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Contests
National and Regional Contests
North Macedonia Contests
Macedonian Team Selection Test
2013 Macedonian Team Selection Test
2013 Macedonian Team Selection Test
Part of
Macedonian Team Selection Test
Subcontests
(6)
Problem 5
1
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Minimal possible value for diagonal of inscribed rectangle
Let
A
B
C
ABC
A
BC
be a triangle with given sides
a
,
b
,
c
a,b,c
a
,
b
,
c
. Determine the minimal possible length of the diagonal of an inscribed rectangle in this triangle.Note: A rectangle is inscribed in the triangle if two of its consecutive vertices lie on one side of the triangle, while the other two vertices lie on the other two sides of the triangle.
Problem 6
1
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Quotient of partial sums in geometric series is an integer
Let
a
a
a
and
n
>
0
n>0
n
>
0
be integers. Define
a
n
=
1
+
a
+
a
2
.
.
.
+
a
n
−
1
a_{n} = 1+a+a^2...+a^{n-1}
a
n
=
1
+
a
+
a
2
...
+
a
n
−
1
. Show that if
p
∣
a
p
−
1
p|a^p-1
p
∣
a
p
−
1
for all prime divisors of
n
2
−
n
1
n_{2}-n_{1}
n
2
−
n
1
, then the number
a
n
2
−
a
n
1
n
2
−
n
1
\frac{a_{n_{2}}-a_{n_{1}}}{n_{2}-n_{1}}
n
2
−
n
1
a
n
2
−
a
n
1
is an integer.
Problem 4
1
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Nonsymmetric inequality for sum of three fractions
Let
a
>
0
,
b
>
0
,
c
>
0
a>0,b>0,c>0
a
>
0
,
b
>
0
,
c
>
0
and
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
. Show the inequality
a
4
+
b
4
a
2
+
b
2
+
b
3
+
c
3
b
+
c
+
2
a
2
+
b
2
+
2
c
2
2
≥
1
2
\frac{a^4+b^4}{a^2+b^2}+\frac{b^3+c^3}{b+c} + \frac{2a^2+b^2+2c^2}{2} \geq \frac{1}{2}
a
2
+
b
2
a
4
+
b
4
+
b
+
c
b
3
+
c
3
+
2
2
a
2
+
b
2
+
2
c
2
≥
2
1
Problem 3
1
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Functional equation with properties similar to degree map for polynomials
Denote by
Z
∗
\mathbb{Z}^{*}
Z
∗
the set of all nonzero integers and denote by
N
0
\mathbb{N}_{0}
N
0
the set of all nonnegative integers. Find all functions
f
:
Z
∗
→
N
0
f:\mathbb{Z}^{*} \rightarrow \mathbb{N}_{0}
f
:
Z
∗
→
N
0
such that:
(
1
)
(1)
(
1
)
For all
a
,
b
∈
Z
∗
a,b \in \mathbb{Z}^{*}
a
,
b
∈
Z
∗
such that
a
+
b
∈
Z
∗
a+b \in \mathbb{Z}^{*}
a
+
b
∈
Z
∗
we have
f
(
a
+
b
)
≥
f(a+b) \geq
f
(
a
+
b
)
≥
min
{
f
(
a
)
,
f
(
b
)
}
\left \{ f(a),f(b) \right \}
{
f
(
a
)
,
f
(
b
)
}
.
(
2
)
(2)
(
2
)
For all
a
,
b
∈
Z
∗
a, b \in \mathbb{Z}^{*}
a
,
b
∈
Z
∗
we have
f
(
a
b
)
=
f
(
a
)
+
f
(
b
)
f(ab) = f(a)+f(b)
f
(
ab
)
=
f
(
a
)
+
f
(
b
)
.
Problem 2
1
Hide problems
Sums and products of digits and irrational numbers
a) Denote by
S
(
n
)
S(n)
S
(
n
)
the sum of digits of a positive integer
n
n
n
. After the decimal point, we write one after the other the numbers
S
(
1
)
,
S
(
2
)
,
.
.
.
S(1),S(2),...
S
(
1
)
,
S
(
2
)
,
...
. Show that the number obtained is irrational.b) Denote by
P
(
n
)
P(n)
P
(
n
)
the product of digits of a positive integer
n
n
n
. After the decimal point, we write one after the other the numbers
P
(
1
)
,
P
(
2
)
,
.
.
.
P(1),P(2),...
P
(
1
)
,
P
(
2
)
,
...
. Show that the number obtained is irrational.
Problem 1
1
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Triangle sides divided in 3 parts and concurrent lines
The points
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}
A
1
,
A
2
,
B
1
,
B
2
,
C
1
,
C
2
are on the sides
A
B
AB
A
B
,
B
C
BC
BC
and
A
C
AC
A
C
of an acute triangle
A
B
C
ABC
A
BC
such that
A
A
1
=
A
1
A
2
=
A
2
B
=
1
3
A
B
AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB
A
A
1
=
A
1
A
2
=
A
2
B
=
3
1
A
B
,
B
B
1
=
B
1
B
2
=
B
2
C
=
1
3
B
C
BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC
B
B
1
=
B
1
B
2
=
B
2
C
=
3
1
BC
and
C
C
1
=
C
1
C
2
=
C
2
A
=
1
3
A
C
CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC
C
C
1
=
C
1
C
2
=
C
2
A
=
3
1
A
C
. Let
k
A
,
k
B
k_{A}, k_{B}
k
A
,
k
B
and
k
C
k_{C}
k
C
be the circumcircles of the triangles
A
A
1
C
2
AA_{1}C_{2}
A
A
1
C
2
,
B
B
1
A
2
BB_{1}A_{2}
B
B
1
A
2
and
C
C
1
B
2
CC_{1}B_{2}
C
C
1
B
2
respectively. Furthermore, let
a
B
a_{B}
a
B
and
a
C
a_{C}
a
C
be the tangents to
k
A
k_{A}
k
A
at
A
1
A_{1}
A
1
and
C
2
C_{2}
C
2
,
b
C
b_{C}
b
C
and
b
A
b_{A}
b
A
the tangents to
k
B
k_{B}
k
B
at
B
1
B_{1}
B
1
and
A
2
A_{2}
A
2
and
c
A
c_{A}
c
A
and
c
B
c_{B}
c
B
the tangents to
k
C
k_{C}
k
C
at
C
1
C_{1}
C
1
and
B
2
B_{2}
B
2
. Show that the perpendicular lines from the intersection points of
a
B
a_{B}
a
B
and
b
A
b_{A}
b
A
,
b
C
b_{C}
b
C
and
c
B
c_{B}
c
B
,
c
A
c_{A}
c
A
and
a
C
a_{C}
a
C
to
A
B
AB
A
B
,
B
C
BC
BC
and
C
A
CA
C
A
respectively are concurrent.