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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1990 All Soviet Union Mathematical Olympiad
1990 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(24)
534
1
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ASU 534 All Soviet Union MO 1990 2n genuine coins and 2n fake
Given
2
n
2n
2
n
genuine coins and
2
n
2n
2
n
fake coins. The fake coins look the same as genuine coins but weigh less (but all fake coins have the same weight). Show how to identify each coin as genuine or fake using a balance at most
3
n
3n
3
n
times.
533
1
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ASU 533 All Soviet Union MO 1990 game with coefficients of cubic, integer roots
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of
x
3
x^3
x
3
is already fixed at
1
1
1
. Can the first player make his choices so that the final cubic has three distinct integer roots?
532
1
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ASU 532 All Soviet Union MO 1990 every altitude of tetrahedron >=1
If every altitude of a tetrahedron is at least
1
1
1
, show that the shortest distance between each pair of opposite edges is more than
2
2
2
.
531
1
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ASU 531 All Soviet Union MO 1990 3^{2n+1} - 2^{2n+1} - 6^n composite
For which positive integers
n
n
n
is
3
2
n
+
1
−
2
2
n
+
1
−
6
n
3^{2n+1} - 2^{2n+1} - 6^n
3
2
n
+
1
−
2
2
n
+
1
−
6
n
composite?
530
1
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ASU 530 All Soviet Union MO 1990 million unit cubes, lattice, decompose
A cube side
100
100
100
is divided into a million unit cubes with faces parallel to the large cube. The edges form a lattice. A prong is any three unit edges with a common vertex. Can we decompose the lattice into prongs with no common edges?
529
1
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ASU 529 All Soviet Union MO 1990 trinomial, coefficients sum 1, product ineq
A quadratic polynomial
p
(
x
)
p(x)
p
(
x
)
has positive real coefficients with sum
1
1
1
. Show that given any positive real numbers with product
1
1
1
, the product of their values under
p
p
p
is at least
1
1
1
.
528
1
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ASU 528 All Soviet Union MO 1990 1990 piles with 1, 2, 3, ... , 1990 stones
Given
1990
1990
1990
piles of stones, containing
1
,
2
,
3
,
.
.
.
,
1990
1, 2, 3, ... , 1990
1
,
2
,
3
,
...
,
1990
stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?
527
1
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ASU 527 All Soviet Union MO 1990 ZX is tangent to the circumcircle of PXQ
Two unequal circles intersect at
X
X
X
and
Y
Y
Y
. Their common tangents intersect at
Z
Z
Z
. One of the tangents touches the circles at
P
P
P
and
Q
Q
Q
. Show that
Z
X
ZX
ZX
is tangent to the circumcircle of
P
X
Q
PXQ
PXQ
.
526
1
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ASU 526 All Soviet Union MO 1990 n vectors with zero sum, interior points
Given a point
X
X
X
and
n
n
n
vectors
x
i
→
\overrightarrow{x_i}
x
i
with sum zero in the plane. For each permutation of the vectors we form a set of
n
n
n
points, by starting at
X
X
X
and adding the vectors in order. For example, with the original ordering we get
X
1
X_1
X
1
such that
X
X
1
=
x
1
→
,
X
2
XX_1 = \overrightarrow{x_1}, X_2
X
X
1
=
x
1
,
X
2
such that
X
1
X
2
=
x
2
→
X_1X_2 = \overrightarrow{x_2}
X
1
X
2
=
x
2
and so on. Show that for some permutation we can find two points
Y
,
Z
Y, Z
Y
,
Z
with angle
∠
Y
X
Z
=
6
0
o
\angle YXZ = 60^o
∠
Y
XZ
=
6
0
o
, so that all the points lie inside or on the triangle
X
Y
Z
XYZ
X
Y
Z
.
525
1
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ASU 525 All Soviet Union MO 1990 graph, n points, n(n-1)/2 edges, coloring
A graph has
n
n
n
points and
n
(
n
−
1
)
2
\frac{n(n-1)}{2}
2
n
(
n
−
1
)
edges. Each edge is colored with one of
k
k
k
colors so that there are no closed monochrome paths. What is the largest possible value of
n
n
n
(given
k
k
k
)?
524
1
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ASU 524 All Soviet Union MO 1990 <XDY = \pi /2n, in a 2n-gon, angle bisector
A
,
B
,
C
A, B, C
A
,
B
,
C
are adjacent vertices of a regular
2
n
2n
2
n
-gon and
D
D
D
is the vertex opposite to
B
B
B
(so that
B
D
BD
B
D
passes through the center of the
2
n
2n
2
n
-gon).
X
X
X
is a point on the side
A
B
AB
A
B
and
Y
Y
Y
is a point on the side
B
C
BC
BC
so that
X
D
Y
=
π
2
n
XDY = \frac{\pi}{2n}
X
D
Y
=
2
n
π
. Show that
D
Y
DY
D
Y
bisects
∠
X
Y
C
\angle XYC
∠
X
Y
C
.
