MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2008 Poland - Second Round
2008 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
3
2
Hide problems
Not very attractive functional equation.
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
for which the equality f(f(x)\minus{}y)\equal{}f(x)\plus{}f(f(y)\minus{}f(\minus{}x))\plus{}x holds for all real
x
,
y
x,y
x
,
y
.
Multiplications and digits
We have a positive integer
n
n
n
such that
n
≠
3
k
n \neq 3k
n
=
3
k
. Prove that there exists a positive integer
m
m
m
such that
∀
k
∈
N
k
≥
m
k
\forall_{k\in N \ k\geq m} \ k
∀
k
∈
N
k
≥
m
k
can be represented as a sum of digits of some multiplication of
n
n
n
.
2
2
Hide problems
Equal angles in the pentagon and perpendicular lines
In the convex pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
following equalities holds: \angle ABD\equal{} \angle ACE, \angle ACB\equal{}\angle ACD, \angle ADC\equal{}\angle ADE and \angle ADB\equal{}\angle AEC. The point
S
S
S
is the intersection of the segments
B
D
BD
B
D
and
C
E
CE
CE
. Prove that lines
A
S
AS
A
S
and
C
D
CD
C
D
are perpendicular.
Geometrical identity
We are given a triangle
A
B
C
ABC
A
BC
such that AC \equal{} BC. There is a point
D
D
D
lying on the segment
A
B
AB
A
B
, and
A
D
<
D
B
AD < DB
A
D
<
D
B
. The point
E
E
E
is symmetrical to
A
A
A
with respect to
C
D
CD
C
D
. Prove that: \frac {AC}{CD} \equal{} \frac {BE}{BD \minus{} AD}
1
2
Hide problems
Consecutive integers of the form x^3+2y^2
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form x^3\plus{}2y^2, with
x
,
y
x, y
x
,
y
being integers.
A board n on n
We have an
n
×
n
n \times n
n
×
n
board, and in every square there is an integer. The sum of all integers on the board is
0
0
0
. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists
n
≥
2
n\geq 2
n
≥
2
such that we can turn all the integers into zeros in a finite number of actions.