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National and Regional Contests
Russia Contests
Soros Olympiad in Mathematics
VI Soros Olympiad 1999 - 2000 (Russia)
VI Soros Olympiad 1999 - 2000 (Russia)
Part of
Soros Olympiad in Mathematics
Subcontests
(40)
8.8
1
Hide problems
S_n + 1 is divisible by 2^{n-2} (VI Soros Olympiad 1990-00 R1 8.8)
Let
p
1
p_1
p
1
,
p
2
p_2
p
2
,
.
.
.
...
...
,
p
n
p_n
p
n
be different prime numbers (
n
≥
2
n\ge 2
n
≥
2
). All possible products containing an even number of coefficients (all coefficients are different) are composed of these numbers. Let
S
n
S_n
S
n
be the sum of all such products. For example,
S
4
=
p
1
p
2
+
p
1
p
3
+
p
1
p
4
+
p
2
p
3
+
p
2
p
4
+
p
3
p
4
+
p
1
p
2
p
3
p
4
.
S_4 = p_1p_2 + p_1p_3 + p_1p_4 + p_2p_3 + p_2p_4 + p_3p_4+ p_1p_2p_3p_4.
S
4
=
p
1
p
2
+
p
1
p
3
+
p
1
p
4
+
p
2
p
3
+
p
2
p
4
+
p
3
p
4
+
p
1
p
2
p
3
p
4
.
Prove that
S
n
+
1
S_n + 1
S
n
+
1
is divisible by
2
n
−
2
2^{n-2}
2
n
−
2
.
8.7
1
Hide problems
1/(x+y)^2+1/x^2+1/y^2>=9/4xy (VI Soros Olympiad 1990-00 R1 8.7)
Prove that for any positive real
x
x
x
and
y
y
y
, holds the inequality
1
(
x
+
y
)
2
+
1
x
2
+
1
y
2
≥
9
4
x
y
\frac{1}{(x+y)^2}+\frac{1}{x^2}+\frac{1}{y^2} \ge \frac{9}{4xy}
(
x
+
y
)
2
1
+
x
2
1
+
y
2
1
≥
4
x
y
9
8.6
1
Hide problems
sum of proper fractions (VI Soros Olympiad 1990-00 R1 8.6)
Two players take turns writing down all proper non-decreasing fractions with denominators from
1
1
1
to
1999
1999
1999
and at the same time writing a "
+
+
+
" sign before each fraction. After all such fractions are written out, their sum is found. If this amount is an integer number, then the one who made the entry last wins, otherwise his opponent wins. Who will be able to secure a win?
8.5
1
Hide problems
3x3 NT system a^4+14ab+1=n^4 (VI Soros Olympiad 1990-00 R1 8.5)
Solve the following system of equations in natural numbers
{
a
4
+
14
a
b
+
1
=
n
4
b
4
+
14
b
c
+
1
=
m
4
c
4
+
14
c
a
+
1
=
k
4
\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}
⎩
⎨
⎧
a
4
+
14
ab
+
1
=
n
4
b
4
+
14
b
c
+
1
=
m
4
c
4
+
14
c
a
+
1
=
k
4
8.4
1
Hide problems
TH bisects BC (VI Soros Olympiad 1990-00 R1 8.4)
Let
C
H
CH
C
H
be the altitude of triangle ABC,
O
O
O
be the center of the circle circumscribed around it. Point
T
T
T
is the projection of point
C
C
C
on the line
T
O
TO
TO
. Prove that the line
T
H
TH
T
H
bisects the side
B
C
BC
BC
.
8.3
1
Hide problems
n+72, has same digts reversely with n (VI Soros Olympiad 1990-00 R1 8.3)
72
72
72
was added to the natural number
n
n
n
and in the sum we got a number written in the same digits as the number
n
n
n
, but in the reverse order. Find all numbers
n
n
n
that satisfy the given condition.
8.2
1
Hide problems
(x+y)/(x-y)+(x-y)/(x+y)=1999 (VI Soros Olympiad 1990-00 R1 8.2)
Real numbers
x
x
x
and
y
y
y
it is know that
x
+
y
x
−
y
+
x
−
y
x
+
y
=
1999
\frac{x+y}{x-y}+\frac{x-y}{x+y}= 1999
x
−
y
x
+
y
+
x
+
y
x
−
y
=
1999
. Find the value of the expression
x
4
+
y
4
x
4
−
y
4
+
x
4
−
y
4
x
4
+
y
4
.
\frac{x^4+y^4}{x^4-y^4}+\frac{x^4-y^4}{x^4+y^4}.
x
4
−
y
4
x
4
+
y
4
+
x
4
+
y
4
x
4
−
y
4
.
