MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1978 All Soviet Union Mathematical Olympiad
1978 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(17)
256
1
Hide problems
ASU 256 All Soviet Union MO 1978 game with 2 heaps of checkers
Given two heaps of checkers. the bigger contains
m
m
m
checkers, the smaller --
n
n
n
(
m
>
n
m>n
m
>
n
). Two players are taking checkers in turn from the arbitrary heap. The players are allowed to take from the heap a number of checkers (not zero) divisible by the number of checkers in another heap. The player that takes the last checker in any heap wins. a) Prove that if
m
>
2
n
m > 2n
m
>
2
n
, than the first can always win. b) Find all
x
x
x
such that if
m
>
x
n
m > xn
m
>
x
n
, than the first can always win.
255
1
Hide problems
ASU 255 All Soviet Union MO 1978 sequence of points in plane and in space
Given a finite set
K
0
K_0
K
0
of points (in the plane or space). The sequence of sets
K
1
,
K
2
,
.
.
.
,
K
n
,
.
.
.
K_1, K_2, ... , K_n, ...
K
1
,
K
2
,
...
,
K
n
,
...
is constructed according to the rule: we take all the points of
K
i
K_i
K
i
, add all the symmetric points with respect to all its points, and, thus obtain
K
i
+
1
K_{i+1}
K
i
+
1
. a) Let
K
0
K_0
K
0
consist of two points
A
A
A
and
B
B
B
with the distance
1
1
1
unit between them. For what
n
n
n
the set
K
n
K_n
K
n
contains the point that is
1000
1000
1000
units far from
A
A
A
? b) Let
K
0
K_0
K
0
consist of three points that are the vertices of the equilateral triangle with the unit square. Find the area of minimal convex polygon containing
K
n
.
K
0
K_n. K_0
K
n
.
K
0
below is the set of the unit volume tetrahedron vertices. c) How many faces contain the minimal convex polyhedron containing
K
1
K_1
K
1
? d) What is the volume of the above mentioned polyhedron? e) What is the volume of the minimal convex polyhedron containing
K
n
K_n
K
n
?
260
1
Hide problems
ASU 260 All Soviet Union MO 1978 three automates and card pairs
Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with (
a
,
b
)
a,b)
a
,
b
)
, produces new card with
(
a
+
1
,
b
+
1
)
(a+1,b+1)
(
a
+
1
,
b
+
1
)
, the second, having got the card with
(
a
,
b
)
(a,b)
(
a
,
b
)
, produces new card with
(
a
/
2
,
b
/
2
)
(a/2,b/2)
(
a
/2
,
b
/2
)
, if both
a
a
a
and
b
b
b
are even and nothing in the opposite case; the third, having got the pair of cards with
(
a
,
b
)
(a,b)
(
a
,
b
)
and
(
b
,
c
)
(b,c)
(
b
,
c
)
produces new card with
(
a
,
c
)
(a,c)
(
a
,
c
)
. All the automates return the initial cards also. Suppose there was
(
5
,
19
)
(5,19)
(
5
,
19
)
card initially. Is it possible to obtain a)
(
1
,
50
)
(1,50)
(
1
,
50
)
? b)
(
1
,
100
)
(1,100)
(
1
,
100
)
? c) Suppose there was
(
a
,
b
)
(a,b)
(
a
,
b
)
card initially
(
a
<
b
)
(a<b)
(
a
<
b
)
. We want to obtain
(
1
,
n
)
(1,n)
(
1
,
n
)
card. For what
n
n
n
is it possible?
262
1
Hide problems
ASU 262 All Soviet Union MO 1978 checkers on a chessboard
The checker is standing on the corner field of a
n
×
n
n\times n
n
×
n
chess-board. Each of two players moves it in turn to the neighbour (i.e. that has the common side) field. It is forbidden to move to the field, the checker has already visited. That who cannot make a move losts. a) Prove that for even
n
n
n
the first can always win, and if
n
n
n
is odd, than the second can always win. b) Who wins if the checker stands initially on the neighbour to the corner field?
