MathDB
Problems
Contests
National and Regional Contests
Switzerland Contests
Switzerland - Final Round
2015 Switzerland - Final Round
2015 Switzerland - Final Round
Part of
Switzerland - Final Round
Subcontests
(6)
7
1
Hide problems
sum a^2/(b + c) =? if sum a/(b + c)=1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers such that:
a
b
+
c
+
b
c
+
a
+
c
a
+
b
=
1
\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1
b
+
c
a
+
c
+
a
b
+
a
+
b
c
=
1
Determine all values which the following expression can take :
a
2
b
+
c
+
b
2
c
+
a
+
c
2
a
+
b
.
\frac{a^2}{b + c} + \frac{b^2}{c + a} + \frac{c^2}{a + b}.
b
+
c
a
2
+
c
+
a
b
2
+
a
+
b
c
2
.
10
1
Hide problems
sum (n + 2)\sqrt{a^2 + b^2} >= n(a + b + c + d)
Find the largest natural number
n
n
n
such that for all real numbers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
the following holds:
(
n
+
2
)
a
2
+
b
2
+
(
n
+
1
)
a
2
+
c
2
+
(
n
+
1
)
a
2
+
d
2
≥
n
(
a
+
b
+
c
+
d
)
(n + 2)\sqrt{a^2 + b^2} + (n + 1)\sqrt{a^2 + c^2} + (n + 1)\sqrt{a^2 + d^2} \ge n(a + b + c + d)
(
n
+
2
)
a
2
+
b
2
+
(
n
+
1
)
a
2
+
c
2
+
(
n
+
1
)
a
2
+
d
2
≥
n
(
a
+
b
+
c
+
d
)
9
1
Hide problems
a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1 is divisible by p
Let
p
p
p
be an odd prime number. Determine the number of tuples
(
a
1
,
a
2
,
.
.
.
,
a
p
)
(a_1, a_2, . . . , a_p)
(
a
1
,
a
2
,
...
,
a
p
)
of natural numbers with the following properties: 1)
1
≤
a
i
≤
p
1 \le ai \le p
1
≤
ai
≤
p
for all
i
=
1
,
.
.
.
,
p
i = 1, . . . , p
i
=
1
,
...
,
p
. 2)
a
1
+
a
2
+
⋅
⋅
⋅
+
a
p
a_1 + a_2 + · · · + a_p
a
1
+
a
2
+
⋅⋅⋅
+
a
p
is not divisible by
p
p
p
. 3)
a
1
a
2
+
a
2
a
3
+
.
.
.
+
a
p
−
1
a
p
+
a
p
a
1
a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1
a
1
a
2
+
a
2
a
3
+
...
+
a
p
−
1
a
p
+
a
p
a
1
is divisible by
p
p
p
.
6
1
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number of ways to cut a 8x8 board
We have an
8
×
8
8\times 8
8
×
8
board. An interior edge is an edge between two
1
×
1
1 \times 1
1
×
1
cells. we cut the board into
1
×
2
1 \times 2
1
×
2
dominoes. For an inner edge
k
k
k
,
N
(
k
)
N(k)
N
(
k
)
denotes the number of ways to cut the board so that it cuts along edge
k
k
k
. Calculate the last digit of the sum we get if we add all
N
(
k
)
N(k)
N
(
k
)
, where
k
k
k
is an inner edge.
2
1
Hide problems
Number Theory
Find all pairs
(
m
,
p
)
(m,p)
(
m
,
p
)
of natural numbers , such that
p
p
p
is a prime and
2
m
p
2
+
27
2^mp^2+27
2
m
p
2
+
27
is the third power of a natural numbers
3
1
Hide problems
Functional equation in real numbers
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
, such that for arbitrary
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
:
(
y
+
1
)
f
(
x
)
+
f
(
x
f
(
y
)
+
f
(
x
+
y
)
)
=
y
.
(y+1)f(x)+f(xf(y)+f(x+y))=y.
(
y
+
1
)
f
(
x
)
+
f
(
x
f
(
y
)
+
f
(
x
+
y
))
=
y
.