8.1
1
Hide problems
p+q+r >= 20 (VI Soros Olympiad 1990-00 R1 8.1)
Let
p
,
q
,
r
p,q,r
p
,
q
,
r
be prime numbers such that
2
p
>
q
2p>q
2
p
>
q
,
q
>
2
r
q > 2r
q
>
2
r
and
q
>
p
+
r
q>p+r
q
>
p
+
r
. Prove that
p
+
q
+
r
≥
20
p+q+r\ge 20
p
+
q
+
r
≥
20
.
11.10
1
Hide problems
angles wanted - VI Soros Olympiad 1999-00 Round 1 11.10
In triangle
A
B
C
ABC
A
BC
, angle
A
A
A
is equal to
a
a
a
and angle
B
B
B
is equal to
2
a
2a
2
a
. A circle with center at point
C
C
C
of radius
C
A
CA
C
A
intersects the line containing the bisector of the exterior angle at vertex
B
B
B
, at points
M
M
M
and
N
N
N
. Find the angles of triangle
M
A
N
MAN
M
A
N
.
11.9
1
Hide problems
sin a + sin (a m ) >= c - VI Soros Olympiad 1999-00 Round 1 11.9
Find the largest
c
c
c
such that for any
λ
≥
1
\lambda \ge 1
λ
≥
1
there is an a that satisfies the inequality
sin
a
+
sin
(
a
λ
)
≥
c
.
\sin a + \sin (a\lambda ) \ge c.
sin
a
+
sin
(
aλ
)
≥
c
.
11.6
2
Hide problems
homothetic polyhedrons
It is known that a
n
n
n
-vertex contains within itself a polyhedron
M
M
M
with a center of symmetry at some point
Q
Q
Q
and is itself contained in a polyhedron homothetic to
M
M
M
with a homothety center at a point
Q
Q
Q
and coefficient
k
k
k
. Find the smallest value of
k
k
k
if a)
n
=
4
n = 4
n
=
4
, b)
n
=
5
n = 5
n
=
5
.
integer polynomial with root \sqrt2+\sqrt3 has also \sqrt2 - \sqrt3
Let
P
(
x
)
P(x)
P
(
x
)
be a polynomial with integer coefficients. It is known that the number
2
+
3
\sqrt2+\sqrt3
2
+
3
is its root. Prove that the number
2
−
3
\sqrt2-\sqrt3
2
−
3
is also its root.
11.8
1
Hide problems
plane that bisects both surface area and volume
Prove that the plane dividing in equal proportions the surface area and volume of the circumscribed polyhedron passes through the center of the sphere inscribed in this polyhedron.
11.7
1
Hide problems
arithmetic progressions with coprimes VI Soros Olympiad 1999-00 Round 1 11.7
Prove that there are arithmetic progressions of arbitrary length, consisting of different pairwise coprime natural numbers.
11.5
3
Hide problems
D=6G, D=25R, D=120T, G=4R, G=21T, R=5T- VI Soros Olympiad 1999-00 Round 1 11.5
At the currency exchange of the island of Luck they sell dinars (D), guilders (G), reals (R) and thalers (T). Stock brokers have the right to make a purchase and sale transaction with any pair of currencies no more than once per day. The exchange rates are as follows:
D
=
6
G
D = 6G
D
=
6
G
,
D
=
25
R
D = 25R
D
=
25
R
,
D
=
120
T
D = 120T
D
=
120
T
,
G
=
4
R
G = 4R
G
=
4
R
,
G
=
21
T
G = 21T
G
=
21
T
,
R
=
5
T
R = 5T
R
=
5
T
. For example, the entry
D
=
6
G
D = 6G
D
=
6
G
means that
1
1
1
dinar can be bought for
6
6
6
guilders (or
6
6
6
guilders can be sold for
1
1
1
dinar). In the morning the broker had
80
80
80
dinars,
100
100
100
guilders,
100
100
100
reals and
50
,
400
50,400
50
,
400
thalers. In the evening he had the same number of dinars and thalers. What is the maximum value of this number?
(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that for all real
x
x
x
holds the equality
(
1
+
2
x
)
P
(
2
x
)
=
(
1
+
2
1999
x
)
P
(
x
)
.
(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .
(
1
+
2
x
)
P
(
2
x
)
=
(
1
+
2
1999
x
)
P
(
x
)
.
sum (\sqrt{x_k^2-1}}{x_{k+1}} <= 1/sqrt2 n
Let
n
≥
2
n \ge 2
n
≥
2
and
x
1
x_1
x
1
,
x
2
x_2
x
2
,
.
.
.
...
...
,
x
n
x_n
x
n
be real numbers from the segment
[
1
,
2
]
[1,\sqrt2]
[
1
,
2
]
. Prove that holds the inequality
x
1
2
−
1
x
2
+
x
2
2
−
1
x
3
+
.
.
.
+
x
n
2
−
1
x
1
≤
2
2
n
.
