MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - First Round
2004 Poland - First Round
2004 Poland - First Round
Part of
Poland - First Round
Subcontests
(4)
4
1
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Optimization problem
4.Given is
n
∈
Z
n \in \mathbb Z
n
∈
Z
and positive reals a,b. Find possible maximal value of the sum:
x
1
y
1
+
x
2
y
2
+
.
.
.
+
x
n
y
n
x_1y_1 + x_2y_2 + ... + x_ny_n
x
1
y
1
+
x
2
y
2
+
...
+
x
n
y
n
when
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
and
y
1
,
y
2
,
.
.
.
,
y
n
y_1,y_2,...,y_n
y
1
,
y
2
,
...
,
y
n
are in
<
0
;
1
>
<0;1>
<
0
;
1
>
and satisfies:
x
1
+
x
2
+
.
.
.
+
x
n
≤
a
x_1 + x_2 + ... + x_n \leq a
x
1
+
x
2
+
...
+
x
n
≤
a
and
y
1
+
y
2
+
.
.
.
+
y
n
≤
b
y_1 + y_2 + ... + y_n \leq b
y
1
+
y
2
+
...
+
y
n
≤
b
3
1
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Perpendicular projections and acute-angled triangle
3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and:
E
F
F
D
=
A
D
D
B
\frac{EF}{FD}=\frac{AD}{DB}
F
D
EF
=
D
B
A
D
Prove that
C
F
⊥
A
E
CF \bot AE
CF
⊥
A
E
2
1
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Find all natural n>1
2. Find all natural
n
>
1
n>1
n
>
1
for which value of the sum
2
2
+
3
2
+
.
.
.
+
n
2
2^2+3^2+...+n^2
2
2
+
3
2
+
...
+
n
2
equals
p
k
p^k
p
k
where p is prime and k is natural
1
1
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Polish equality
1. Solve in real numbers x,y,z :
{
x
2
=
y
z
+
1
y
2
=
z
x
+
2
z
2
=
x
y
+
4
\{\begin{array}{ccc} x^2=yz+1 \\ y^2=zx+2 \\ z^2=xy+4 \\ \end{array}
{
x
2
=
yz
+
1
y
2
=
z
x
+
2
z
2
=
x
y
+
4