MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
1999 Ukraine Team Selection Test
1999 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(11)
7
1
Hide problems
sum of assigned angles is multiple of 720 then odd number of self-intersections
Let
P
1
P
2
.
.
.
P
n
P_1P_2...P_n
P
1
P
2
...
P
n
be an oriented closed polygonal line with no three segments passing through a single point. Each point
P
i
P_i
P
i
is assinged the angle
18
0
o
−
∠
P
i
−
1
P
i
P
i
+
1
≥
0
180^o - \angle P_{i-1}P_iP_{i+1} \ge 0
18
0
o
−
∠
P
i
−
1
P
i
P
i
+
1
≥
0
if
P
i
+
1
P_{i+1}
P
i
+
1
lies on the left from the ray
P
i
−
1
P
i
P_{i-1}P_i
P
i
−
1
P
i
, and the angle
−
(
18
0
o
−
∠
P
i
−
1
P
i
P
i
+
1
)
<
0
-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0
−
(
18
0
o
−
∠
P
i
−
1
P
i
P
i
+
1
)
<
0
if
P
i
+
1
P_{i+1}
P
i
+
1
lies on the right. Prove that if the sum of all the assigned angles is a multiple of
72
0
o
720^o
72
0
o
, then the number of self-intersections of the polygonal line is odd
4
1
Hide problems
sum sin(n+1)x/sin nx < 2cos x/sin^2 x
If
n
∈
N
n \in N
n
∈
N
and
0
<
x
<
π
2
n
0 < x <\frac{\pi}{2n}
0
<
x
<
2
n
π
, prove the inequality
sin
2
x
sin
x
+
sin
3
x
sin
2
x
+
.
.
.
+
sin
(
n
+
1
)
x
sin
n
x
<
2
cos
x
sin
2
x
\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}
s
i
n
x
s
i
n
2
x
+
s
i
n
2
x
s
i
n
3
x
+
...
+
s
i
n
n
x
s
i
n
(
n
+
1
)
x
<
2
s
i
n
2
x
c
o
s
x
. .
12
1
Hide problems
number of persons having no signal <= [n+3 -\frac{18m}{n}]
In a group of
n
≥
4
n \ge 4
n
≥
4
persons, every three who know each other have a common signal. Assume that these signals are not repeated and that there are
m
≥
1
m \ge 1
m
≥
1
signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed
[
n
+
3
−
18
m
n
]
\left[n+3 -\frac{18m}{n}\right]
[
n
+
3
−
n
18
m
]
10
1
Hide problems
2^{w(n)} \le k\sqrt[4]{n}. no of (positive) prime divisors related inequality
For a natural number
n
n
n
, let
w
(
n
)
w(n)
w
(
n
)
denote the number of (positive) prime divisors of
n
n
n
. Find the smallest positive integer
k
k
k
such that
2
w
(
n
)
≤
k
n
4
2^{w(n)} \le k \sqrt[4]{ n}
2
w
(
n
)
≤
k
4
n
for each
n
∈
N
n \in N
n
∈
N
.
9
1
Hide problems
f(x+y) = f(x)u(y)+ f(y) , strictly increasing function f : R \to R
Find all functions
u
:
R
→
R
u : R \to R
u
:
R
→
R
for which there is a strictly increasing function
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
+
y
)
=
f
(
x
)
u
(
y
)
+
f
(
y
)
f(x+y) = f(x)u(y)+ f(y)
f
(
x
+
y
)
=
f
(
x
)
u
(
y
)
+
f
(
y
)
for all
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
.
8
1
Hide problems
x^n + 2^n + 1 divides x^{n+1} +2^{n+1} +1
Find all pairs
(
x
,
n
)
(x,n)
(
x
,
n
)
of positive integers for which
x
n
+
2
n
+
1
x^n + 2^n + 1
x
n
+
2
n
+
1
divides
x
n
+
1
+
2
n
+
1
+
1
x^{n+1} +2^{n+1} +1
x
n
+
1
+
2
n
+
1
+
1
.
6
1
Hide problems
f(x) = (x^2 +x)^2n+1 is irreducible over Z[x]
Show that for any
n
∈
N
n \in N
n
∈
N
the polynomial
f
(
x
)
=
(
x
2
+
x
)
2
n
+
1
f(x) = (x^2 +x)^{2^n}+1
f
(
x
)
=
(
x
2
+
x
)
2
n
+
1
is irreducible over
Z
[
x
]
Z[x]
Z
[
x
]
.
5
1
Hide problems
pairs of equal angles in a convex pentagon wanted
A convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
with
D
C
=
D
E
DC = DE
D
C
=
D
E
and
∠
D
C
B
=
∠
D
E
A
=
9
0
o
\angle DCB = \angle DEA = 90^o
∠
D
CB
=
∠
D
E
A
=
9
0
o
is given. Let
F
F
F
be a point on the segment
A
B
AB
A
B
such that
A
F
:
B
F
=
A
E
:
B
C
AF : BF = AE : BC
A
F
:
BF
=
A
E
:
BC
. Prove that
∠
F
C
E
=
∠
A
D
E
\angle FCE = \angle ADE
∠
FCE
=
∠
A
D
E
and
∠
F
E
C
=
∠
B
D
C
\angle FEC = \angle BDC
∠
FEC
=
∠
B
D
C
.
3
1
Hide problems
maxi number of elements in F, a family of m elements of 1,2,...,n
Let
m
,
n
m,n
m
,
n
be positive integers with
m
≤
n
m \le n
m
≤
n
, and let
F
F
F
be a family of
m
m
m
-element subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1,2,...,n\}
{
1
,
2
,
...
,
n
}
satisfying
A
∩
B
≠
∅
A \cap B \ne \varnothing
A
∩
B
=
∅
for all
A
,
B
∈
F
A,B \in F
A
,
B
∈
F
. Determine the maximum possible number of elements in
F
F
F
.
2
1
Hide problems
j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12 , exist integers >100
Show that there exist integers
j
,
k
,
l
,
m
,
n
j,k,l,m,n
j
,
k
,
l
,
m
,
n
greater than
100
100
100
such that
j
2
+
k
2
+
l
2
+
m
2
+
n
2
=
j
k
l
m
n
−
12
j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12
j
2
+
k
2
+
l
2
+
m
2
+
n
2
=
jk
l
mn
−
12
.
1
1
Hide problems
locus of intersection of perpendicular lines, cyclic given
A triangle
A
B
C
ABC
A
BC
is given. Points
E
,
F
,
G
E,F,G
E
,
F
,
G
are arbitrarily selected on the sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
, respectively, such that
A
F
⊥
E
G
AF\perp EG
A
F
⊥
EG
and the quadrilateral
A
E
F
G
AEFG
A
EFG
is cyclic. Find the locus of the intersection point of
A
F
AF
A
F
and
E
G
EG
EG
.