\frac{\sqrt{x_1^2-1}}{x_2}+\frac{\sqrt{x_2^2-1}}{x_3}+...+\frac{\sqrt{x_n^2-1}}{x_1} \le \frac{\sqrt2}{2} n.
x
2
x
1
2
−
1
+
x
3
x
2
2
−
1
+
...
+
x
1
x
n
2
−
1
≤
2
2
n
.
11.4
3
Hide problems
3 spheres touch plane and line - VI Soros Olympiad 1999-00 Round 1 11.4
Let the line
L
L
L
be perpendicular to the plane
P
P
P
. Three spheres touch each other in pairs so that each sphere touches the plane
P
P
P
and the line
L
L
L
. The radius of the larger sphere is
1
1
1
. Find the minimum radius of the smallest sphere.
p^{2k}+q^{2n}=r^2 (VI Soros Olympiad 1990-00 R2 11.4)
For prime numbers
p
p
p
and
q
q
q
, natural numbers
n
n
n
,
k
k
k
,
r
r
r
, the equality
p
2
k
+
q
2
n
=
r
2
p^{2k}+q^{2n}=r^2
p
2
k
+
q
2
n
=
r
2
holds. Prove that the number
r
r
r
is prime.
circumcenter of (BCD) lies on (ABD), AB=AC
Given isosceles triangle
A
B
C
ABC
A
BC
(
A
B
=
A
C
AB = AC
A
B
=
A
C
). A straight line
ℓ
\ell
ℓ
is drawn through its vertex
B
B
B
at a right angle with
A
B
AB
A
B
. On the straight line
A
C
AC
A
C
, an arbitrary point
D
D
D
is taken, different from the vertices, and a straight line is drawn through it at a right angle with
A
C
AC
A
C
, intersecting
ℓ
\ell
ℓ
at the point
F
F
F
. Prove that the center of the circle circumscribed around the triangle
B
C
D
BCD
BC
D
lies on the circumscribed circle of triangle
A
B
D
ABD
A
B
D
.
11.3
3
Hide problems
sum a^4 + 2(sum ab^2 )^2 < 1 - VI Soros Olympiad 1999-00 Round 1 11.3
The numbers
a
,
b
a, b
a
,
b
and
c
c
c
are such that
a
2
+
b
2
+
c
2
=
1
a^2 + b^2 + c^2 = 1
a
2
+
b
2
+
c
2
=
1
. Prove that
a
4
+
b
4
+
c
4
+
2
(
a
b
2
+
b
c
2
+
c
a
2
)
2
≤
1.
a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1.
a
4
+
b
4
+
c
4
+
2
(
a
b
2
+
b
c
2
+
c
a
2
)
2
≤
1.
At what
a
,
b
a, b
a
,
b
and
c
c
c
does inequality turn into equality?
2S <=OA OC+ OBxOB for tangential ABCD
A convex quadrilateral
A
B
C
D
ABCD
A
BC
D
has an inscribed circle touching its sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
,
D
A
DA
D
A
at the points
M
M
M
,
N
N
N
,
P
P
P
,
K
K
K
, respectively. Let
O
O
O
be the center of the inscribed circle, the area of the quadrilateral
M
N
P
K
MNPK
MNP
K
is equal to
8
8
8
. Prove the inequality
2
S
≤
O
A
⋅
O
C
+
O
B
⋅
O
D
.
2S \le OA \cdot OC+ OB \cdot OD.
2
S
≤
O
A
⋅
OC
+
OB
⋅
O
D
.
3 spheres intersect alone one circle
Three spheres
s
1
s_1
s
1
,
s
2
s_2
s
2
,
s
3
s_3
s
3
intersect along one circle
ω
\omega
ω
. Let
A
A
A
be an arbitrary point lying on the circle
ω
\omega
ω
. Ray
A
B
AB
A
B
intersects spheres
s
1
s_1
s
1
,
s
2
s_2
s
2
,
s
3
s_3
s
3
at points
B
1
B_1
B
1
,
B
2
B_2
B
2
,
B
3
B_3
B
3
, respectively, ray
A
C
AC
A
C
intersects spheres
s
1
s_1
s
1
,
s
2
s_2
s
2
,
s
3
s_3
s
3
at points
C
1
C_1
C
1
,
C
2
C_2
C
2
,
C
3
C_3
C
3
, respectively (
B
i
≠
A
i
B_i \ne A_i
B
i
=
A
i
,
C
i
≠
A
i
C_i \ne A_i
C
i
=
A
i
,
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
). It is known that
B
2
B_2
B
2
is the midpoint of the segment
B
1
B
3
B_1B_3
B
1
B
3
. Prove that
C
2
C_2
C
2
is the midpoint of the segment
C
1
C
3
C_1C_3
C
1
C
3
.
