MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
1993 All-Russian Olympiad Regional Round
1993 All-Russian Olympiad Regional Round
Part of
All-Russian Olympiad Regional Round
Subcontests
(22)
11.7
1
Hide problems
loci, equailaterals on sides All-Russian MO 1993 Regional (R4) 11.7
Let
A
B
C
ABC
A
BC
be an equilateral triangle. For an arbitrary line
ℓ
\ell
ℓ
. through
B
B
B
, the orthogonal projections of
A
A
A
and
C
C
C
on
ℓ
\ell
ℓ
are denoted by
D
D
D
and
E
E
E
respectively. If
D
≠
E
D\ne E
D
=
E
, equilateral triangles
D
E
P
DEP
D
EP
and
D
E
T
DET
D
ET
are constructed on different sides of
ℓ
\ell
ℓ
. Find the loci of
P
P
P
and
T
T
T
.
11.2
1
Hide problems
8| ](\sqrt{3}{n}+\sqrt{3}{n+2})^3]+1
Prove that, for every integer
n
>
2
n > 2
n
>
2
, the number
[
(
n
3
+
n
+
2
3
)
3
]
+
1
\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1
[
(
3
n
+
3
n
+
2
)
3
]
+
1
is divisible by
8
8
8
.
11.1
1
Hide problems
sum of digits of 5^n = 2^n All-Russian MO 1993 Regional (R4) 11.1
Find all natural numbers
n
n
n
for which the sum of digits of
5
n
5^n
5
n
equals
2
n
2^n
2
n
.
10.8
1
Hide problems
4 2x994 from 1000x1000 All-Russian MO 1993 Regional (R4) 10.8
From a square board
1000
×
1000
1000\times 1000
1000
×
1000
four rectangles
2
×
994
2\times 994
2
×
994
have been cut off as shown on the picture. Initially, on the marked square there is a centaur - a piece that moves to the adjacent square to the left, up, or diagonally up-right in each move. Two players alternately move the centaur. The one who cannot make a move loses the game. Who has a winning strategy? https://cdn.artofproblemsolving.com/attachments/c/6/f61c186413b642b5b59f3947bc7a108c772d27.png
10.1
1
Hide problems
AN = 2KM if AK = BK - All-Russian MO 1993 Regional (R4) 10.1
Point
D
D
D
is chosen on the side
A
C
AC
A
C
of an acute-angled triangle
A
B
C
ABC
A
BC
. The median
A
M
AM
A
M
intersects the altitude
C
H
CH
C
H
and the segment
B
D
BD
B
D
at points
N
N
N
and
K
K
K
respectively. Prove that if
A
K
=
B
K
AK = BK
A
K
=
B
K
, then
A
N
=
2
K
M
AN = 2KM
A
N
=
2
K
M
.
9.5
1
Hide problems
x^3 +y^3 = 4(x^2y+xy^2 +1) NT All-Russian MO 1993 Regional (R4) 9.5
Show that the equation
x
3
+
y
3
=
4
(
x
2
y
+
x
y
2
+
1
)
x^3 +y^3 = 4(x^2y+xy^2 +1)
x
3
+
y
3
=
4
(
x
2
y
+
x
y
2
+
1
)
has no integer solutions.
9.3
1
Hide problems
ABC necessarily isosceles ? All-Russian MO 1993 Regional (R4) 9.3
Points
M
M
M
and
N
N
N
are chosen on the sides
A
B
AB
A
B
and BC of a triangle
A
B
C
ABC
A
BC
. The segments
A
N
AN
A
N
and
C
M
CM
CM
meet at
O
O
O
such that
A
O
=
C
O
AO =CO
A
O
=
CO
. Is the triangle
A
B
C
ABC
A
BC
necessarily isosceles, if(a)
A
M
=
C
N
AM = CN
A
M
=
CN
?(b)
B
M
=
B
N
BM = BN
BM
=
BN
?
9.2
1
Hide problems
not a multiple of 11 All-Russian MO 1993 Regional (R4) 9.2
Find the largest natural number which cannot be turned into a multiple of
11
11
11
by reordering its (decimal) digits.
9.1
1
Hide problems
a^2 +ab+b^2>= 3(a+b-1).All-Russian MO 1993 Regional (R4) 9.1
If
a
a
a
and
b
b
b
are positive numbers, prove the inequality
a
2
+
a
b
+
b
2
≥
3
(
a
+
b
−
1
)
.
a^2 +ab+b^2\ge 3(a+b-1).
a
2
+
ab
+
b
2
≥
3
(
a
+
b
−
1
)
.
