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Problems
Contests
National and Regional Contests
Serbia Contests
Federal Math Competition of Serbia and Montenegro
2001 Federal Math Competition of S&M
2001 Federal Math Competition of S&M
Part of
Federal Math Competition of Serbia and Montenegro
Subcontests
(4)
Problem 4
3
Hide problems
removing coins from a pile
There are
n
n
n
coins in the pile. Two players play a game by alternately performing a move. A move consists of taking
5
,
7
5,7
5
,
7
or
11
11
11
coins away from the pile. The player unable to perform a move loses the game. Which player - the one playing first or second - has the winning strategy if:(a)
n
=
2001
n=2001
n
=
2001
; (b)
n
=
5000
n=5000
n
=
5000
?
arithmetic means given, find max/min(max)
Let
S
S
S
be the set of all
n
n
n
-tuples of real numbers, with the property that among the numbers
x
1
,
x
1
+
x
2
2
,
…
,
x
1
+
x
2
+
…
+
x
n
n
x_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n
x
1
,
2
x
1
+
x
2
,
…
,
n
x
1
+
x
2
+
…
+
x
n
the least is equal to
0
0
0
, and the greatest is equal to
1
1
1
. Determine
max
(
x
1
,
x
2
,
…
,
x
n
)
∈
S
max
1
≤
i
,
j
≤
n
(
x
i
−
x
j
)
and
min
(
x
1
,
x
2
,
…
,
x
n
)
∈
S
max
1
≤
i
,
j
≤
n
(
x
i
−
x
j
)
.
\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).
(
x
1
,
x
2
,
…
,
x
n
)
∈
S
max
1
≤
i
,
j
≤
n
max
(
x
i
−
x
j
)
and
(
x
1
,
x
2
,
…
,
x
n
)
∈
S
min
1
≤
i
,
j
≤
n
max
(
x
i
−
x
j
)
.
pyramid, parallelogram base
Parallelogram
A
B
C
D
ABCD
A
BC
D
is the base of a pyramid
S
A
B
C
D
SABCD
S
A
BC
D
. Planes determined by triangles
A
S
C
ASC
A
SC
and
B
S
D
BSD
BS
D
are mutually perpendicular. Find the area of the side
A
S
D
ASD
A
S
D
, if areas of sides
A
S
B
,
B
S
C
ASB,BSC
A
SB
,
BSC
and
C
S
D
CSD
CS
D
are equal to
x
,
y
x,y
x
,
y
and
z
z
z
, respectively.
Problem 2
3
Hide problems
square on a sphere
Vertices of a square
A
B
C
D
ABCD
A
BC
D
of side
25
4
\frac{25}4
4
25
lie on a sphere. Parallel lines passing through points
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
intersect the sphere at points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
and
D
1
D_1
D
1
, respectively. Given that
A
A
1
=
2
AA_1=2
A
A
1
=
2
,
B
B
1
=
10
BB_1=10
B
B
1
=
10
,
C
C
1
=
6
CC_1=6
C
C
1
=
6
, determine the length of the segment
D
D
1
DD_1
D
D
1
.
5 segments, triangles formed, one must be acute
Given are
5
5
5
segments, such that from any three of them one can form a triangle. Prove that from some three of them one can form an acute-angled triangle.
inequality, sum of (x_n^2)/(n^3)
Let
x
1
,
x
2
,
…
,
x
2001
x_1,x_2,\ldots,x_{2001}
x
1
,
x
2
,
…
,
x
2001
be positive numbers such that
x
i
2
≥
x
1
2
+
x
2
2
2
3
+
x
3
2
3
3
+
…
+
x
i
−
1
2
(
i
−
1
)
3
for
2
≤
i
≤
2001.
x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.
x
i
2
≥
x
1
2
+
2
3
x
2
2
+
3
3
x
3
2
+
…
+
(
i
−
1
)
3
x
i
−
1
2
for
2
≤
i
≤
2001.
Prove that
∑
i
=
2
2001
x
i
x
1
+
x
2
+
…
+
x
i
−
1
>
1.999
\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999
∑
i
=
2
2001
x
1
+
x
2
+
…
+
x
i
−
1
x
i
>
1.999
.
Problem 3
3
Hide problems
Prime number ineq
Let
p
1
,
p
2
,
.
.
.
,
p
n
p_{1}, p_{2},...,p_{n}
p
1
,
p
2
,
...
,
p
n
, where
n
>
2
n>2
n
>
2
, be the first
n
n
n
prime numbers. Prove that
1
p
1
2
+
1
p
2
2
+
.
.
.
+
1
p
n
2
+
1
p
1
p
2
.
.
.
p
n
<
1
2
\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}
p
1
2
1
+
p
2
2
1
+
...
+
p
n
2
1
+
p
1
p
2
...
p
n
1
<
2
1
{0,1,...,2k+1}, odd zeroes
Let
k
k
k
be a positive integer and
N
k
N_k
N
k
be the number of sequences of length
2001
2001
2001
, all members of which are elements of the set
{
0
,
1
,
2
,
…
,
2
k
+
1
}
\{0,1,2,\ldots,2k+1\}
{
0
,
1
,
2
,
…
,
2
k
+
1
}
, and the number of zeroes among these is odd. Find the greatest power of
2
2
2
which divides
N
k
N_k
N
k
.
paint in plane
Determine all positive integers
n
n
n
for which there is a coloring of all points in space so that each of the following conditions is satisfied: (i) Each point is painted in exactly one color. (ii) Exactly
n
n
n
colors are used. (iii) Each line is painted in at most two different colors.
Problem 1
3
Hide problems
SSSS and diagonals perpendicular
Let
A
B
C
D
ABCD
A
BC
D
and
A
1
B
1
C
1
D
1
A_1B_1C_1D_1
A
1
B
1
C
1
D
1
be convex quadrangles in a plane, such that
A
B
=
A
1
B
1
AB=A_1B_1
A
B
=
A
1
B
1
,
B
C
=
B
1
C
1
BC=B_1C_1
BC
=
B
1
C
1
,
C
D
=
C
1
D
1
CD=C_1D_1
C
D
=
C
1
D
1
and
D
A
=
D
1
A
1
DA=D_1A_1
D
A
=
D
1
A
1
. Given that diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular to each other, prove that the same holds for diagonals
A
1
C
1
A_1C_1
A
1
C
1
and
B
1
D
1
B_1D_1
B
1
D
1
.
3a=x^2+2y^2 implies a=b^2+2c^2
Let
S
=
{
x
2
+
2
y
2
∣
x
,
y
∈
Z
}
S=\{x^2+2y^2\mid x,y\in\mathbb Z\}
S
=
{
x
2
+
2
y
2
∣
x
,
y
∈
Z
}
. If
a
a
a
is an integer with the property that
3
a
3a
3
a
belongs to
S
S
S
, prove that then
a
a
a
belongs to
S
S
S
as well.
x^y + y = y^x + x
Solve in positive integers
x
y
+
y
=
y
x
+
x
x^y + y = y^x + x
x
y
+
y
=
y
x
+
x