MathDB
Problems
Contests
National and Regional Contests
Russia Contests
Soros Olympiad in Mathematics
VII Soros Olympiad 2000 - 01
VII Soros Olympiad 2000 - 01
Part of
Soros Olympiad in Mathematics
Subcontests
(33)
11.7
1
Hide problems
sets of sequences, M<= 2n-4 (VII Soros Olympiad 2000-01 R1 11.7)
Consider all possible functions defined for
x
=
1
,
2
,
.
.
.
,
M
x = 1, 2, ..., M
x
=
1
,
2
,
...
,
M
and taking values
y
=
1
,
2
,
.
.
.
,
n
y = 1, 2, ..., n
y
=
1
,
2
,
...
,
n
. We denote the set of such functions by
T
.
T.
T
.
By
T
0
T_0
T
0
we denote the subset of
T
T
T
consisting of functions whose value changes exactly by
1
1
1
(in one direction or another) when the argument changes by
1
1
1
. Prove that if
M
≥
2
n
−
4
M\ge 2n-4
M
≥
2
n
−
4
, then among the functions from of the set
T
T
T
, there is a function that coincides at least at one point with any function from
T
0
T_0
T
0
. Specify at least one such function. Prove that if
M
<
2
n
−
4
M <2n-4
M
<
2
n
−
4
, then there is no such function.
11.2
1
Hide problems
3 parameter trinomial (VII Soros Olympiad 2000-01 R1 11.2)
For all valid values of
a
,
b
a, b
a
,
b
, and
c
c
c
, solve the equation
a
(
x
−
b
)
(
x
−
c
)
(
a
−
b
)
(
a
−
c
)
+
b
(
x
−
c
)
(
x
−
a
)
(
b
−
c
)
(
b
−
a
)
+
c
(
x
−
a
)
(
x
−
b
)
(
c
−
a
)
(
c
−
b
)
=
x
2
\frac{a (x-b) (x-c) }{(a-b) (a-c)} + \frac{b (x-c) (x-a)}{(b-c) (b-a)} +\frac{c (x-a) (x-b) }{(c-a ) (c-b)} = x^2
(
a
−
b
)
(
a
−
c
)
a
(
x
−
b
)
(
x
−
c
)
+
(
b
−
c
)
(
b
−
a
)
b
(
x
−
c
)
(
x
−
a
)
+
(
c
−
a
)
(
c
−
b
)
c
(
x
−
a
)
(
x
−
b
)
=
x
2
11.1
1
Hide problems
y(x)=cos (cos x)+ a sin (sin x) is periodic (VII Soros Olympiad 2000-01 R1 11.1)
Prove that for any
a
a
a
the function
y
(
x
)
=
cos
(
cos
x
)
+
a
⋅
sin
(
sin
x
)
y (x) = \cos (\cos x) + a \cdot \sin (\sin x)
y
(
x
)
=
cos
(
cos
x
)
+
a
⋅
sin
(
sin
x
)
is periodic. Find its smallest period in terms of
a
a
a
.
11.3
1
Hide problems
exists x_o in R: F(x_0) F'''(x_0)>=0 (VII Soros Olympiad 2000-01 R1 11.3)
The function
F
(
x
)
F (x)
F
(
x
)
is defined on
R
R
R
and has a second derivative for each value of the variable. Prove that there is a point
x
0
x_0
x
0
such that the product
F
(
x
0
)
F
′
′
(
x
0
)
F(x_0) F''(x_0)
F
(
x
0
)
F
′′
(
x
0
)
is non-negative.PS. In my [url=http://www.1543.su/olympiads/soros/20002001/1/1soros00.htm]source, it is not clear if it means
F
(
x
0
)
F
′
′
(
x
0
)
F(x_0) F''(x_0)
F
(
x
0
)
F
′′
(
x
0
)
or
F
(
x
0
)
F
′
(
x
0
)
F(x_0) F'(x_0)
F
(
x
0
)
F
′
(
x
0
)
.
11.4
1
Hide problems
first 200 decimal digits of a^2000, cubic (VII Soros Olympiad 2000-01 R1 11.4)
Let
a
a
a
be the largest root of the equation
x
3
−
3
x
2
+
1
=
0
x^3 - 3x^2 + 1 = 0
x
3
−
3
x
2
+
1
=
0
. Find the first
200
200
200
decimal digits for the number
a
2000
a^{2000}
a
2000
.