11.1
3
Hide problems
x^3 + ax^2 + bx + c.=0, 3 real roots - VI Soros Olympiad 1999-00 Round 1 11.1
The game involves two players
A
A
A
and
B
B
B
. Player
A
A
A
sets the value of one of the coefficients
a
,
b
a, b
a
,
b
or
c
c
c
of the polynomial
x
3
+
a
x
2
+
b
x
+
c
.
x^3 + ax^2 + bx + c.
x
3
+
a
x
2
+
b
x
+
c
.
Player
B
B
B
indicates the value of any of the two remaining coefficients . Player
A
A
A
then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player
B
B
B
plays, the equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3 + ax^2 + bx + c = 0
x
3
+
a
x
2
+
b
x
+
c
=
0
to have three different (real) solutions?
2x2 system with arcsin and radical (VI Soros Olympiad 1990-00 R2 11.1)
Solve the system of equations
{
x
2
+
a
r
c
s
i
n
y
=
y
2
+
a
r
c
s
i
n
x
x
2
+
y
2
−
3
x
=
2
y
x
2
−
2
x
−
y
+
1
\begin{cases} x^2+arc siny =y^2+arcsin x \\ x^2+y^2-3x=2y\sqrt{x^2-2x-y}+1 \end{cases}
{
x
2
+
a
rcs
in
y
=
y
2
+
a
rcs
in
x
x
2
+
y
2
−
3
x
=
2
y
x
2
−
2
x
−
y
+
1
2 coprime among 16 naturals (VI Soros Olympiad 1990-00 R3 11.1)
16
16
16
different natural numbers are written on the board, none of which exceeds
30
30
30
. Prove that there must be two coprime numbers among the written numbers.
11.2
3
Hide problems
f''(0) , (...((x - 2)^2 - 2)^2 - 2)^2..- VI Soros Olympiad 1999-00 Round 1 11.2
Let
f
(
x
)
=
(
.
.
.
(
(
x
−
2
)
2
−
2
)
2
−
2
)
2
.
.
.
−
2
)
2
f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2
f
(
x
)
=
(
...
((
x
−
2
)
2
−
2
)
2
−
2
)
2
...
−
2
)
2
(here there are
n
n
n
brackets
(
)
( )
(
)
). Find
f
′
′
(
0
)
f''(0)
f
′′
(
0
)
bus, cyclist, motorcyclist, pedestrian (VI Soros Olympiad 1990-00 R2 11.2)
A bus and a cyclist left town
A
A
A
at
10
10
10
o'clock in the same direction, and a motorcyclist left town
B
B
B
to meet them
15
15
15
minutes later. The bus drove past the pedestrian at
10
10
10
o'clock
30
30
30
minutes, met the motorcyclist at
11
11
11
o'clock and arrived in the city of
B
B
B
at
12
12
12
o'clock. The motorcyclist met the cyclist
15
15
15
minutes after meeting the bus and another
15
15
15
minutes later caught up with the pedestrian. At what time did the cyclist and the pedestrian meet? (The speeds and directions of movement of all participants were equal, the pedestrian and the motorcyclist were moving in the direction of city
A
A
A
.)
sum of all products of k_1k_2...k_{999} (VI Soros Olympiad 1990-00 R3 11.2)
Find the sum of all possible products of natural numbers of the form
k
1
k
2
.
.
.
k
999
k_1k_2...k_{999}
k
1
k
2
...
k
999
, where in each product
k
1
<
k
2
<
.
.
.
<
k
999
<
1999
k_1 < k_2 < ... < k_{999} <1999
k
1
<
k
2
<
...
<
k
999
<
1999
, and there are no
k
i
k_i
k
i
and
k
j
k_j
k
j
such that
k
i
+
k
j
=
1999
k_i + k_j =1999
k
i
+
k
j
=
1999
.
10.10
2
Hide problems
fixed point for circumcircles - - VI Soros Olympiad 1999-00 Round 1 10.10
Take an arbitrary point
D
D
D
on side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
and draw a circle through point
D
D
D
and the centers of the circles inscribed in triangles
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
. Prove that all circles obtained for different points
D
D
D
of side
B
C
BC
BC
have a common point.
[a^m] + 1 is divisible by n
Prove that for every integer
n
≥
1
n \ge 1
n
≥
1
there exists a real number
a
a
a
such that for any integer
m
≥
1
m \ge 1
m
≥
1
the number
[
a
m
]
+
1
[a^m] + 1
[
a
m
]
+
1
is divisible by
n
n
n
(
[
x
]
[x]
[
x
]
denotes the largest integer that does not exceed
x
x
x
).
10.9
2
Hide problems
sin x + sin (m x) >= 1.8 - VI Soros Olympiad 1999-00 Round 1 10.9
Prove that for any
λ
>
3
\lambda > 3
λ
>
3
there is a number
x
x
x
for which
sin
x
+
sin
(
λ
x
)
≥
1.8.
