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National and Regional Contests
Moldova Contests
Chisinau City MO
1973 Chisinau City MO
1973 Chisinau City MO
Part of
Chisinau City MO
Subcontests
(8)
70
1
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Chisinau MO p70 1973 p = q if naturals, p^p + q^q = p^q + q^p
The natural numbers
p
,
q
p, q
p
,
q
satisfy the relation
p
p
+
q
q
=
p
q
+
q
p
p^p + q^q = p^q + q^p
p
p
+
q
q
=
p
q
+
q
p
. Prove that
p
=
q
p = q
p
=
q
.
69
1
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Chisinau MO p69 1973 compare 1 with 0.99999^{1.00001}x 1.00001^{0.99999}
Greater or less than one is the number
0.9999
9
1.00001
⋅
1.0000
1
0.99999
0.99999^{1.00001} \cdot 1.00001^{0.99999}
0.9999
9
1.00001
⋅
1.0000
1
0.99999
?
68
1
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Chisinau MO p68 1973 ABC equilater if AOB, BOC, COA similar
Inside the triangle
A
B
C
ABC
A
BC
, point
O
O
O
was chosen so that the triangles
A
O
B
,
B
O
C
,
C
O
A
AOB, BOC, COA
A
OB
,
BOC
,
CO
A
turned out to be similar. Prove that triangle
A
B
C
ABC
A
BC
is equilateral.
67
1
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Chisinau MO p67 1973 max sum of naturals with product 10^{10}
The product of
10
10
10
natural numbers is equal to
1
0
10
10^{10}
1
0
10
. What is the largest possible sum of these numbers?
66
1
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Chisinau MO p66 1973 symmetry wrt point, square
If
A
A
A
and
B
B
B
are points of the plane, then by
A
∗
B
A * B
A
∗
B
we denote a point symmetric to
A
A
A
with respect to
B
B
B
. Is it possible, by applying the operation
∗
*
∗
several times, to obtain from the three vertices of a given square its fourth vertex?
65
1
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Chisinau MO p65 1973 combo geo with N chords, sum of lenghts <=3.15N
A finite number of chords is drawn in a circle
1
1
1
cm in diameter so that any diameter of the circle intersects at most
N
N
N
of these chords. Prove that the sum of the lengths of all chords is less than
3.15
⋅
N
3.15 \cdot N
3.15
⋅
N
cm.
64
1
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Chisinau MO p64 1973, 1973 zeros, in decimal (5+\sqrt{26})^{-1973}
Prove that in the decimal notation of the number
(
5
+
26
)
−
1973
(5+\sqrt{26})^{-1973}
(
5
+
26
)
−
1973
immediately after the decimal point there are at least
1973
1973
1973
zeros.
63
1
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Chisinau MO p63 1973 coloring all points in 4 colors
Each point in space is colored in one of four different colors. Prove that there is a segment
1
1
1
cm long with endpoints of the same color.