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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1992 All Soviet Union Mathematical Olympiad
1992 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(23)
565
1
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ASU 565 Commonwealth of Independent States 1991 gnomons in mxn table
An
m
×
n
m \times n
m
×
n
rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a
2
×
2
2 \times 2
2
×
2
square. For what
m
,
n
m, n
m
,
n
can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?
580
1
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ASU 580 Commonwealth of Independent States 1991 (a-d)^2>=4d+8 if ad=bc
If
a
>
b
>
c
>
d
>
0
a > b > c > d > 0
a
>
b
>
c
>
d
>
0
are integers such that
a
d
=
b
c
ad = bc
a
d
=
b
c
, show that
(
a
−
d
)
2
≥
4
d
+
8
(a - d)^2 \ge 4d + 8
(
a
−
d
)
2
≥
4
d
+
8
579
1
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ASU 579 Commonwealth of Independent States 1991 vectors game
1992
1992
1992
vectors are given in the plane. Two players pick unpicked vectors alternately. The winner is the one whose vectors sum to a vector with larger magnitude (or they draw if the magnitudes are the same). Can the first player always avoid losing?
578
1
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ASU 578 Commonwealth of Independent States 1991 100 equilaterals
An equilateral triangle side
10
10
10
is divided into
100
100
100
equilateral triangles of side
1
1
1
by lines parallel to its sides. There are m equilateral tiles of
4
4
4
unit triangles and
25
−
m
25 - m
25
−
m
straight tiles of
4
4
4
unit triangles (as shown below). For which values of
m
m
m
can they be used to tile the original triangle. [The straight tiles may be turned over.]
577
1
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ASU 577 Commonwealth of Independent States 1991 digits ^a+1, k^b+1
Find all integers
k
>
1
k > 1
k
>
1
such that for some distinct positive integers
a
,
b
a, b
a
,
b
, the number
k
a
+
1
k^a + 1
k
a
+
1
can be obtained from
k
b
+
1
k^b + 1
k
b
+
1
by reversing the order of its (decimal) digits.
576
1
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ASU 576 Commonwealth of Independent States 1991 cubics x=p(y), y=p(x)
If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to
x
=
p
(
y
)
,
y
=
p
(
x
)
x = p(y), y = p(x)
x
=
p
(
y
)
,
y
=
p
(
x
)
, where
p
p
p
is a cubic polynomial?
575
1
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ASU 575 Commonwealth of Independent States 1991 BP·BQ indepedent plane
A plane intersects a sphere in a circle
C
C
C
. The points
A
A
A
and
B
B
B
lie on the sphere on opposite sides of the plane. The line joining
A
A
A
to the center of the sphere is normal to the plane. Another plane
p
p
p
intersects the segment
A
B
AB
A
B
and meets
C
C
C
at
P
P
P
and
Q
Q
Q
. Show that
B
P
⋅
B
Q
BP\cdot BQ
BP
⋅
BQ
is independent of the choice of
p
p
p
.
574
1
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ASU 574 Commonwealth of Independent States 91 f(x)=acos(x+1)+bcos(x+2)+ccos(x+3)
Let
f
(
x
)
=
a
cos
(
x
+
1
)
+
b
cos
(
x
+
2
)
+
c
cos
(
x
+
3
)
f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)
f
(
x
)
=
a
cos
(
x
+
1
)
+
b
cos
(
x
+
2
)
+
c
cos
(
x
+
3
)
, where
a
,
b
,
c
a, b, c
a
,
b
,
c
are real. Given that
f
(
x
)
f(x)
f
(
x
)
has at least two zeros in the interval
(
0
,
π
)
(0, \pi)
(
0
,
π
)
, find all its real zeros.
573
1
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ASU 573 Commonwealth of Independent States 1991 graph,17 p. 4 edges
A graph has
17
17
17
points and each point has
4
4
4
edges. Show that there are two points which are not joined and which are not both joined to the same point.
572
1
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ASU 572 Commonwealth of Independent States 1991 bw cells in 2mxn board
Half the cells of a
2
m
×
n
2m \times n
2
m
×
n
board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.
571
1
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ASU 571 Commonwealth of Independent States 1991 # and 2 excircles
A
B
C
D
ABCD
A
BC
D
is a parallelogram. The excircle of
A
B
C
ABC
A
BC
opposite
A
A
A
has center
E
E
E
and touches the line
A
B
AB
A
B
at
X
X
X
. The excircle of
A
D
C
ADC
A
D
C
opposite
A
A
A
has center
F
F
F
and touches the line
A
D
AD
A
D
at
Y
Y
Y
. The line
F
C
FC
FC
meets the line
A
B
AB
A
B
at
W
W
W
, and the line
E
C
EC
EC
meets the line
A
D
AD
A
D
at
Z
Z
Z
. Show that
W
X
=
Y
Z
WX = YZ
W
X
=
Y
Z
.
570
1
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ASU 570 Commonwealth of Independent States 1991 a_n= sum a_i^2 +n
Define the sequence
a
1
=
1
,
a
2
,
a
3
,
.
.
