MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria EGMO TST
2017 Bulgaria EGMO TST
2017 Bulgaria EGMO TST
Part of
Bulgaria EGMO TST
Subcontests
(3)
3
1
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Linearization bruh
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
be positive real numbers with
a
+
b
+
c
+
d
=
4
a+b+c+d = 4
a
+
b
+
c
+
d
=
4
. Prove that
a
b
2
+
1
+
b
c
2
+
1
+
c
d
2
+
1
+
d
a
2
+
1
≥
2
\frac{a}{b^2 + 1} + \frac{b}{c^2+1} + \frac{c}{d^2+1} + \frac{d}{a^2+1} \geq 2
b
2
+
1
a
+
c
2
+
1
b
+
d
2
+
1
c
+
a
2
+
1
d
≥
2
.
2
2
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Challenging geometry with incircles
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. The line
A
I
AI
A
I
intersects
B
C
BC
BC
and the circumcircle of
A
B
C
ABC
A
BC
at the points
T
T
T
and
S
S
S
, respectively. Let
K
K
K
and
L
L
L
be the incenters of
S
B
T
SBT
SBT
and
S
C
T
SCT
SCT
, respectively,
M
M
M
be the midpoint of
B
C
BC
BC
and
P
P
P
be the reflection of
I
I
I
with respect to
K
L
KL
K
L
. a) Prove that
M
M
M
,
T
T
T
,
K
K
K
and
L
L
L
are concyclic. b) Determine the measure of
∠
B
P
C
\angle BPC
∠
BPC
.
Polychromatic colouring of a table
Let
n
n
n
be a positive integer. Determine the smallest positive integer
k
k
k
such that for any colouring of the cells of a
2
n
×
k
2n\times k
2
n
×
k
table with
n
n
n
colours there are two rows and two columns which intersect in four squares of the same colour.
1
2
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A triangulation too acute to be cute
Prove that every convex polygon has at most one triangulation consisting entirely of acute triangles.
Functional equation
Let
Q
+
\mathbb{Q^+}
Q
+
denote the set of positive rational numbers. Determine all functions
f
:
Q
+
→
Q
+
f: \mathbb{Q^+} \to \mathbb{Q^+}
f
:
Q
+
→
Q
+
that satisfy the conditions
f
(
x
x
+
1
)
=
f
(
x
)
x
+
1
and
f
(
1
x
)
=
f
(
x
)
x
3
f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}
f
(
x
+
1
x
)
=
x
+
1
f
(
x
)
and
f
(
x
1
)
=
x
3
f
(
x
)
for all
x
∈
Q
+
.
x \in \mathbb{Q^+}.
x
∈
Q
+
.