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Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1962 All-Soviet Union Olympiad
1962 All-Soviet Union Olympiad
Part of
All-Russian Olympiad
Subcontests
(14)
14
1
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Two sets with the same sum
Given are two sets of positive numbers with the same sum. The first set has
m
m
m
numbers and the second
n
n
n
. Prove that you can find a set of less than
m
+
n
m+n
m
+
n
positive numbers which can be arranged to part fill an
m
×
n
m \times n
m
×
n
array, so that the row and column sums are the two given sets.
13
1
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Sequence with Recurrent Inequality
Given are
a
0
,
a
1
,
.
.
.
,
a
n
a_0,a_1, ... , a_n
a
0
,
a
1
,
...
,
a
n
, satisfying
a
0
=
a
n
=
0
a_0=a_n = 0
a
0
=
a
n
=
0
, and
a
k
−
1
−
2
a
k
+
a
k
+
1
≥
0
a_{k-1} - 2a_k+a_{k+1}\ge 0
a
k
−
1
−
2
a
k
+
a
k
+
1
≥
0
for
k
=
0
,
1
,
.
.
.
,
n
−
1
k=0, 1, ... , n-1
k
=
0
,
1
,
...
,
n
−
1
. Prove that all the numbers are negative or zero.
12
1
Hide problems
Three variable expression divisibility
Given unequal integers
x
,
y
,
z
x, y, z
x
,
y
,
z
prove that
(
x
−
y
)
5
+
(
y
−
z
)
5
+
(
z
−
x
)
5
(x-y)^5 + (y-z)^5 + (z-x)^5
(
x
−
y
)
5
+
(
y
−
z
)
5
+
(
z
−
x
)
5
is divisible by
5
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
5(x-y)(y- z)(z-x)
5
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
.
11
1
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Maximum Area of a Triangle
The triangle
A
B
C
ABC
A
BC
satisfies
0
≤
A
B
≤
1
≤
B
C
≤
2
≤
C
A
≤
3
0\le AB\le 1\le BC\le 2\le CA\le 3
0
≤
A
B
≤
1
≤
BC
≤
2
≤
C
A
≤
3
. What is the maximum area it can have?
10
1
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Isosceles Triangle
In a triangle,
A
B
=
B
C
AB=BC
A
B
=
BC
and
M
M
M
is the midpoint of
A
C
AC
A
C
.
H
H
H
is chosen on
B
C
BC
BC
so that
M
H
MH
M
H
is perpendicular to
B
C
BC
BC
.
P
P
P
is the midpoint of
M
H
MH
M
H
. Prove that
A
H
AH
A
H
is perpendicular to
B
P
BP
BP
.
9
1
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Number with 1998 digits.
Given is a number with
1998
1998
1998
digits which is divisible by
9
9
9
. Let
x
x
x
be the sum of its digits, let
y
y
y
be the sum of the digits of
x
x
x
, and
z
z
z
the sum of the digits of
y
y
y
. Find
z
z
z
.
8
1
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Point inside a Pentagon
Given is a fixed regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
with side
1
1
1
. Let
M
M
M
be an arbitrary point inside or on it. Let the distance from
M
M
M
to the closest vertex be
r
1
r_1
r
1
, to the next closest be
r
2
r_2
r
2
and so on, so that the distances from
M
M
M
to the five vertices satisfy
r
1
≤
r
2
≤
r
3
≤
r
4
≤
r
5
r_1\le r_2\le r_3\le r_4\le r_5
r
1
≤
r
2
≤
r
3
≤
r
4
≤
r
5
. Find (a) the locus of
M
M
M
which gives
r
3
r_3
r
3
the minimum possible value, and (b) the locus of
M
M
M
which gives
r
3
r_3
r
3
the maximum possible value.
6
1
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Construct triangle using perpendicularity of medians
Given the lengths
A
B
AB
A
B
and
B
C
BC
BC
and the fact that the medians to those two sides are perpendicular, construct the triangle
A
B
C
ABC
A
BC
.
5
1
Hide problems
Array of numbers with 1 and -1
An
n
×
n
n \times n
n
×
n
array of numbers is given.
n
n
n
is odd and each number in the array is
1
1
1
or
−
1
-1
−
1
. Prove that the number of rows and columns containing an odd number of
−
1
-1
−
1
s cannot total
n
n
n
.
4
1
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Cubic with Integer Coefficients
Prove that there are no integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that the polynomial
a
x
3
+
b
x
2
+
c
x
+
d
ax^3+bx^2+cx+d
a
x
3
+
b
x
2
+
c
x
+
d
equals
1
1
1
at
x
=
19
x=19
x
=
19
and
2
2
2
at
x
=
62
x=62
x
=
62
.
3
1
Hide problems
Sequence of Integers
Given integers
a
0
,
a
1
,
.
.
.
,
a
100
a_0,a_1, ... , a_{100}
a
0
,
a
1
,
...
,
a
100
, satisfying
a
1
>
a
0
a_1>a_0
a
1
>
a
0
,
a
1
>
0
a_1>0
a
1
>
0
, and
a
r
+
2
=
3
a
r
+
1
−
2
a
r
a_{r+2}=3 a_{r+1}-2a_r
a
r
+
2
=
3
a
r
+
1
−
2
a
r
for
r
=
0
,
1
,
.
.
.
,
98
r=0, 1, ... , 98
r
=
0
,
1
,
...
,
98
. Prove
a
100
>
299
a_{100}>299
a
100
>
299
2
1
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Locus of a tangent point
Given a fixed circle
C
C
C
and a line L through the center
O
O
O
of
C
C
C
. Take a variable point
P
P
P
on
L
L
L
and let
K
K
K
be the circle with center
P
P
P
through
O
O
O
. Let
T
T
T
be the point where a common tangent to
C
C
C
and
K
K
K
meets
K
K
K
. What is the locus of
T
T
T
?
1
1
Hide problems
Constructing Quadrilateral
A
B
C
D
ABCD
A
BC
D
is any convex quadrilateral. Construct a new quadrilateral as follows. Take
A
′
A'
A
′
so that
A
A
A
is the midpoint of
D
A
′
DA'
D
A
′
; similarly,
B
′
B'
B
′
so that
B
B
B
is the midpoint of
A
B
′
AB'
A
B
′
;
C
′
C'
C
′
so that
C
C
C
is the midpoint of
B
C
′
BC'
B
C
′
; and
D
′
D'
D
′
so that
D
D
D
is the midpoint of
C
D
′
CD'
C
D
′
. Show that the area of
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
is five times the area of
A
B
C
D
ABCD
A
BC
D
.
7
1
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Inequality with 4 variable
Let
a
;
b
;
c
;
d
>
0
a;b;c;d>0
a
;
b
;
c
;
d
>
0
such that
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
a
2
+
b
2
+
c
2
+
d
2
+
a
(
b
+
c
)
+
b
(
c
+
d
)
+
c
(
d
+
a
)
≥
10
a^2+b^2+c^2+d^2+a(b+c)+b(c+d)+c(d+a)\ge 10
a
2
+
b
2
+
c
2
+
d
2
+
a
(
b
+
c
)
+
b
(
c
+
d
)
+
c
(
d
+
a
)
≥
10