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International Contests
European Mathematical Cup
2013 European Mathematical Cup
2013 European Mathematical Cup
Part of
European Mathematical Cup
Subcontests
(4)
4
2
Hide problems
A good looking inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals satisfying :
a
1
+
b
+
c
+
b
1
+
c
+
a
+
c
1
+
a
+
b
≥
a
b
1
+
a
+
b
+
b
c
1
+
b
+
c
+
c
a
1
+
c
+
a
\frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a}
1
+
b
+
c
a
+
1
+
c
+
a
b
+
1
+
a
+
b
c
≥
1
+
a
+
b
ab
+
1
+
b
+
c
b
c
+
1
+
c
+
a
c
a
Then prove that :
a
2
+
b
2
+
c
2
a
b
+
b
c
+
c
a
+
a
+
b
+
c
+
2
≥
2
(
a
b
+
b
c
+
c
a
)
\frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})
ab
+
b
c
+
c
a
a
2
+
b
2
+
c
2
+
a
+
b
+
c
+
2
≥
2
(
ab
+
b
c
+
c
a
)
Proposed by Dimitar Trenevski
Problem 4
Given a triangle
A
B
C
ABC
A
BC
let
D
D
D
,
E
E
E
,
F
F
F
be orthogonal projections from
A
A
A
,
B
B
B
,
C
C
C
to the opposite sides respectively. Let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
denote midpoints of
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
respectively. Prove that perpendiculars from
D
D
D
to
Y
Z
YZ
Y
Z
, from
E
E
E
to
X
Z
XZ
XZ
and from
F
F
F
to
X
Y
XY
X
Y
are concurrent.
3
2
Hide problems
Lock-breaking codes
We are given a combination lock consisting of
6
6
6
rotating discs. Each disc consists of digits
0
,
1
,
2
,
…
,
9
0, 1, 2,\ldots , 9
0
,
1
,
2
,
…
,
9
in that order (after digit
9
9
9
comes
0
0
0
). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combination is correct. Discs are initially put in the position
000000
000000
000000
, and we know that this combination is not correct.a) What is the least number of moves necessary to ensure that we have found the correct combination? b) What is the least number of moves necessary to ensure that we have found the correct combination, if we know that none of the combinations
000000
,
111111
,
222222
,
…
,
999999
000000, 111111, 222222, \ldots , 999999
000000
,
111111
,
222222
,
…
,
999999
is correct?Proposed by Ognjen Stipetić and Grgur Valentić
Problem 3
We call a sequence of
n
n
n
digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code
(
1101
)
(1101)
(
1101
)
contains
10
10
10
palindromes, of which
6
6
6
are nice.)a) What is the least number of palindromes in a code?b) What is the least number of nice palindromes in a code?
2
2
Hide problems
An area equality
Let
P
P
P
be a point inside a triangle
A
B
C
ABC
A
BC
. A line through
P
P
P
parallel to
A
B
AB
A
B
meets
B
C
BC
BC
and
C
A
CA
C
A
at points
L
L
L
and
F
F
F
, respectively. A line through
P
P
P
parallel to
B
C
BC
BC
meets
C
A
CA
C
A
and
B
A
BA
B
A
at points
M
M
M
and
D
D
D
respectively, and a line through
P
P
P
parallel to
C
A
CA
C
A
meets
A
B
AB
A
B
and
B
C
BC
BC
at points
N
N
N
and
E
E
E
respectively. Prove \begin{align*} [PDBL] \cdot [PECM] \cdot [PFAN]=8\cdot [PFM] \cdot [PEL] \cdot [PDN] \\ \end{align*}Proposed by Steve Dinh
Problem 2
Palindrome is a sequence of digits which doesn't change if we reverse the order of its digits. Prove that a sequence
(
x
n
)
n
=
0
∞
(x_n)^{\infty}_{n=0}
(
x
n
)
n
=
0
∞
defined as
x
n
=
2013
+
317
n
x_n=2013+317n
x
n
=
2013
+
317
n
contains infinitely many numbers with their decimal expansions being palindromes.
1
2
Hide problems
Product of first m primes
For
m
∈
N
m\in \mathbb{N}
m
∈
N
define
m
?
m?
m
?
be the product of first
m
m
m
primes. Determine if there exists positive integers
m
,
n
m,n
m
,
n
with the following property :
m
?
=
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
m?=n(n+1)(n+2)(n+3)
m
?
=
n
(
n
+
1
)
(
n
+
2
)
(
n
+
3
)
Proposed by Matko Ljulj
Tables in which each entry is the product of its neighbors
In each field of a table there is a real number. We call such
n
×
n
n \times n
n
×
n
table silly if each entry equals the product of all the numbers in the neighbouring fields.a) Find all
2
×
2
2 \times 2
2
×
2
silly tables. b) Find all
3
×
3
3 \times 3
3
×
3
silly tables.