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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2020 Switzerland - Final Round
2020 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(5)
6
1
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k stones among squares of n x n chessboard
Let
n
≥
2
n \ge 2
n
≥
2
be an integer. Consider the following game: Initially,
k
k
k
stones are distributed among the
n
2
n^2
n
2
squares of an
n
×
n
n\times n
n
×
n
chessboard. A move consists of choosing a square containing at least as many stones as the number of its adjacent squares (two squares are adjacent if they share a common edge) and moving one stone from this square to each of its adjacent squares. Determine all positive integers
k
k
k
such that: (a) There is an initial configuration with
k
k
k
stones such that no move is possible. (b) There is an initial configuration with
k
k
k
stones such that an infinite sequence of moves is possible.
5
1
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a! b! = a! + b! + c! , factorial diophantine
Find all the positive integers
a
,
b
,
c
a, b, c
a
,
b
,
c
such that
a
!
⋅
b
!
=
a
!
+
b
!
+
c
!
a! \cdot b! = a! + b! + c!
a
!
⋅
b
!
=
a
!
+
b
!
+
c
!
1
1
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Natural number divisibility FE
Let
N
\mathbb N
N
be the set of positive integers. Find all functions
f
:
N
→
N
f\colon\mathbb N\to \mathbb N
f
:
N
→
N
such that for every
m
,
n
∈
N
m,n\in \mathbb N
m
,
n
∈
N
,
f
(
m
)
+
f
(
n
)
∣
m
+
n
.
f(m)+f(n)\mid m+n.
f
(
m
)
+
f
(
n
)
∣
m
+
n
.
3
1
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Lower bound on number of angles
We are given
n
n
n
distinct rectangles in the plane. Prove that between the
4
n
4n
4
n
interior angles formed by these rectangles at least
4
n
4\sqrt n
4
n
are distinct.
4
1
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Tasty Number Theory
Let
φ
\varphi
φ
denote the Euler phi-function. Prove that for every positive integer
n
n
n
2
n
(
n
+
1
)
∣
32
⋅
φ
(
2
2
n
−
1
)
.
2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).
2
n
(
n
+
1
)
∣32
⋅
φ
(
2
2
n
−
1
)
.