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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2002 Poland - Second Round
2002 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
2
2
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Right triangle ABC is base of pyramid ABCD, AD=BD and AB=CD
Triangle
A
B
C
ABC
A
BC
with
∠
B
A
C
=
9
0
∘
\angle BAC=90^{\circ}
∠
B
A
C
=
9
0
∘
is the base of the pyramid
A
B
C
D
ABCD
A
BC
D
. Moreover,
A
D
=
B
D
AD=BD
A
D
=
B
D
and
A
B
=
C
D
AB=CD
A
B
=
C
D
. Prove that
∠
A
C
D
≥
3
0
∘
\angle ACD\ge 30^{\circ}
∠
A
C
D
≥
3
0
∘
.
Quadraliteral with strange angles
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, both
∠
A
D
B
=
2
∠
A
C
B
\angle ADB=2\angle ACB
∠
A
D
B
=
2∠
A
CB
and
∠
B
D
C
=
2
∠
B
A
C
\angle BDC=2\angle BAC
∠
B
D
C
=
2∠
B
A
C
. Prove that
A
D
=
C
D
AD=CD
A
D
=
C
D
.
3
2
Hide problems
Poland 2002
A positive integer
n
n
n
is given. In an association consisting of
n
n
n
members work
6
6
6
commissions. Each commission contains at least
n
4
\large \frac{n}{4}
4
n
persons. Prove that there exist two commissions containing at least
n
30
\large \frac{n}{30}
30
n
persons in common.
Find n for which inequality with x_i,y_i holds
Find all positive integers
n
n
n
such that for all real numbers
x
1
,
x
2
,
…
,
x
n
,
y
1
,
y
2
,
…
,
y
n
x_1,x_2,\ldots ,x_n,y_1,y_2,\ldots ,y_n
x
1
,
x
2
,
…
,
x
n
,
y
1
,
y
2
,
…
,
y
n
the following inequality holds:
x
1
x
2
…
x
n
+
y
1
y
2
…
y
n
≤
x
1
2
+
y
1
2
⋅
x
2
2
+
y
2
2
⋅
⋯
x
n
2
+
y
n
2
⋅
x_1x_2\ldots x_n+y_1y_2\ldots y_n\le\sqrt{x_1^2+y_1^2}\cdot\sqrt{x_2^2+y_2^2}\cdot \cdots \sqrt{x_n^2+y_n^2}\cdot
x
1
x
2
…
x
n
+
y
1
y
2
…
y
n
≤
x
1
2
+
y
1
2
⋅
x
2
2
+
y
2
2
⋅
⋯
x
n
2
+
y
n
2
⋅
1
2
Hide problems
Periodic
Prove that all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
satisfying, for all real
x
x
x
,
f
(
x
)
=
f
(
2
x
)
=
f
(
1
−
x
)
f(x)=f(2x)=f(1-x)
f
(
x
)
=
f
(
2
x
)
=
f
(
1
−
x
)
are periodic.
All expressions of p,q,r are primes
Find all numbers
p
≤
q
≤
r
p\le q\le r
p
≤
q
≤
r
such that all the numbers
p
q
+
r
,
p
q
+
r
2
,
q
r
+
p
,
q
r
+
p
2
,
r
p
+
q
,
r
p
+
q
2
pq+r,pq+r^2,qr+p,qr+p^2,rp+q,rp+q^2
pq
+
r
,
pq
+
r
2
,
q
r
+
p
,
q
r
+
p
2
,
r
p
+
q
,
r
p
+
q
2
are prime.