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Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2023 Switzerland - Final Round
2023 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(5)
5
1
Hide problems
Functional equation with -1 missing from domain
Let
D
D
D
be the set of real numbers excluding
−
1
-1
−
1
. Find all functions
f
:
D
→
D
f: D \to D
f
:
D
→
D
such that for all
x
,
y
∈
D
x,y \in D
x
,
y
∈
D
satisfying
x
≠
0
x \neq 0
x
=
0
and
y
≠
−
x
y \neq -x
y
=
−
x
, the equality
(
f
(
f
(
x
)
)
+
y
)
f
(
y
x
)
+
f
(
f
(
y
)
)
=
x
(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x
(
f
(
f
(
x
))
+
y
)
f
(
x
y
)
+
f
(
f
(
y
))
=
x
holds.
4
1
Hide problems
Inequality subject to strange conditions
Determine the smallest possible value of the expression
a
b
+
1
a
+
b
+
b
c
+
1
b
+
c
+
c
a
+
1
c
+
a
\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}
a
+
b
ab
+
1
+
b
+
c
b
c
+
1
+
c
+
a
c
a
+
1
where
a
,
b
,
c
∈
R
a,b,c \in \mathbb{R}
a
,
b
,
c
∈
R
satisfy
a
+
b
+
c
=
−
1
a+b+c = -1
a
+
b
+
c
=
−
1
and
a
b
c
⩽
−
3
abc \leqslant -3
ab
c
⩽
−
3
3
1
Hide problems
Gcd of some consecutive terms of a recursive sequence
Let
x
,
y
x,y
x
,
y
and
a
0
,
a
1
,
a
2
,
⋯
a_0, a_1, a_2, \cdots
a
0
,
a
1
,
a
2
,
⋯
be integers satisfying
a
0
=
a
1
=
0
a_0 = a_1 = 0
a
0
=
a
1
=
0
, and
a
n
+
2
=
x
a
n
+
1
+
y
a
n
+
1
a_{n+2} = xa_{n+1}+ya_n+1
a
n
+
2
=
x
a
n
+
1
+
y
a
n
+
1
for all integers
n
≥
0
n \geq 0
n
≥
0
. Let
p
p
p
be any prime number. Show that
gcd
(
a
p
,
a
p
+
1
)
\gcd(a_p,a_{p+1})
g
cd
(
a
p
,
a
p
+
1
)
is either equal to
1
1
1
or greater than
p
\sqrt{p}
p
.
2
1
Hide problems
Albus and Brian play on a square near lava
The wizard Albus and Brian are playing a game on a square of side length
2
n
+
1
2n+1
2
n
+
1
meters surrounded by lava. In the centre of the square there sits a toad. In a turn, a wizard chooses a direction parallel to a side of the square and enchants the toad. This will cause the toad to jump
d
d
d
meters in the chosen direction, where
d
d
d
is initially equal to
1
1
1
and increases by
1
1
1
after each jump. The wizard who sends the toad into the lava loses. Albus begins and they take turns. Depending on
n
n
n
, determine which wizard has a winning strategy.
1
1
Hide problems
Reflection lies on circumcircle
Let
A
B
C
ABC
A
BC
be an acute triangle with incenter
I
I
I
. On its circumcircle, let
M
A
M_A
M
A
,
M
B
M_B
M
B
and
M
C
M_C
M
C
be the midpoints of minor arcs
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively. Prove that the reflection
M
A
M_A
M
A
over the line
I
M
B
IM_B
I
M
B
lies on the circumcircle of the triangle
I
M
B
M
C
IM_BM_C
I
M
B
M
C
.