523
1
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ASU 523 All Soviet Union MO 1990 [n/1!] + [n/2!] + ... + [n/10!] = 1001
Find all integers
n
n
n
such that
[
n
1
!
]
+
[
n
2
!
]
+
.
.
.
+
[
n
10
!
]
=
1001
\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001
[
1
!
n
]
+
[
2
!
n
]
+
...
+
[
10
!
n
]
=
1001
.
522
1
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ASU 522 All Soviet Union MO 1990 grasshoppers game, points, interval [0, 1]
Two grasshoppers sit at opposite ends of the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest
n
n
n
such that if each grasshopper makes
n
n
n
moves or less, then they end up with no marked points between them?
521
1
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ASU 521 All Soviet Union MO 1990 concurrent circumcircles
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral.
X
X
X
is a point on the side
A
B
.
A
C
AB. AC
A
B
.
A
C
and
D
X
DX
D
X
intersect at
Y
Y
Y
. Show that the circumcircles of
A
B
C
,
C
D
Y
ABC, CDY
A
BC
,
C
D
Y
and
B
D
X
BDX
B
D
X
have a common point.
520
1
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ASU 520 All Soviet Union MO 1990 sum x_i^2/(x_i + x_{i+1}) >=1/2 if sum x_1=1
Let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ..., x_n
x
1
,
x
2
,
...
,
x
n
be positive reals with sum
1
1
1
. Show that
x
1
2
x
1
+
x
2
+
x
2
2
x
2
+
x
3
+
.
.
.
+
x
n
−
1
2
x
n
−
1
+
x
n
+
x
n
2
x
n
+
x
1
≥
1
2
\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12
x
1
+
x
2
x
1
2
+
x
2
+
x
3
x
2
2
+
...
+
x
n
−
1
+
x
n
x
n
−
1
2
+
x
n
+
x
1
x
n
2
≥
2
1
519
1
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ASU 519 All Soviet Union MO 1990 bw color in 1990 x 1990 chessboard
Can the squares of a
1990
×
1990
1990 \times 1990
1990
×
1990
chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?
518
1
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ASU 518 All Soviet Union 1990 n^2 equilateral triangle of side 1, acute in path
An equilateral triangle of side
n
n
n
is divided into
n
2
n^2
n
2
equilateral triangles of side
1
1
1
. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least
n
n
n
vertices.
517
1
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ASU 517 All Soviet Union MO 1990 max |...| |a_1 - a_2| - a_3| - ... - a_{1990}|
What is the largest possible value of
∣
.
.
.
∣
∣
a
1
−
a
2
∣
−
a
3
∣
−
.
.
.
−
a
1990
∣
|...| |a_1 - a_2| - a_3| - ... - a_{1990}|
∣...∣∣
a
1
−
a
2
∣
−
a
3
∣
−
...
−
a
1990
∣
, where
a
1
,
a
2
,
.
.
.
,
a
1990
a_1, a_2, ... , a_{1990}
a
1
,
a
2
,
...
,
a
1990
is a permutation of
1
,
2
,
3
,
.
.
.
,
1990
1, 2, 3, ... , 1990
1
,
2
,
3
,
...
,
1990
?
516
1
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ASU 516 All Soviet Union MO 1990 distinct rational roots in integer quadratics
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
515
1
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ASU 515 All Soviet Union MO 1990 abc = a'b'c' = a"b"c", lengths of segments
The point
P
P
P
lies inside the triangle
A
B
C
ABC
A
BC
. A line is drawn through
P
P
P
parallel to each side of the triangle. The lines divide
A
B
AB
A
B
into three parts length
c
,
c
′
,
c
"
c, c', c"
c
,
c
′
,
c
"
(in that order), and
B
C
BC
BC
into three parts length
a
,
a
′
,
a
"
a, a', a"
a
,
a
′
,
a
"
(in that order), and
C
A
CA
C
A
into three parts length
b
,
b
′
,
b
"
b, b', b"
b
,
b
′
,
b
"
(in that order). Show that
a
b
c
=
a
′
b
′
c
′
=
a
"
b
"
c
"
abc = a'b'c' = a"b"c"
ab
c
=
a
′
b
′
c
′
=
a
"
b
"
c
"
.
514
1
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ASU 514 All Soviet Union MO 1990 rectangle disect 15 congruent polygons
Does there exist a rectangle which can be dissected into
15
15
15
congruent polygons which are not rectangles? Can a square be dissected into
15
15
15
congruent polygons which are not rectangles?
513
1
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ASU 513 All Soviet Union MO 1990 graph, 30 points, each point has 6 edges
A graph has
30
30
30
points and each point has
6
6
6
edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.
512
1
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ASU 512 All Soviet Union MO 1990 equal angles given, equal diagonals wanted
The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.
511
1
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ASU 511 All Soviet Union MO 1990 x^4 > x - 1/2
Show that
x
4
>
x
−
1
2
x^4 > x - \frac12
x
4
>
x
−
2
1
for all real
x
x
x
.