MathDB
Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2009 Ukraine Team Selection Test
2009 Ukraine Team Selection Test
Part of
Ukraine Team Selection Test
Subcontests
(8)
12
1
Hide problems
sum h_a^{2}/(a ^2-CH_a ^2) >= 3, altitudes
Denote an acute-angle
△
A
B
C
\vartriangle ABC
△
A
BC
with sides
a
,
b
,
c
a, b, c
a
,
b
,
c
respectively by
H
a
,
H
b
,
H
c
{{H}_{a}}, {{H}_{b}}, {{H}_{c}}
H
a
,
H
b
,
H
c
the feet of altitudes
h
a
,
h
b
,
h
c
{{h}_{a}}, {{h}_{b}}, {{h}_{c}}
h
a
,
h
b
,
h
c
. Prove the inequality:
h
a
2
a
2
−
C
H
a
2
+
h
b
2
b
2
−
A
H
b
2
+
h
c
2
c
2
−
B
H
c
2
≥
3
\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3
a
2
−
C
H
a
2
h
a
2
+
b
2
−
A
H
b
2
h
b
2
+
c
2
−
B
H
c
2
h
c
2
≥
3
(Dmitry Petrovsky)
11
1
Hide problems
integers 1 to n^2 in a nxn , yellow / blue the largest in each row / column
Suppose that integers are given
m
<
n
m <n
m
<
n
. Consider a spreadsheet of size
n
×
n
n \times n
n
×
n
, whose cells arbitrarily record all integers from
1
1
1
to
n
2
{{n} ^ {2}}
n
2
. Each row of the table is colored in yellow
m
m
m
the largest elements. Similarly, the blue colors the
m
m
m
of the largest elements in each column. Find the smallest number of cells that are colored yellow and blue at a time
6
1
Hide problems
equal sets modulo p
Find all odd prime numbers
p
p
p
for which there exists a natural number
g
g
g
for which the sets
A
=
{
(
k
2
+
1
)
m
o
d
p
∣
k
=
1
,
2
,
…
,
p
−
1
2
}
A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}
A
=
{
(
k
2
+
1
)
mod
p
∣
k
=
1
,
2
,
…
,
2
p
−
1
}
and
B
=
{
g
k
m
o
d
p
∣
k
=
1
,
2
,
.
.
.
,
p
−
1
2
}
B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}
B
=
{
g
k
mod
p
∣
k
=
1
,
2
,
...
,
2
p
−
1
}
are equal.
4
1
Hide problems
f(x^{n+1}+(y^{n+1})=x^n f(x)+y^n f(y) when x,y>0
Let
n
n
n
be some positive integer. Find all functions
f
:
R
+
→
R
f:{{R}^{+}}\to R
f
:
R
+
→
R
(i.e., functions defined by the set of all positive real numbers with real values) for which equality holds
f
(
x
n
+
1
+
y
n
+
1
)
=
x
n
f
(
x
)
+
y
n
f
(
y
)
f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)
f
(
x
n
+
1
+
y
n
+
1
)
=
x
n
f
(
x
)
+
y
n
f
(
y
)
for any positive real numbers
x
,
y
x, y
x
,
y
3
1
Hide problems
intersection of 2^{n-1} subsets of F is not empty
Let
S
S
S
be a set consisting of
n
n
n
elements,
F
F
F
a set of subsets of
S
S
S
consisting of
2
n
−
1
2^{n-1}
2
n
−
1
subsets such that every three such subsets have a non-empty intersection. a) Show that the intersection of all subsets of
F
F
F
is not empty. b) If you replace the number of sets from
2
n
−
1
2^{n-1}
2
n
−
1
with
2
n
−
1
−
1
2^{n-1}-1
2
n
−
1
−
1
, will the previous answer change?
8
1
Hide problems
fixed point's revenge, 4 circles related, tangent circles related
Two circles
γ
1
,
γ
2
\gamma_1, \gamma_2
γ
1
,
γ
2
are given, with centers at points
O
1
,
O
2
O_1, O_2
O
1
,
O
2
respectively. Select a point
K
K
K
on circle
γ
2
\gamma_2
γ
2
and construct two circles, one
γ
3
\gamma_3
γ
3
that touches circle
γ
2
\gamma_2
γ
2
at point
K
K
K
and circle
γ
1
\gamma_1
γ
1
at a point
A
A
A
, and the other
γ
4
\gamma_4
γ
4
that touches circle
γ
2
\gamma_2
γ
2
at point
K
K
K
and circle
γ
1
\gamma_1
γ
1
at a point
B
B
B
. Prove that, regardless of the choice of point K on circle
γ
2
\gamma_2
γ
2
, all lines
A
B
AB
A
B
pass through a fixed point of the plane.
5
1
Hide problems
triangle of orthocenters is right, AC=BD=CE=DO, O circumcenter of ABCDE
Let
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
be consecutive points on a circle with center
O
O
O
such that
A
C
=
B
D
=
C
E
=
D
O
AC=BD=CE=DO
A
C
=
B
D
=
CE
=
D
O
. Let
H
1
,
H
2
,
H
3
H_1,H_2,H_3
H
1
,
H
2
,
H
3
be the orthocenters triangles
A
C
D
,
B
C
D
,
B
C
E
ACD,BCD,BCE
A
C
D
,
BC
D
,
BCE
respectively. Prove that the triangle
H
1
H
2
H
3
H_1H_2H_3
H
1
H
2
H
3
is right.
2
1
Hide problems
Inequality with non-trivial answer
Let
a
a
a
,
b
b
b
,
c
c
c
are sides of a triangle. Find the least possible value
k
k
k
such that the following inequality always holds: \left|\frac{a\minus{}b}{a\plus{}b}\plus{}\frac{b\minus{}c}{b\plus{}c}\plus{}\frac{c\minus{}a}{c\plus{}a}\right|