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Contests
National and Regional Contests
Singapore Contests
Singapore MO Open
2022 Singapore MO Open
2022 Singapore MO Open
Part of
Singapore MO Open
Subcontests
(2)
Q4
1
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Classic array sorting problem with colouring
Let
n
,
k
n,k
n
,
k
,
1
≤
k
≤
n
1\le k\le n
1
≤
k
≤
n
be fixed integers. Alice has
n
n
n
cards in a row, where the card has position
i
i
i
has the label
i
+
k
i+k
i
+
k
(or
i
+
k
−
n
i+k-n
i
+
k
−
n
if
i
+
k
>
n
i+k>n
i
+
k
>
n
). Alice starts by colouring each card either red or blue. Afterwards, she is allowed to make several moves, where each move consists of choosing two cards of different colours and swapping them. Find the minimum number of moves she has to make (given that she chooses the colouring optimally) to put the cards in order (i.e. card
i
i
i
is at position
i
i
i
).NOTE: edited from original phrasing, which was ambiguous.
Q2
1
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non-integer distances from point to corners of odd rectangle
Prove that if the length and breadth of a rectangle are both odd integers, then there does not exist a point
P
P
P
inside the rectangle such that each of the distances from
P
P
P
to the 4 corners of the rectangle is an integer.