11.6
1
Hide problems
exists plane cutting all 7 tetrahedra in triangles of equal areas
Seven tetrahedra are placed on the table. For any three of them there exists a horizontal plane cutting them in triangles of equal areas. Show that there exists a plane cutting all seven tetrahedra in triangles of equal areas.
11.3
1
Hide problems
collinearity wanted, sphere and pyramid
Point
O
O
O
is the foot of the altitude of a quadrilateral pyramid. A sphere with center
O
O
O
is tangent to all lateral faces of the pyramid. Points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
are taken on successive lateral edges so that segments
A
B
AB
A
B
,
B
C
BC
BC
, and
C
D
CD
C
D
pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment
A
D
AD
A
D
passes through the fourth tangency point
9.6
1
Hide problems
exists line \\ ta a given cutting 3 right triangles into congruent segments
Three right-angled triangles have been placed in a halfplane determined by a line
ℓ
\ell
ℓ
, each with one leg lying on
ℓ
\ell
ℓ
. Assume that there is a line parallel to
ℓ
\ell
ℓ
cutting the triangles in three congruent segments. Show that, if each of the triangles is rotated so that its other leg lies on
ℓ
\ell
ℓ
, then there still exists a line parallel to
ℓ
\ell
ℓ
cutting them in three congruent segments.
9.4
1
Hide problems
n playing cards some turned up and some turned down
We have a deck of
n
n
n
playing cards, some of which are turned up and some are turned down. In each step we are allowed to take a set of several cards from the top, turn the set and place it back on the top of the deck. What is the smallest number of steps necessary to make all cards in the deck turned down, independent of the initial configuration?
9.7
1
Hide problems
collinear wanted, AE = NE, CE = ME, rhombus related
On the diagonal
A
C
AC
A
C
of the rhombus
A
B
C
D
ABCD
A
BC
D
, a point
E
E
E
is taken, which is different from points
A
A
A
and
C
C
C
, and on the lines
A
B
AB
A
B
and
B
C
BC
BC
are points
N
N
N
and
M
M
M
, respectively, with
A
E
=
N
E
AE = NE
A
E
=
NE
and
C
E
=
M
E
CE = ME
CE
=
ME
. Let
K
K
K
be the intersection point of lines
A
M
AM
A
M
and
C
N
CN
CN
. Prove that points
K
,
E
K, E
K
,
E
and
D
D
D
are collinear.
11.5
1
Hide problems
roots of cubic game
The expression x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0 is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?
10.6
1
Hide problems
deceptively simple inequality
Prove the inequality \sqrt {2 \plus{} \sqrt [3]{3 \plus{} ... \plus{} \sqrt [{2008}]{2008}}} < 2
10.3
1
Hide problems
nice system of equations
Solve in positive numbers the system x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1
9.8
1
Hide problems
game with + and -
Number
0
0
0
is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either \plus{} or \minus{} sign, while the second player writes one of the numbers
1
,
2
,
.
.
.
,
1993
1, 2, ... , 1993
1
,
2
,
...
,
1993
,writing each of these numbers exactly once. The game ends after
1993
1993
1993
moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?
10.4
1
Hide problems
arranging an election
Each citizen in a town knows at least
30
30
30
% of the remaining citizens. A citizen votes in elections if he/she knows at least one candidate. Prove that it is possible to schedule elections with two candidates for the mayor of the city so that at least
50
50
50
% of the citizen can vote.
11.4
1
Hide problems
Combinatorial Geometry: assigning vectors to a regular k-gon
Given a regular
2
n
2n
2
n
-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.
11.8
1
Hide problems
town connection limit
There are
1993
1993
1993
towns in a country, and at least
93
93
93
roads going out of each town. It's known that every town can be reached from any other town. Prove that this can always be done with no more than
62
62
62
transfers.
10.7
1
Hide problems
equal area condition
Points
M
,
N
M,N
M
,
N
are taken on sides
B
C
,
C
D
BC,CD
BC
,
C
D
respectively of parallelogram
A
B
C
D
ABCD
A
BC
D
. Let E\equal{}BD\cap AM, F\equal{}BD\cap AN. Diagonal
B
D
BD
B
D
cuts triangle
A
M
N
AMN
A
MN
into two parts. Prove that these two parts have equal area if and only if the point
K
K
K
given by
E
K
∥
A
D
,
F
K
∥
A
B
EK\parallel{}AD, FK\parallel{}AB
E
K
∥
A
D
,
F
K
∥
A
B
lies on segment
M
N
MN
MN
.