10.7
1
Hide problems
lucky gablings on dollar's exchange rate (VII Soros Olympiad 2000-01 R1 10.7)
The President of the Bank "Glavny Central" Gerasim Shchenkov announced that from January
2
2
2
,
2001
2001
2001
until January
31
31
31
of the same year, the dollar exchange rate would not go beyond the boundaries of the corridor of
27
27
27
rubles
50
50
50
kopecks. and
28
28
28
rubles
30
30
30
kopecks for the dollar. On January
2
2
2
, the rate will be a multiple of
5
5
5
kopecks, and starting from January 3, each day will differ from the rate of the previous day by exactly
5
5
5
kopecks. Mr. Shchenkov suggested that citizens try to guess what the dollar exchange rate will be during the specified period. Anyone who can give an accurate forecast for at least one day, he promised to give a cash prize. One interesting person lives in our house, a tireless arguer. For his passion for arguments and constant winnings, he was even nicknamed Zhora Sporos. Zhora claims that he can give such a forecast of the dollar exchange rate for every day from January 424 to January 4314, which he will surely guess at least once, if, of course, the banker strictly acts in accordance with the announced rules. Is Zhora right?Note: 1 ruble =100 kopecks [hide=original wording]10-I-7. Президент банка "Главный централ" Герасим Щенков объявил, что со 2-го января 2001 года и до 31-го января этого же года курс доллара не будет выходить за границы коридора 27 руб. 50 коп. и 28 руб. 30 коп. за доллар. 2-го января курс будет кратен 5 копейкам, а, начиная с 3-го января, каждый день будет отличаться от курса предыдущего дня ровно на 5 копеек. Господин Щенков предложил гражданам попробовать угадать, каким будет курс доллара в течение указанного периода. Тому, кто сумеет дать точный прогноз хотя бы на один день, он обещал выдать денежный приз. В нашем доме живет один интересный человек, неутомимый спорщик. За страсть к спорам и постоянные выигрыши его даже прозвали Жора Спорос. Жора утверждает, что может дать такой прогноз курса доллара на каждый день со 2-го по 31-е января, что обязательно хотя бы один раз угадает, если, конечно, банкир будет строго действовать в соответствии с объявленными правилами. Прав ли Жора?
10.3
1
Hide problems
draw points of convex function on [0,1] (VII Soros Olympiad 2000-01 R1 10.3)
Let
y
=
f
(
x
)
y = f (x)
y
=
f
(
x
)
be a convex function defined on
[
0
,
1
]
[0,1]
[
0
,
1
]
,
f
(
0
)
=
0
,
f (0) = 0,
f
(
0
)
=
0
,
f
(
1
)
=
0
f (1) = 0
f
(
1
)
=
0
. It is also known that the area of the segment bounded by this function and the segment
[
0
,
1
]
[0, 1]
[
0
,
1
]
is equal to
1
1
1
. Find and draw the set of points of the coordinate plane through which the graph of such a function can pass. (A function is called convex if all points of the line segment connecting any two points on its graph are located no higher than the graph of this function.)
10.2
1
Hide problems
cos of a,b,(a+b) rational if sines rational (VII Soros Olympiad 2000-01 R1 10.2)
Let
a
a
a
and
b
b
b
be acute corners. Prove that if
sin
a
\sin a
sin
a
,
sin
b
\sin b
sin
b
, and
sin
(
a
+
b
)
\sin (a + b)
sin
(
a
+
b
)
are rational numbers, then
cos
a
\cos a
cos
a
,
cos
b
\cos b
cos
b
, and
cos
(
a
+
b
)
\cos (a + b)
cos
(
a
+
b
)
are also rational numbers.
10.1
1
Hide problems
(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0 (VII Soros Olympiad 2000-01 R1 10.1)
Find all values of the parameter
a
a
a
for which the equation
(
a
−
1
)
2
x
4
+
(
a
2
−
a
)
x
3
+
3
x
−
1
=
0
(a-1)^2x^4 + (a^2-a) x^3 + 3x - 1 = 0
(
a
−
1
)
2
x
4
+
(
a
2
−
a
)
x
3
+
3
x
−
1
=
0
has a unique solution and for these
a
a
a
solve the equation.
9.5
1
Hide problems
2 parameter cubic equation (VII Soros Olympiad 2000-01 R1 9.5)
For all valid values of
a
a
a
and
b
b
b
, solve the equation
x
3
(
x
−
a
)
(
x
−
b
)
+
a
3
(
a
−
b
)
(
a
−
x
)
+
b
3
(
b
−
x
)
(
b
−
a
)
=
x
2
+
a
+
b
\frac{x^3}{(x-a) (x-b)} +\frac{a^3}{(a-b) (a-x)} + \frac{b^3}{ (b-x) (b-a)}= x^2 + a + b
(
x
−
a
)
(
x
−
b
)
x
3
+
(
a
−
b
)
(
a
−
x
)
a
3
+
(
b
−
x
)
(
b
−
a
)
b
3
=
x
2
+
a
+
b
9.4
1
Hide problems
cyclic and bus meet among 2 cities (VII Soros Olympiad 2000-01 R1 9.4)
The distance between cities
A
A
A
and
B
B
B
is
30
30
30
km. A bus departed from
A
A
A
, which makes a stop every
5
5
5
km for
2
2
2
minutes. The bus moves between stops at a speed of
80
80
80
km / h. Simultaneously with the departure of the bus from
A
A
A
, a cyclist leaves
B
B
B
to meet it, traveling at a speed of
27
27
27
km / h. How far from
A
A
A
will the cyclist meet the bus?
9.3
1
Hide problems
102 =sum of max no distinct primes (VII Soros Olympiad 2000-01 R1 9.3)
Write
102
102
102
as the sum of the largest number of distinct primes.