265
1
Hide problems
ASU 265 All Soviet Union MO 1978 marking lattice points
Given a simple number
p
>
3
p>3
p
>
3
. Consider the set
M
M
M
of the pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
with the integer coordinates in the plane such that
0
≤
x
<
p
,
0
≤
y
<
p
0 \le x < p, 0 \le y < p
0
≤
x
<
p
,
0
≤
y
<
p
. Prove that it is possible to mark
p
p
p
points of
M
M
M
such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points.
267
1
Hide problems
ASU 267 All Soviet Union MO 1978, &Sigma;(a_n - b_n)^2
Given
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
. Define
b
k
=
a
1
+
a
2
+
.
.
.
+
a
k
k
b_k = \frac{a_1 + a_2 + ... + a_k}{k}
b
k
=
k
a
1
+
a
2
+
...
+
a
k
for
1
≤
k
≤
n
.
1 \le k\le n.
1
≤
k
≤
n
.
Let
C
=
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
.
.
.
+
(
a
n
−
b
n
)
2
,
D
=
(
a
1
−
b
n
)
2
+
(
a
2
−
b
n
)
2
+
.
.
.
+
(
a
n
−
b
n
)
2
C = (a_1 - b_1)^2 + (a_2 - b_2)^2 + ... + (a_n - b_n)^2, D = (a_1 - b_n)^2 + (a_2 - b_n)^2 + ... + (a_n - b_n)^2
C
=
(
a
1
−
b
1
)
2
+
(
a
2
−
b
2
)
2
+
...
+
(
a
n
−
b
n
)
2
,
D
=
(
a
1
−
b
n
)
2
+
(
a
2
−
b
n
)
2
+
...
+
(
a
n
−
b
n
)
2
Prove that
C
≤
D
≤
2
C
C \le D \le 2C
C
≤
D
≤
2
C
.
268
1
Hide problems
ASU 268 All Russian MO 1978 x_n=(1+\sqrt(2)+\sqrt(3))^n , limits
Consider a sequence
x
n
=
(
1
+
2
+
3
)
n
x_n=(1+\sqrt2+\sqrt3)^n
x
n
=
(
1
+
2
+
3
)
n
Each member can be represented as
x
n
=
q
n
+
r
n
2
+
s
n
3
+
t
n
6
x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6
x
n
=
q
n
+
r
n
2
+
s
n
3
+
t
n
6
where
q
n
,
r
n
,
s
n
,
t
n
q_n, r_n, s_n, t_n
q
n
,
r
n
,
s
n
,
t
n
are integers. Find the limits of the fractions
r
n
/
q
n
,
s
n
/
q
n
,
t
n
/
q
n
r_n/q_n, s_n/q_n, t_n/q_n
r
n
/
q
n
,
s
n
/
q
n
,
t
n
/
q
n
.
264
1
Hide problems
ASU 264 All Soviet Union MO 1978 Sx_i &Sigma;1/x_i <= ((a+b)^2/4ab)n^2
Given
0
<
a
≤
x
1
≤
x
2
≤
.
.
.
≤
x
n
≤
b
0 < a \le x_1\le x_2\le ... \le x_n \le b
0
<
a
≤
x
1
≤
x
2
≤
...
≤
x
n
≤
b
. Prove that
(
x
1
+
x
2
+
.
.
.
+
x
n
)
(
1
x
1
+
1
x
2
+
.
.
.
+
1
x
n
)
≤
(
a
+
b
)
2
4
a
b
n
2
(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2
(
x
1
+
x
2
+
...
+
x
n
)
(
x
1
1
+
x
2
1
+
...