\sin x + \sin (\lambda x) \ge 1.8.
sin
x
+
sin
(
λ
x
)
≥
1.8.
orthocenter wanted
Given an acute-angled triangle
A
B
C
ABC
A
BC
, in which
P
P
P
,
M
M
M
,
N
N
N
are the midpoints of the sides
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
, respectively. A point
H
H
H
is taken inside the triangle and perpendiculars
H
K
HK
HK
,
H
S
HS
H
S
,
H
Q
HQ
H
Q
are lowered from it to the sides
A
B
AB
A
B
,
B
C
BC
BC
,
A
C
AC
A
C
, respectively (
K
∈
A
B
K \in AB
K
∈
A
B
,
S
∈
B
C
S \in BC
S
∈
BC
,
Q
∈
A
C
Q \in AC
Q
∈
A
C
). It turned out that
M
K
=
M
Q
MK = MQ
M
K
=
MQ
,
N
S
=
N
K
NS = NK
NS
=
N
K
,
P
S
=
P
Q
PS=PQ
PS
=
PQ
. Prove that
H
H
H
is the point of intersection of the altitudes of triangle
A
B
C
ABC
A
BC
.
10.8
2
Hide problems
product a_i >= 99^{100} - VI Soros Olympiad 1999-00 Round 1 10.8
There are
100
100
100
positive numbers
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
100
a_{100}
a
100
such that
1
a
1
+
1
+
1
a
2
+
1
+
.
.
.
+
1
a
100
+
1
≤
1.
\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.
a
1
+
1
1
+
a
2
+
1
1
+
...
+
a
100
+
1
1
≤
1.
Prove that
a
1
⋅
a
2
⋅
.
.
.
⋅
a
100
≥
9
9
100
.
a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.
a
1
⋅
a
2
⋅
...
⋅
a
100
≥
9
9
100
.
f(x)=sin 1998x+ sin2000x (VI Soros Olympiad 1990-00 R1 10.8)
Find the smallest positive period of the function
f
(
x
)
=
sin
(
1998
x
)
+
sin
(
2000
x
)
f(x)=\sin (1998x)+ \sin (2000x)
f
(
x
)
=
sin
(
1998
x
)
+
sin
(
2000
x
)
10.7
2
Hide problems
perpendicular bisects MN - VI Soros Olympiad 1999-00 Round 1 10.7
Let a line, perpendicular to side
A
D
AD
A
D
of parallelogram
A
B
C
D
ABCD
A
BC
D
passing through point
B
B
B
, intersect line
C
D
CD
C
D
at point
M
M
M
, and a line, passing through point
B
B
B
and perpendicular to side
C
D
CD
C
D
, intersect line
A
D
AD
A
D
at point
N
N
N
. Prove that the line passing through point
B
B
B
perpendicular to the diagonal
A
C
AC
A
C
, passes through the midpoint of the segment
M
N
MN
MN
.
1-100 in 10x10 grid
The numbers
1
,
2
,
3
,
.
.
.
,
99
,
100
1, 2, 3, ..., 99, 100
1
,
2
,
3
,
...
,
99
,
100
are randomly arranged in the cells of a square table measuring
10
×
10
10\times 10
10
×
10
(each number is used only once). Prove that there are three cells in the table whose sum of numbers does not exceed 1
82
82
82
. The centers of these cells form an isosceles right triangle, the legs of which are parallel to the edges of the table.
10.6
3
Hide problems
a^3 - a- 1 = 0 - VI Soros Olympiad 1999-00 Round 1 10.6
Let
a
3
−
a
−
1
=
0
a^3 - a- 1 = 0
a
3
−
a
−
1
=
0
. Find the exact value of the expression
3
a
2
−
4
a
3
+
a
2
a
2
+
3
a
+
2
4
.
\sqrt[3]{3a^2-4a} + a\sqrt[4]{2a^2+3a+2}.
3
3
a
2
−
4
a
+
a
4
2
a
2
+
3
a
+
2
.
3 points construction for max pentagon area
Points
A
A
A
and
B
B
B
are given on a circle. With the help of a compass and a ruler, construct on this circle the points
C
,
C,
C
,
D
D
D
,
E
E
E
that lie on one side of the straight line
A
B
AB
A
B
and for which the pentagon with vertices
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
,
E
E
E
has the largest possible area
x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0
A natural number
n
n
n
is given. Find the longest interval of a real line such that for numbers taken arbitrarily from it
a
0
a_0
a
0
,
a
1
a_1
a
1
,
a
2
a_2
a
2
,
.
.
.
...
...
,
a
2
n
−
1
a_{2n-1}
a
2
n
−
1
the polynomial
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
.
.
.
+
a
1
x
+
a
0
x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x + a_0
x
2
n
+
a
2
n
−
1
x
2
n
−
1
+
...
