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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1983 Polish MO Finals
1983 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(6)
6
1
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dihedral angles of tetrahedron are acute, then faces are acute-angled triangles
Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.
5
1
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length of the vector c_1a_1+c_2a_2+c_3a_3 is at least 2
On the plane are given unit vectors
a
1
→
,
a
2
→
,
a
3
→
\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}
a
1
,
a
2
,
a
3
. Show that one can choose numbers
c
1
,
c
2
,
c
3
∈
{
−
1
,
1
}
c_1,c_2,c_3 \in \{-1,1\}
c
1
,
c
2
,
c
3
∈
{
−
1
,
1
}
such that the length of the vector
c
1
a
1
→
+
c
2
a
2
→
+
c
3
a
3
→
c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}
c
1
a
1
+
c
2
a
2
+
c
3
a
3
is at least
2
2
2
.
4
1
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\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a when ab = cd
Prove that if natural numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy the equality
a
b
=
c
d
ab = cd
ab
=
c
d
, then
g
c
d
(
a
,
c
)
g
c
d
(
a
,
d
)
g
c
d
(
a
,
b
,
c
,
d
)
=
a
\frac{gcd(a,c)gcd(a,d)}{gcd(a,b,c,d)}= a
g
c
d
(
a
,
b
,
c
,
d
)
g
c
d
(
a
,
c
)
g
c
d
(
a
,
d
)
=
a
2
1
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a irrational in (0,1), p/q < a <r/s, r/s -p/q< 1/N, rq- ps = 1
Let be given an irrational number
a
a
a
in the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
and a positive integer
N
N
N
. Prove that there exist positive integers
p
,
q
,
r
,
s
p,q,r,s
p
,
q
,
r
,
s
such that
p
q
<
a
<
r
s
,
r
s
−
p
q
<
1
N
\frac{p}{q} < a <\frac{r}{s}, \frac{r}{s} -\frac{p}{q}<\frac{1}{N}
q
p
<
a
<
s
r
,
s
r
−
q
p
<
N
1
, and
r
q
−
p
s
=
1
rq- ps = 1
r
q
−
p
s
=
1
.
3
1
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one-player game on an infinite chessboard
Consider the following one-player game on an infinite chessboard. If two horizontally or vertically adjacent squares are occupied by a pawn each, and a square on the same line that is adjacent to one of them is empty, then it is allowed to remove the two pawns and place a pawn on the third (empty) square. Prove that if in the initial position all the pawns were forming a rectangle with the number of squares divisible by
3
3
3
, then it is not possible to end the game with only one pawn left on the board.
1
1
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even no of triangles P_iP_jP_k containing interior P in convex n-gon
On the plane are given a convex
n
n
n
-gon
P
1
P
2
.
.
.
.
P
n
P_1P_2....P_n
P
1
P
2
....
P
n
and a point
Q
Q
Q
inside it, not lying on any of its diagonals. Prove that if
n
n
n
is even, then the number of triangles
P
i
P
j
P
k
P_iP_jP_k
P
i
P
j
P
k
containing the point
Q
Q
Q
is even.