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Problems
Contests
National and Regional Contests
Serbia Contests
Federal Math Competition of Serbia and Montenegro
2003 Federal Math Competition of S&M
2003 Federal Math Competition of S&M
Part of
Federal Math Competition of Serbia and Montenegro
Subcontests
(4)
Problem 1
3
Hide problems
2011 as sum of 10 4th powers
Find the number of solutions to the equation
x
1
4
+
x
2
4
+
…
+
x
10
4
=
2011
x_1^4+x_2^4+\ldots+x_{10}^4=2011
x
1
4
+
x
2
4
+
…
+
x
10
4
=
2011
in the set of positive integers.
existence of triangle with sides √a,√b,√c
Given a
△
A
B
C
\triangle ABC
△
A
BC
with the edges
a
,
b
a,b
a
,
b
and
c
c
c
and the area
S
S
S
:(a) Prove that there exists
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
with the sides
a
,
b
\sqrt a,\sqrt b
a
,
b
and
c
\sqrt c
c
. (b) If
S
1
S_1
S
1
is the area of
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
, prove that
S
1
2
≥
S
3
4
S_1^2\ge\frac{S\sqrt3}4
S
1
2
≥
4
S
3
.
10^n|floor((5+√35)^(2n-1))
Prove that the number
⌊
(
5
+
35
)
2
n
−
1
⌋
\left\lfloor\left(5+\sqrt{35}\right)^{2n-1}\right\rfloor
⌊
(
5
+
35
)
2
n
−
1
⌋
is divisible by
1
0
n
10^n
1
0
n
for each
n
∈
N
n\in\mathbb N
n
∈
N
.
Problem 2
3
Hide problems
not rational
Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.
maximum number of lattice unit squares intersecting segment
Given a segment
A
B
AB
A
B
of length
2003
2003
2003
in a coordinate plane, determine the maximal number of unit squares with vertices in the lattice points whose intersection with the given segment is non-empty.
A functional inequality
Let
f
:
[
0
,
1
]
→
R
f : [0, 1] \to\ R
f
:
[
0
,
1
]
→
R
be a function such that :-
1.
)
1.)
1.
)
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for all
x
x
x
in
[
0
,
1
]
[0,1]
[
0
,
1
]
.
2.
)
2.)
2.
)
f
(
1
)
=
1
f(1) = 1
f
(
1
)
=
1
.
3.
)
3.)
3.
)
If
x
1
,
x
2
x_1 , x_2
x
1
,
x
2
are in
[
0
,
1
]
[0,1]
[
0
,
1
]
such that
x
1
+
x
2
≤
1
x_1 + x_2 \le 1
x
1
+
x
2
≤
1
, then
f
(
x
1
)
+
f
(
x
2
)
≤
f
(
x
1
+
x
2
)
f(x_1) + f(x_2) \le f(x_1 + x_2)
f
(
x
1
)
+
f
(
x
2
)
≤
f
(
x
1
+
x
2
)
. Show that
f
(
x
)
≤
2
x
f(x) \le 2x
f
(
x
)
≤
2
x
for all
x
x
x
in
[
0
,
1
]
[0,1]
[
0
,
1
]
.
Problem 4
3
Hide problems
locus of tangency points as center moves
An acute angle with the vertex
O
O
O
and the rays
O
p
1
Op_1
O
p
1
and
O
p
2
Op_2
O
p
2
is given in a plane. Let
k
1
k_1
k
1
be a circle with the center on
O
p
1
Op_1
O
p
1
which is tangent to
O
p
2
Op_2
O
p
2
. Let
k
2
k_2
k
2
be the circle that is tangent to both rays
O
p
1
Op_1
O
p
1
and
O
p
2
Op_2
O
p
2
and to the circle
k
1
k_1
k
1
from outside. Find the locus of tangency points of
k
1
k_1
k
1
and
k
2
k_2
k
2
when center of
k
1
k_1
k
1
moves along the ray
O
p
1
Op_1
O
p
1
.
nice problem
Let
S
S
S
be the subset of
N
N
N
(
N
N
N
is the set of all natural numbers) satisfying: i)Among each
2003
2003
2003
consecutive natural numbers there exist at least one contained in
S
S
S
; ii)If
n
∈
S
n \in S
n
∈
S
and
n
>
1
n>1
n
>
1
then
[
n
2
]
∈
S
[\frac{n}{2}] \in S
[
2
n
]
∈
S
Prove that:
S
=
N
S=N
S
=
N
I hope it hasn't posted before. :lol: :lol:
subset partition
Let
n
n
n
be an even number, and
S
S
S
be the set of all arrays of length
n
n
n
whose elements are from the set
{
0
,
1
}
\left\{0,1\right\}
{
0
,
1
}
. Prove that
S
S
S
can be partitioned into disjoint three-element subsets such that for each three arrays \left(a_i\right)_{i \equal{} 1}^n, \left(b_i\right)_{i \equal{} 1}^n, \left(c_i\right)_{i \equal{} 1}^n which belong to the same subset and for each
i
∈
{
1
,
2
,
.
.
.
,
n
}
i\in\left\{1,2,...,n\right\}
i
∈
{
1
,
2
,
...
,
n
}
, the number a_i \plus{} b_i \plus{} c_i is divisible by
2
2
2
.
Problem 3
3
Hide problems
side equality in triangle gives angles
Let
a
,
b
a,b
a
,
b
and
c
c
c
be the lengths of the edges of a triangle whose angles are
α
=
4
0
∘
,
β
=
6
0
∘
\alpha=40^\circ,\beta=60^\circ
α
=
4
0
∘
,
β
=
6
0
∘
and
γ
=
8
0
∘
\gamma=80^\circ
γ
=
8
0
∘
. Prove that
a
(
a
+
b
+
c
)
=
b
(
b
+
c
)
.
a(a+b+c)=b(b+c).
a
(
a
+
b
+
c
)
=
b
(
b
+
c
)
.
Determine the set of all points P....very easy
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle. Determine the set of all points
P
P
P
from the region between the parallel lines
A
B
AB
A
B
and
C
D
CD
C
D
such that
∠
A
P
B
=
∠
C
P
D
\angle APB=\angle CPD
∠
A
PB
=
∠
CP
D
.
easy and nice
Given a circle
k
k
k
and the point
P
P
P
outside it, an arbitrary line
s
s
s
passing through
P
P
P
intersects
k
k
k
at the points
A
A
A
and
B
B
B
. Let
M
M
M
and
N
N
N
be the midpoints of the arcs determined by the points
A
A
A
and
B
B
B
and let
C
C
C
be the point on
A
B
AB
A
B
such that
P
C
2
=
P
A
⋅
P
B
PC^2=PA\cdot PB
P
C
2
=
P
A
⋅
PB
. Prove that
∠
M
C
N
\angle MCN
∠
MCN
doesn't depend on the choice of
s
s
s
. [Moderator edit: This problem has also been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=56295 .]