+
a
1
x
+
a
0
has no roots on the entire real axis. (The left and right ends of the interval do not belong to the interval.)
10.5
4
Show problems
10.4
4
Show problems
10.3
4
Show problems
10.2
4
Show problems
10.1
4
Show problems
9.10
2
Hide problems
at least 11 new mistakes were made - VI Soros Olympiad 1999-00 Round 1 9.10
The schoolboy wrote a homework essay on the topic “How I spent my summer.” Two of his comrades from a neighboring school decided not to bother themselves with work and rewrote his essay. But while rewriting they made several mistakes - each their own. Before submitting their work, both students gave their essays to four other friends to rewrite (each gave them to two acquaintances). These four schoolchildren do the same, and so on. With each rewrite, all previous mistakes are saved and, possibly, new ones are made. It is known that on some day each new essay contained at least
10
10
10
errors. Prove that there was a day when at least
11
11
11
new mistakes were made in total.
sum 1/x(1-y ) >= 3 / ( xyz+(1-x)(1-y)(1-z) ) , for 0<x,y,z<1
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be real numbers from interval
(
0
,
1
)
(0, 1)
(
0
,
1
)
. Prove that
1
x
(
1
−
y
)
+
1
y
(
1
−
x
)
+
1
z
(
1
−
x
)
≥
3
x
y
z
+
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
\frac{1}{x(1-y)}+\frac{1}{y(1-x)}+\frac{1}{z(1-x)}\ge \frac{3}{xyz+(1-x)(1-y)(1-z)}
x
(
1
−
y
)
1
+
y
(
1
−
x
)
1
+
z
(
1
−
x
)
1
≥
x
yz
+
(
1
−
x
)
(
1
−
y
)
(
1
−
z
)
3
9.9
2
Hide problems
square , 4 right isosceles triangles -VI Soros Olympiad 1999-00 Round 1 9.9
On the plane there are two isosceles non-intersecting right triangles
A
B
C
ABC
A
BC
and
D
E
C
DEC
D
EC
(
A
B
AB
A
B
and
D
E
DE
D
E
are the hypotenuses,
A
B
D
E
ABDE
A
B
D
E
is a convex quadrilateral), and
A
B
=
2
D
E
AB = 2 DE
A
B
=
2
D
E
. Let's construct two more isosceles right triangles:
B
D
F
BDF
B
D
F
(with hypotenuse
B
F
BF
BF
located outside triangle
B
D
C
BDC
B
D
C
) and
A
E
G
AEG
A
EG
(with hypotenuse
A
G
AG
A
G
located outside triangle
A
E
C
AEC
A
EC
). Prove that the line
F
G
FG
FG
passes through a point
N
N
N
such that
D
C
E
N
DCEN
D
CEN
is a square.
min area of a triangle inscribed in a right angle
The center of a circle, the radius of which is
r
r
r
, lies on the bisector of the right angle
A
A
A
at a distance
a
a
a
from its sides (
a
>
r
a > r
a
>
r
). A tangent to the circle intersects the sides of the angle at points
B
B
B
and
C
C
C
. Find the smallest possible value of the area of triangle
A
B
C
ABC
A
BC
.
9.8
2
Hide problems
locus of circumcenters - VI Soros Olympiad 1999-00 Round 1 9.8
Given a line
ℓ
\ell
ℓ
and a ray
p
p
p
on a plane with its origin on this line. Two fixed circles (not necessarily equal) are constructed, inscribed in the two formed angles. On ray
p
p
p
, point
A
A
A
is taken so that the tangents from
A
A
A
to the given circles, different from
p
p
p
, intersect line
ℓ
\ell
ℓ
at points
B
B
B
and
C
C
C
, and at the same time triangle
A
B
C
ABC
A
BC
contains the given circles. Find the locus of the centers of the circles inscribed in triangle
A
B
C
ABC
A
BC
(as
A
A
A
moves).
sum \sqrt{1+k^2} sin(a_k-a_{1000}) (VI Soros Olympiad 1990-00 R1 9.8)
Let
a
n
a_n
a
n
denote an angle from the interval for each
(
0
,
π
2
)
\left( 0, \frac{\pi}{2}\right)
(
0
,
2
π
)
, the tangent of which is equal to
n
n
n
. Prove that
1
+
1
2
sin
(
a
1
−
a
1000
)
+
1
+
2
2
sin
(
a
2
−
a
1000
)
+
.
.
.
+
1
+
200
0
2
sin
(
a
2000
−
a
1000
)
=
sin
a
1000
\sqrt{1+1^2} \sin(a_1-a_{1000}) + \sqrt{1+2^2} \sin(a_2-a_{1000})+...+\sqrt{1+2000^2} \sin(a_{2000}-a_{1000}) = \sin a_{1000}
1
+
1
2
sin
(
a
1
−
a
1000
)
+
1
+
2
2
sin
(
a
2
−
a
1000
)
+
...
