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National and Regional Contests
Poland Contests
Poland - Second Round
1980 Poland - Second Round
1
1
Part of
1980 Poland - Second Round
Problems
(1)
2player game with vectors
Source: Polish MO Recond Round 1980 p1
9/9/2024
Students
A
A
A
and
B
B
B
play according to the following rules: student
A
A
A
selects a vector
a
1
→
\overrightarrow{a_1}
a
1
of length 1 in the plane, then student
B
B
B
gives the number
s
1
s_1
s
1
, equal to
1
1
1
or
−
-
−
1; then the student
A
A
A
chooses a vector
a
1
→
\overrightarrow{a_1}
a
1
of length
1
1
1
, and in turn the student
B
B
B
gives a number
s
2
s_2
s
2
equal to
1
1
1
or
−
1
-1
−
1
etc.
B
B
B
wins if for a certain
n
n
n
vector
∑
j
=
1
n
ε
j
a
j
→
\sum_{j=1}^n \varepsilon_j \overrightarrow{a_j}
∑
j
=
1
n
ε
j
a
j
has a length greater than the number
R
R
R
determined before the start of the game. Prove that student
B
B
B
can achieve a win in no more than
R
2
+
1
R^2 + 1
R
2
+
1
steps regardless of partner
A
A
A
's actions.
combinatorics
vector