
∣
=
∣
C
∣
|A|=|B|=|C|
∣
A
∣
=
∣
B
∣
=
∣
C
∣
and
∑
a
∈
A
a
=
∑
b
∈
B
b
=
∑
c
∈
C
c
\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c
∑
a
∈
A
a
=
∑
b
∈
B
b
=
∑
c
∈
C
c
3
4
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1
4
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4
2
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China Team Selection Test 2015 TST 1 Day 2 Q1
Prove that : For each integer
n
≥
3
n \ge 3
n
≥
3
, there exists the positive integers
a
1
<
a
2
<
⋯
<
a
n
a_1<a_2< \cdots <a_n
a
1
<
a
2
<
⋯
<
a
n
, such that for
i
=
1
,
2
,
⋯
,
n
−
2
i=1,2,\cdots,n-2
i
=
1
,
2
,
⋯
,
n
−
2
, With
a
i
,
a
i
+
1
,
a
i
+
2
a_{i},a_{i+1},a_{i+2}
a
i
,
a
i
+
1
,
a
i
+
2
may be formed as a triangle side length , and the area of the triangle is a positive integer.
Bounded functions on reals
Let
n
n
n
be a positive integer, let
f
1
(
x
)
,
…
,
f
n
(
x
)
f_1(x),\ldots,f_n(x)
f
1
(
x
)
,
…
,
f
n
(
x
)
be
n
n
n
bounded real functions, and let
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
be
n
n
n
distinct reals. Show that there exists a real number
x
x
x
such that
∑
i
=
1
n
f
i
(
x
)
−
∑
i
=
1
n
f
i
(
x
−
a
i
)
<
1
\sum^n_{i=1}f_i(x)-\sum^n_{i=1}f_i(x-a_i)<1
∑
i
=
1
n
f
i
(
x
)
−
∑
i
=
1
n
f
i
(
x
−
a
i
)
<
1
.
2
4
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