+
1
+
200
0
2
sin
(
a
2000
−
a
1000
)
=
sin
a
1000
9.7
2
Hide problems
ratio chasing, AB/BC=AC/CD=l -- VI Soros Olympiad 1999-00 Round 1 9.7
Points
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
are located on line
ℓ
\ell
ℓ
so that
A
B
B
C
=
A
C
C
D
=
λ
\frac{AB}{BC}=\frac{AC}{CD}=\lambda
BC
A
B
=
C
D
A
C
=
λ
. A certain circle is tangent to line
ℓ
\ell
ℓ
at point
C
C
C
. A line is drawn through
A
A
A
that intersects this circle at points
M
M
M
and
N
N
N
such that the bisector perpendiculars to segments
B
M
BM
BM
and
D
N
DN
D
N
intersect at point
Q
Q
Q
on line
ℓ
\ell
ℓ
. In what ratio does point
Q
Q
Q
divide segment
A
D
AD
A
D
?
equilateral wanted if 3 projections are equal
In the acute-angled triangle
A
B
C
ABC
A
BC
, the points
P
P
P
,
N
N
N
,
M
M
M
are the feet of the altitudes drawn from the vertices
C
C
C
,
A
A
A
,
B
B
B
, respectively. The lengths of the projections of the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
on straight lines
M
N
MN
MN
,
P
M
PM
PM
,
N
P
NP
NP
respectively, are equal to each other. Prove that triangle
A
B
C
ABC
A
BC
is regular.
9.6
3
Hide problems
radicals simplify -- VI Soros Olympiad 1999-00 Round 1 9.6
For all valid values of
a
a
a
and
b
b
b
, simplify the expression
4
b
−
a
2
+
2
a
b
+
4
+
a
4
a
b
−
10
b
2
−
8
+
b
.
\frac{\sqrt{4b-a^2+2ab+4}+a}{\sqrt{4ab-10b^2-8}+b}.
4
ab
−
10
b
2
−
8
+
b
4
b
−
a
2
+
2
ab
+
4
+
a
.
battleship game, revisited (VI Soros Olympiad 1990-00 R1 9.6)
On the "battleship" field (a square of
10
×
10
10\times 10
10
×
10
cells),
10
10
10
"ships" are placed in the following sequence: first one "ship" of size
1
×
4
1\times 4
1
×
4
, then two - of size
1
×
3
1\times 3
1
×
3
, three - of size
1
×
2
1\times 2
1
×
2
, and, finally, four -
1
×
1
1\times 1
1
×
1
. The rules do not allow "ships" to touch each other even with their tops. Can it happen that when part of the "ships" have already been displayed, there is nowhere to place the next one?
recurrence relation, parity of a_{n+1}a_n
The sequence of integers
a
1
,
a
2
,
a
3
,
.
.
a_1,a_2,a_3 ,..
a
1
,
a
2
,
a
3
,
..
such that
a
1
=
1
a_1 = 1
a
1
=
1
,
a
2
=
2
a_2 = 2
a
2
=
2
and for every natural
n
≥
1
n \ge 1
n
≥
1
a
n
+
2
=
{
2001
a
n
+
1
−
1999
a
n
,
if
the
product
a
n
+
1
a
n
is
an
even
number
/
a
n
+
1
−
a
n
,
if
the
product
a
n
+
1
a
n
is
an
odd
number
a_{n+2}=\begin{cases} 2001a_{n+1} - 1999a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,even\,\,number} /\\ a_{n+1}-a_n , \text{\,\,if\,\,the\,\,product\,\,} a_{n+1}a_n \text{is\,\,an\,\,odd\,\,number} \end{cases}
a
n
+
2
=
{
2001
a
n
+
1
−
1999
a
n
,
if
the
product
a
n
+
1
a
n
is
an
even
number
/
a
n
+
1
−
a
n
,
if
the
product
a
n
+
1
a
n
is
an
odd
number
Is there such a natural
m
m
m
that
a
m
=
2000
a_m= 2000
a
m
=
2000
?
