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Problems
Contests
International Contests
EGMO
2014 EGMO
2014 EGMO
Part of
EGMO
Subcontests
(6)
6
1
Hide problems
Function related to (x+y)^2
Determine all functions
f
:
R
→
R
f:\mathbb R\rightarrow\mathbb R
f
:
R
→
R
satisfying the condition
f
(
y
2
+
2
x
f
(
y
)
+
f
(
x
)
2
)
=
(
y
+
f
(
x
)
)
(
x
+
f
(
y
)
)
f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))
f
(
y
2
+
2
x
f
(
y
)
+
f
(
x
)
2
)
=
(
y
+
f
(
x
))
(
x
+
f
(
y
))
for all real numbers
x
x
x
and
y
y
y
.
5
1
Hide problems
Moving pebbles in n boxes
Let
n
n
n
be a positive integer. We have
n
n
n
boxes where each box contains a non-negative number of pebbles. In each move we are allowed to take two pebbles from a box we choose, throw away one of the pebbles and put the other pebble in another box we choose. An initial configuration of pebbles is called solvable if it is possible to reach a configuration with no empty box, in a finite (possibly zero) number of moves. Determine all initial configurations of pebbles which are not solvable, but become solvable when an additional pebble is added to a box, no matter which box is chosen.
4
1
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n divides 2i+j determines a sequence
Determine all positive integers
n
≥
2
n\geq 2
n
≥
2
for which there exist integers
x
1
,
x
2
,
…
,
x
n
−
1
x_1,x_2,\ldots ,x_{n-1}
x
1
,
x
2
,
…
,
x
n
−
1
satisfying the condition that if
0
<
i
<
n
,
0
<
j
<
n
,
i
≠
j
0<i<n,0<j<n, i\neq j
0
<
i
<
n
,
0
<
j
<
n
,
i
=
j
and
n
n
n
divides
2
i
+
j
2i+j
2
i
+
j
, then
x
i
<
x
j
x_i<x_j
x
i
<
x
j
.
3
1
Hide problems
On existence of infinitely many positive integers satisfying
We denote the number of positive divisors of a positive integer
m
m
m
by
d
(
m
)
d(m)
d
(
m
)
and the number of distinct prime divisors of
m
m
m
by
ω
(
m
)
\omega(m)
ω
(
m
)
. Let
k
k
k
be a positive integer. Prove that there exist infinitely many positive integers
n
n
n
such that
ω
(
n
)
=
k
\omega(n) = k
ω
(
n
)
=
k
and
d
(
n
)
d(n)
d
(
n
)
does not divide
d
(
a
2
+
b
2
)
d(a^2+b^2)
d
(
a
2
+
b
2
)
for any positive integers
a
,
b
a, b
a
,
b
satisfying
a
+
b
=
n
a + b = n
a
+
b
=
n
.
2
1
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Incentre of ABC, orthocentre of DEF,midpointof Arc collinear
Let
D
D
D
and
E
E
E
be points in the interiors of sides
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, of a triangle
A
B
C
ABC
A
BC
, such that
D
B
=
B
C
=
C
E
DB = BC = CE
D
B
=
BC
=
CE
. Let the lines
C
D
CD
C
D
and
B
E
BE
BE
meet at
F
F
F
. Prove that the incentre
I
I
I
of triangle
A
B
C
ABC
A
BC
, the orthocentre
H
H
H
of triangle
D
E
F
DEF
D
EF
and the midpoint
M
M
M
of the arc
B
A
C
BAC
B
A
C
of the circumcircle of triangle
A
B
C
ABC
A
BC
are collinear.
1
1
Hide problems
If a,b,c are sides of triangle then so are are <a^2+bct>
Determine all real constants
t
t
t
such that whenever
a
a
a
,
b
b
b
and
c
c
c
are the lengths of sides of a triangle, then so are
a
2
+
b
c
t
a^2+bct
a
2
+
b
c
t
,
b
2
+
c
a
t
b^2+cat
b
2
+
c
a
t
,
c
2
+
a
b
t
c^2+abt
c
2
+
ab
t
.