.
a_1 = 1, a_2, a_3, ...
a
1
=
1
,
a
2
,
a
3
,
...
by
a
n
+
1
=
a
1
2
+
a
2
2
+
a
3
2
+
.
.
.
+
a
n
2
+
n
a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n
a
n
+
1
=
a
1
2
+
a
2
2
+
a
3
2
+
...
+
a
n
2
+
n
Show that
1
1
1
is the only square in the sequence.
569
1
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ASU 569 Commonwealth of Independent States 1991 right angle, inters.circles
Circles
C
C
C
and
C
′
C'
C
′
intersect at
O
O
O
and
X
X
X
. A circle center
O
O
O
meets
C
C
C
at
Q
Q
Q
and
R
R
R
and meets
C
′
C'
C
′
at
P
P
P
and
S
S
S
.
P
R
PR
PR
and
Q
S
QS
QS
meet at
Y
Y
Y
distinct from
X
X
X
. Show that
∠
Y
X
O
=
9
0
o
\angle YXO = 90^o
∠
Y
XO
=
9
0
o
.
568
1
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ASU 568 Commonwealth of Independent States 1991 cinema with mxn seats
A cinema has its seats arranged in
n
n
n
rows
×
m
\times m
×
m
columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum
k
k
k
such that he could have always put every ticket holder in the correct row or column and at least
k
k
k
people in the correct seat?
567
1
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ASU 567 Commonwealth of Independent States 1991 15 no in 2 to1992,coprime pairs
Show that if
15
15
15
numbers lie between
2
2
2
and
1992
1992
1992
and each pair is coprime, then at least one is prime.
566
1
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ASU 566 Commonwealth of Independent States 1991 x^2/(y-1) + y^2/(x-1)>= 8.
Show that for any real numbers
x
,
y
>
1
x, y > 1
x
,
y
>
1
, we have
x
2
y
−
1
+
y
2
x
−
1
≥
8
\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge 8
y
−
1
x
2
+
x
−
1
y
2
≥
8
564
1
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ASU 564 Commonwealth of Independent States 1991 (1+x)(1+x^2)(1+x^4)=1+y^7
Find all real
x
,
y
x, y
x
,
y
such that
{
(
1
+
x
)
(
1
+
x
2
)
(
1
+
x
4
)
=
1
+
y
7
(
1
+
y
)
(
1
+
y
2
)
(
1
+
y
4
)
=
1
+
x
7
\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\ (1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}
{
(
1
+
x
)
(
1
+
x
2
)
(
1
+
x
4
)
=
1
+
y
7
(
1
+
y
)
(
1
+
y
2
)
(
1
+
y
4
)
=
1
+
x
7
563
1
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ASU 563 Commonwealth of Independent States 1991 symmetric wrt line
A
A
A
and
B
B
B
lie on a circle.
P
P
P
lies on the minor arc
A
B
AB
A
B
.
Q
Q
Q
and
R
R
R
(distinct from
P
P
P
) also lie on the circle, so that
P
P
P
and
Q
Q
Q
are equidistant from
A
A
A
, and
P
P
P
and
R
R
R
are equidistant from
B
B
B
. Show that the intersection of
A
R
AR
A
R
and
B
Q
BQ
BQ
is the reflection of
P
P
P
in
A
B
AB
A
B
.
562
1
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ASU 562 Commonwealth of Independent States 1992 4-digit not multiple 1992
Does there exist a
4
4
4
-digit integer which cannot be changed into a multiple of
1992
1992
1992
by changing
3
3
3
of its digits?
561
1
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ASU 561 Commonwealth of Independent States 1992 infiinte board, mxn rectangle
Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an
m
x
n
m x n
m
x
n
rectangle (
m
,
n
>
1
m, n > 1
m
,
n
>
1
) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?
560
1
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ASU 560 Commonwealth of Independent States 1992 n cities, gold route
A country contains
n
n
n
cities and some towns. There is at most one road between each pair of towns and at most one road between each town and each city, but all the towns and cities are connected, directly or indirectly. We call a route between a city and a town a gold route if there is no other route between them which passes through fewer towns. Show that we can divide the towns and cities between
n
n
n
republics, so that each belongs to just one republic, each republic has just one city, and each republic contains all the towns on at least one of the gold routes between each of its towns and its city.
559
1
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ASU 559 Commonwealth of Independent States 1992 square inside a square
E
E
E
is a point on the diagonal
B
D
BD
B
D
of the square
A
B
C
D
ABCD
A
BC
D
. Show that the points
A
,
E
A, E
A
,
E
and the circumcenters of
A
B
E
ABE
A
BE
and
A
D
E
ADE
A
D
E
form a square.
558
1
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ASU 558 Commonwealth of Independent States 1992 x^4 +y^4+z^2>=xyz√8
Show that
x
4
+
y
4
+
z
2
≥
x
y
z
8
x^4 + y^4 + z^2\ge xyz \sqrt8
x
4
+
y
4
+
z
2
≥
x
yz
8
for all positive reals
x
,
y
,
z
x, y, z
x
,
y
,
z
.