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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1973 All Soviet Union Mathematical Olympiad
1973 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(15)
174
1
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ASU 174 All Soviet Union MO 1973 14 coins, false and true, weighings
Fourteen coins are submitted to the judge. An expert knows, that the coins from number one to seven are false, and from
8
8
8
to
14
14
14
-- normal. The judge is sure only that all the true coins have the same weight and all the false coins weights equal each other, but are less then the weight of the true coins. The expert has the scales without weights. a) The expert wants to prove, that the coins
1
−
−
7
1--7
1
−
−
7
are false. How can he do it in three weighings? b) How can he prove, that the coins
1
−
−
7
1--7
1
−
−
7
are false and the coins
8
−
−
14
8--14
8
−
−
14
are true in three weighings?
177
1
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ASU 177 All Soviet Union MO 1973 equal segments, circle tangent to angle
Given an angle with the vertex
O
O
O
and a circle touching its sides in the points
A
A
A
and
B
B
B
. A ray is drawn from the point
A
A
A
parallel to
[
O
B
)
[OB)
[
OB
)
. It intersects with the circumference in the point
C
C
C
. The segment
[
O
C
]
[OC]
[
OC
]
intersects the circumference in the point
E
E
E
. The straight lines
(
A
E
)
(AE)
(
A
E
)
and
(
O
B
)
(OB)
(
OB
)
intersect in the point
K
K
K
. Prove that
∣
O
K
∣
=
∣
K
B
∣
|OK| = |KB|
∣
O
K
∣
=
∣
K
B
∣
.
181
1
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ASU 181 All Soviet Union MO 1973 black colour in an infinite board
n
n
n
squares of the infinite cross-lined sheet of paper are painted with black colour (others are white). Every move all the squares of the sheet change their colour simultaneously. The square gets the colour, that had the majority of three ones: the square itself, its neighbour from the right side and its neighbour from the upper side. a) Prove that after the finite number of the moves all the black squares will disappear. b) Prove that it will happen not later than on the
n
n
n
-th move
179
1
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ASU 179 All Soviet Union MO 1973 1024 sportmen in tennis tournament
The tennis federation has assigned numbers to
1024
1024
1024
sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains
512
512
512
participants, in the third --
256
256
256
, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on
2
2
2
always win that whose number is less. What is the greatest possible number of the winner?
182
1
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ASU 182 All Soviet Union MO 1973 similar triangles on sides of triangle
Three similar acute-angled triangles
A
C
1
B
,
B
A
1
C
AC_1B, BA_1C
A
C
1
B
,
B
A
1
C
and
C
B
1
A
CB_1A
C
B
1
A
are constructed on the outer side of the acute-angled triangle
A
B
C
ABC
A
BC
. (Equal triples of the angles are
A
B
1
C
,
A
B
C
1
,
A
1
B
C
AB_1C, ABC_1, A_1BC
A
B
1
C
,
A
B
C
1
,
A
1
BC
and
B
A
1
C
,
B
A
C
1
,
B
1
A
C
BA_1C, BAC_1, B_1AC
B
A
1
C
,
B
A
C
1
,
B
1
A
C
.) a) Prove that the circles circumscribed around the outer triangles intersect in one point. b) Prove that the straight lines
A
A
1
,
B
B
1
AA_1, BB_1
A
A
1
,
B
B
1
and
C
C
1
CC_1
C
C
1
intersect in the same point
184
1
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ASU 184 All Soviet Union MO 1973 king moves in a 8x8 chessboard
The king have revised the chess-board
8
×
8
8\times 8
8
×
8
having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared. a) Give an example that the king could make
28
28
28
steps parallel the sides of the board only. b) Prove that he could not make less than
28
28
28
such a steps. c) What is the maximal and minimal length of the broken line if the side of a field is
1
1
1
?
186
1
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ASU 186 All Soviet Union MO 1973 half area of a convex polygon
Given a convex
n
n
n
-gon with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than
n
n
n
straight lines coming through that point and halving the area of the
n
n
n
-gon.
185
1
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ASU 185 All Soviet Union MO 1973 triangle of area 1, a>=b>=c, b^2 \ge 2
Given a triangle with
a
,
b
,
c
a,b,c
a
,
b
,
c
sides and with the area
1
1
1
(
a
≥
b
≥
c
a \ge b \ge c
a
≥
b
≥
c
). Prove that
b
2
≥
2
b^2 \ge 2
b
2
≥
2
.
187
1
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ASU 187 All Soviet Union MO 1973 (Σx_i)^2 \ge 4(x_1x_2+x_3x_4+x_5x_1+..)
Prove that for every positive
x
1
,
x
2
,
x
3
,
x
4
,
x
5
x_1, x_2, x_3, x_4, x_5
x
1
,
x
2
,
x
3
,
x
4
,
x
5
holds inequality:
(
x
1
+
x
2
+
x
3
+
x
4
+
x
5
)
2
≥
4
(
x
1
x
2
+
x
3
x
4
+
x
5
x
1
+
x
2
x
3
+
x
4
x
5
)
(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)
(
x
1
+
x
2
+
x
3
+
x
4
+
x
5
)
2
≥
4
(
x
1
x
2
+
x
3
x
4
+
x
5
x
1
+
x
2
x
3
+
x
4
x
5
)
188
1
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ASU 188 All Soviet Union MO 1973 parallelepipeds with vertex 1 of 4 points
Given
4
4
4
points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?
183
1
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ASU 183 All Soviet Union MO 1973 no 3 men have same no of acquaintances of N
N
N
N
men are not acquainted each other. You need to introduce some of them to some of them in such a way, that no three men will have equal number of acquaintances. Prove that it is possible for all
N
N
N
.
180
1
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ASU 180 All Soviet Union MO 1973 ax^2+bx+c=x has not real roots
The square polynomial
f
(
x
)
=
a
x
2
+
b
x
+
c
f(x)=ax^2+bx+c
f
(
x
)
=
a
x
2
+
b
x
+
c
is of such a sort, that the equation
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
does not have real roots. Prove that the equation
f
(
f
(
x
)
)
=
0
f(f(x))=0
f
(
f
(
x
))
=
0
does not have real roots also.
178
1
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ASU 178 All Soviet Union MO 1973 |ax^2+bx+c|\le 1 => |cx^2+bx+a| \le 2
The real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy the condition: for all
x
x
x
, such that for
−
1
≤
x
≤
1
-1 \le x \le 1
−
1
≤
x
≤
1
, the inequality
∣
a
x
2
+
b
x
+
c
∣
≤
1
| ax^2 + bx + c | \le 1
∣
a
x
2
+
b
x
+
c
∣
≤
1
is held. Prove that for the same
x
x
x
,
∣
c
x
2
+
b
x
+
a
∣
≤
2
| cx^2 + bx + a | \le 2
∣
c
x
2
+
b
x
+
a
∣
≤
2
176
1
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ASU 176 All Soviet Union MO 1973 arrows between n points
Given
n
n
n
points,
n
>
4
n > 4
n
>
4
. Prove that tou can connect them with arrows, in such a way, that you can reach every point from every other point, having passed through one or two arrows. (You can connect every pair with one arrow only, and move along the arrow in one direction only.)
175
1
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ASU 175 All Soviet Union MO 1973 9-digit perfect square does not end in 5
Prove that
9
9
9
-digit number, that contains all the decimal digits except zero and does not ends with
5
5
5
can not be exact square.