+
x
n
1
)
≤
4
ab
(
a
+
b
)
2
n
2
261
1
Hide problems
ASU 261 All Soviet Union MO 1978 perimeter of cyclic polygon >=2S/R
Given a circle with radius
R
R
R
and inscribed
n
n
n
-gon with area
S
S
S
. We mark one point on every side of the given polygon. Prove that the perimeter of the polygon with the vertices in the marked points is not less than
2
S
/
R
2S/R
2
S
/
R
.
263
1
Hide problems
ASU 263 All Soviet Union MO 1978 n nonintersecting segments in plane
Given
n
n
n
nonintersecting segments in the plane. Not a pair of those belong to the same straight line. We want to add several segments, connecting the ends of given ones, to obtain one nonselfintersecting broken line. Is it always possible?
266
1
Hide problems
ASU 266 All Soviet Union MO 1978 projection area of tetrahedron on 2 planes
Prove that for every tetrahedron there exist two planes such that the projection areas on those planes ratio is not less than
2
\sqrt 2
2
.
259
1
Hide problems
ASU 259 All Soviet Union MO 1978 1978 inscibed squares in plot of y=Asin(x)
Prove that there exists such a number
A
A
A
that you can inscribe
1978
1978
1978
different size squares in the plot of the function
y
=
A
s
i
n
(
x
)
y = A sin(x)
y
=
A
s
in
(
x
)
. (The square is inscribed if all its vertices belong to the plot.)
258
1
Hide problems
ASU 258 All Soviet Union MO 1978 f(x)=x^2-x+1, m, f(m), f(f(m)), ... rel.prime
Let
f
(
x
)
=
x
2
−
x
+
1
f(x) = x^2 - x + 1
f
(
x
)
=
x
2
−
x
+
1
. Prove that for every natural
m
>
1
m>1
m
>
1
the numbers
m
,
f
(
m
)
,
f
(
f
(
m
)
)
,
.
.
.
m, f(m), f(f(m)), ...
m
,
f
(
m
)
,
f
(
f
(
m
))
,
...
are relatively prime.
257
1
Hide problems
ASU 257 All Soviet Union MO 1978 |x_m-x_k|>1/|m-k|
Prove that there exists such an infinite sequence
{
x
i
}
\{x_i\}
{
x
i
}
, that for all
m
m
m
and all
k
k
k
(
m
≠
k
m\ne k
m
=
k
) holds the inequality
∣
x
m
−
x
k
∣
>
1
/
∣
m
−
k
∣
|x_m-x_k|>1/|m-k|
∣
x
m
−
x
k
∣
>
1/∣
m
−
k
∣
254
1
Hide problems
ASU 254 All Soviet Union MO 1978 (1978^m-1) not divisible by (1000^m-1)
Prove that there is no
m
m
m
such that (
197
8
m
−
1
1978^m - 1
197
8
m
−
1
) is divisible by (
100
0
m
−
1
1000^m - 1
100
0
m
−
1
).
253
1
Hide problems
ASU 253 All Soviet Union MO 1978 equal angles given and wanted, parallelogram
Given a quadrangle
A
B
C
D
ABCD
A
BC
D
and a point
M
M
M
inside it such that
A
B
M
D
ABMD
A
BM
D
is a parallelogram.
∠
C
B
M
=
∠
C
D
M
\angle CBM = \angle CDM
∠
CBM
=
∠
C
D
M
. Prove that the
∠
A
C
D
=
∠
B
C
M
\angle ACD = \angle BCM
∠
A
C
D
=
∠
BCM
.
252
1
Hide problems
ASU 252 All Soviet Union MO 1978 a_n the closest to \sqrt n integer, sum 1/a_i
Let
a
n
a_n
a
n
be the closest to
n
\sqrt n
n
integer. Find the sum
1
/
a
1
+
1
/
a
2
+
.
.
.
+
1
/
a
1980
1/a_1 + 1/a_2 + ... + 1/a_{1980}
1/
a
1
+
1/
a
2
+
...
+
1/
a
1980