MathDB
Problems
Contests
National and Regional Contests
Lithuania Contests
Grand Duchy of Lithuania
2010 Grand Duchy of Lithuania
2010 Grand Duchy of Lithuania
Part of
Grand Duchy of Lithuania
Subcontests
(5)
1
1
Hide problems
16 points are placed in the centers of a 4x4 chess table
Sixteen points are placed in the centers of a
4
×
4
4 \times 4
4
×
4
chess table in the following way: • • • • • • • • • • • • • • • • (a) Prove that one may choose
6
6
6
points such that no isoceles triangle can be drawn with the vertices at these points. (b) Prove that one cannot choose
7
7
7
points with the above property.
2
1
Hide problems
1/a_1 + 2/a_2 +...+ n/a_n = (a_1 + a_2 +... + a_n)/ 2
Find all positive integers
n
n
n
for which there are distinct integer numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
such that
1
a
1
+
2
a
2
+
.
.
.
+
n
a
n
=
a
1
+
a
2
+
.
.
.
+
a
n
2
\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}
a
1
1
+
a
2
2
+
...
+
a
n
n
=
2
a
1
+
a
2
+
...
+
a
n
5
1
Hide problems
quotient if n+36 is divided by 4 is a 4digit number, the digits beeing 2,0,0,9
Find positive integers n that satisfy the following two conditions: (a) the quotient obtained when
n
n
n
is divided by
9
9
9
is a positive three digit number, that has equal digits. (b) the quotient obtained when
n
+
36
n + 36
n
+
36
is divided by
4
4
4
is a four digit number, the digits beeing
2
,
0
,
0
,
9
2, 0, 0, 9
2
,
0
,
0
,
9
in some order.
3
1
Hide problems
at a strange party, each person knew exactly 22 others
At a strange party, each person knew exactly
22
22
22
others. For any pair of people
X
X
X
and
Y
Y
Y
who knew each other, there was no other person at the party that they both knew. For any pair of people
X
X
X
and
Y
Y
Y
who did not know one another, there were exactly 6 other people that they both knew. How many people were at the party?
4
1
Hide problems
BK = 2DC wanted, < ABC = < KAD = < AKD, < CAB = 90^o
In the triangle
A
B
C
ABC
A
BC
angle
C
C
C
is a right angle. On the side
A
C
AC
A
C
point
D
D
D
has been found, and on the segment
B
D
BD
B
D
point K has been found such that
∠
A
B
C
=
∠
K
A
D
=
∠
A
K
D
\angle ABC = \angle KAD = \angle AKD
∠
A
BC
=
∠
K
A
D
=
∠
A
KD
. Prove that
B
K
=
2
D
C
BK = 2DC
B
K
=
2
D
C
.