MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2023 China Second Round
3
3
Part of
2023 China Second Round
Problems
(1)
Problem 3
Source: 2023 China Second Round P3
9/10/2023
Find the smallest positive integer
k
{k}
k
with the following properties
:
:{}{}{}{}{}
:
If each positive integer is arbitrarily colored red or blue
,
{}{}{},
,
there may be
9
{}{}{}{}9
9
distinct red positive integers
x
1
,
x
2
,
⋯
,
x
9
,
x_1,x_2,\cdots ,x_9,
x
1
,
x
2
,
⋯
,
x
9
,
satisfying
x
1
+
x
2
+
⋯
+
x
8
<
x
9
,
x_1+x_2+\cdots +x_8<x_9,
x
1
+
x
2
+
⋯
+
x
8
<
x
9
,
or there are
10
10{}{}{}{}{}{}
10
distinct blue positive integers
y
1
,
y
2
,
⋯
,
y
10
y_1,y_2,\cdots ,y_{10}
y
1
,
y
2
,
⋯
,
y
10
satisfiying
y
1
+
y
2
+
⋯
+
y
9
<
y
10
.
{y_1+y_2+\cdots +y_9<y_{10}}.
y
1
+
y
2
+
⋯
+
y
9
<
y
10
.
combinatorics
chang09