9.5
4
Show problems
9.4
4
Show problems
9.3
4
Show problems
9.2
4
Show problems
9.1
4
Show problems
grade8
1
Hide problems
grade 8 problems (VI Soros Olympiad 1999-00 Round 1)
p1. Can a number ending in
1999
1999
1999
be the square of a natural number? p2. The Three-Headed Snake Gorynych celebrated his birthday. His heads took turns feasting on birthday cakes and ate two identical cakes in
15
15
15
minutes. It is known that each head ate as much time as it would take the other two to eat the same pie together. In how many minutes would the three heads of the Serpent Gorynych eat one pie together? p3. Find the sum of the coefficients of the polynomial obtained after opening the brackets and bringing similar terms into the expression: a)
(
7
x
−
6
)
4
−
1
(7x - 6)^4 - 1
(
7
x
−
6
)
4
−
1
b)
(
7
x
−
6
)
1999
−
1
(7x - 6)^{1999}-1
(
7
x
−
6
)
1999
−
1
p4. The general wants to arrange seven anti-aircraft installations so that among any three of them there are two installations, the distance between which is exactly
10
10
10
kilometers. Help the general solve this problem. p5. Gulliver, whose height is
999
999
999
millimeters, is building a tower of cubes. The first cube has a height of
1
/
2
1/2
1/2
a lilikilometer, the second -
1
/
4
1/4
1/4
a lilikilometer, the third -
1
/
8
1/8
1/8
a lilikilometer, etc. How many cubes will be in the tower when its height exceeds Gulliver's height. (
1
1
1
lilikilometer is equal to
1000
1000
1000
lilimeters). p6. It is known that in any pentagon you can choose three diagonals from which you can form a triangle. Is there a pentagon in which such diagonals can be chosen in a unique way? p7. It is known that for natural numbers
a
a
a
and
b
b
b
the equality
19
a
=
99
b
19a = 99b
19
a
=
99
b
holds. Can
a
+
b
a + b
a
+
b
be a prime number? p8. Vitya thought of
5
5
5
integers and told Vanya all their pairwise sums:
0
,
1
,
5
,
7
,
11
,
12
,
18
,
24
,
25
,
29.
0, 1, 5, 7, 11, 12, 18, 24, 25, 29.
0
,
1
,
5
,
7
,
11
,
12
,
18
,
24
,
25
,
29.
Help Vanya guess the numbers he has in mind. p9. In a
3
×
3
3 \times 3
3
×
3
square, numbers are arranged so that the sum of the numbers in each row, in each column and on each major diagonal is equal to
0
0
0
. It is known that the sum of the squares of the numbers in the top row is
n
n
n
. What can be the sum of the squares of the numbers in the bottom line? p10.
N
N
N
points are marked on a circle. Two players play this game: the first player connects two of these points with a chord, from the end of which the second player draws a chord to one of the remaining points so as not to intersect the already drawn chord. Then the first player makes the same “move” - draws a new chord from the end of the second chord to one of the remaining points so that it does not intersect any of the already drawn ones. The one who cannot make such a “move” loses. Who wins when played correctly? (A chord is a segment whose ends lie on a given circle) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade7
1
Hide problems
grade 7 problems (VI Soros Olympiad 1999-00 Round 1)
p1. Cities A, B, C, D and E are located next to each other along the highway at a distance of
5
5
5
km from each other. The bus runs along the highway from city A to city E and back. The bus consumes
20
20
20
liters of gasoline for every
100
100
100
kilometers. In which city will a bus run out of gas if it initially had
150
150
150
liters of gasoline in its tank? p2. Find the minimum four-digit number whose product of all digits is
729
729
729
. Explain your answer. p3. At the parade, soldiers are lined up in two lines of equal length, and in the first line the distance between adjacent soldiers is
20
%
20\%
20%
greater than in the second (there is the same distance between adjacent soldiers in the same line). How many soldiers are in the first rank if there are
85
85
85
soldiers in the second rank? p4. It is known about three numbers that the sum of any two of them is not less than twice the third number, and the sum of all three is equal to
300
300
300
. Find all triplets of such (not necessarily integer) numbers. p5. The tourist fills two tanks of water using two hoses.
2.9
2.9
2.9
liters of water flow out per minute from the first hose,
8.7
8.7
8.7
liters from the second. At that moment, when the smaller tank was half full, the tourist swapped the hoses, after which both tanks filled at the same time. What is the capacity of the larger tank if the capacity of the smaller one is
12.5
12.5
12.5
liters? p6. Is it possible to mark 6 points on a plane and connect them with non-intersecting segments (with ends at these points) so that exactly four segments come out of each point? p7. Petya wrote all the natural numbers from
1
1
1
to
1000
1000
1000
and circled those that are represented as the difference of the squares of two integers. Among the circled numbers, which numbers are more even or odd? p8. On a sheet of checkered paper, draw a circle of maximum radius that intersects the grid lines only at the nodes. Explain your answer. p9. Along the railway there are kilometer posts at a distance of
1
1
1
km from each other. One of them was painted yellow and six were painted red. The sum of the distances from the yellow pillar to all the red ones is
14
14
14
km. What is the maximum distance between the red pillars? p10. The island nation is located on
100
100
100
islands connected by bridges, with some islands also connected to the mainland by a bridge. It is known that from each island you can travel to each (possibly through other islands). In order to improve traffic safety, one-way traffic was introduced on all bridges. It turned out that from each island you can leave only one bridge and that from at least one of the islands you can go to the mainland. Prove that from each island you can get to the mainland, and along a single route. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.