9.2
1
Hide problems
||x| -1| = a |x|+b |x-1|+c |x+1| + d (VII Soros Olympiad 2000-01 R1 9.2)
Find
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
such that for all
x
x
x
the equality
∣
∣
x
∣
−
1
∣
=
a
∣
x
∣
+
b
∣
x
−
1
∣
+
c
∣
x
+
1
∣
+
d
|| x | -1 | = a | x | + b | x-1 | + c | x + 1 | + d
∣∣
x
∣
−
1∣
=
a
∣
x
∣
+
b
∣
x
−
1∣
+
c
∣
x
+
1∣
+
d
holds.
9.1
1
Hide problems
draw points of x^3 + y^3 = x^2y^2 + xy (VII Soros Olympiad 2000-01 R1 9.1)
Draw on the plane a set of points whose coordinates
(
x
,
y
)
(x,y)
(
x
,
y
)
satisfy the equation
x
3
+
y
3
=
x
2
y
2
+
x
y
x^3 + y^3 = x^2y^2 + xy
x
3
+
y
3
=
x
2
y
2
+
x
y
.
8.10
1
Hide problems
numbers 0,1,-1 in 2nx2n board (VII Soros Olympiad 2000-01 R1 8.10)
Place in the cells the boards measuring: a)
2
×
2
2 \times 2
2
×
2
, b)
4
×
4
4 \times 4
4
×
4
, c)
2
n
×
2
n
2n \times 2n
2
n
×
2
n
, numbers
0
0
0
,
1
1
1
and
−
1
-1
−
1
so that in each case all the sums of numbers in rows and columns are different.
8.9
1
Hide problems
(a+b)(b+c)(a+c)/(abc) (VII Soros Olympiad 2000-01 R1 8.9)
It is known about the numbers
a
,
b
a, b
a
,
b
and
c
c
c
that
a
b
+
c
−
a
=
b
a
+
c
−
b
=
c
a
+
b
−
c
\frac{a}{b+c-a}=\frac{b}{a + c-b}= \frac{c}{a + b-c}
b
+
c
−
a
a
=
a
+
c
−
b
b
=
a
+
b
−
c
c
.What values can an expression take
(
a
+
b
)
(
b
+
c
)
(
a
+
c
)
a
b
c
\frac{(a + b) (b + c) (a + c)}{abc}
ab
c
(
a
+
b
)
(
b
+
c
)
(
a
+
c
)
?
8.7
1
Hide problems
(x+100) (x+99) ... (x-99)(x-100)=x^{201}+... (VII Soros Olympiad 2000-01 R1 8.7)
In the expression
(
x
+
100
)
(
x
+
99
)
.
.
.
(
x
−
99
)
(
x
−
100
)
(x + 100) (x + 99) ... (x-99) (x-100)
(
x
+
100
)
(
x
+
99
)
...
(
x
−
99
)
(
x
−
100
)
, the brackets were expanded and similar terms were given. The expression
x
201
+
.
.
.
+
a
x
2
+
b
x
+
c
x^{201} + ...+ ax^2 + bx + c
x
201
+
...
+
a
x
2
+
b
x
+
c
turned out. Find the numbers
a
a
a
and
c
c
c
.
8.5
1
Hide problems
1+- brackets , sum 10 (VII Soros Olympiad 2000-01 R1 8.5)
Vanya was asked to write on the board an expression equal to
10
10
10
, using only the numbers
1
1
1
, the signs
+
+
+
and
−
-
−
and brackets (you cannot make up the numbers
11
11
11
,
111
111
111
, etc., as well as
(
−
1
)
(-1)
(
−
1
)
). He knows that the bully Anton will then correct all the
+
+
+
signs to
−
-
−
and vice versa. Help Vanya compose the required expression, which will remain equal to
10
10
10
even after Anton's actions.
8.4
1
Hide problems
max no of vertices of cube, equilateral (VII Soros Olympiad 2000-01 R1 8.4)
Paint the maximum number of vertices of the cube red so that you cannot select three of the red vertices that form an equilateral triangle.
8.3
1
Hide problems
sum of no 1-500 not divisible by 5 or 7 (VII Soros Olympiad 2000-01 R1 8.3)
Find the sum of all such natural numbers from
1
1
1
to
500
500
500
that are not divisible by
5
5
5
or
7
7
7
.
8.2
1
Hide problems
2 brothers walk, run, bus related (VII Soros Olympiad 2000-01 R1 8.2)
The two brothers, without waiting for the bus, decided to walk to the next stop. After passing
1
/
3
1/3
1/3
of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is
30
30
30
km / h, neglect the bus stop time.
8.1
1
Hide problems
percentage of boys and girls (VII Soros Olympiad 2000-01 R1 8.1)
If there are as many boys in the class as there are girls in the class now, the percentage of girls will decrease by
1.4
1.4
1.4
times. Find out what percentage of the students in the class were boys.
9.8
1
Hide problems
locus of circumcenters 2<B_1A_1C_1+<BAC = 180^o(VII Soros Olympiad R1 9.8)
Given a triangle
A
B
C
ABC
A
BC
. On its sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, the points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
are taken, respectively , such that
2
∠
B
1
A
1
C
1
+
∠
B
A
C
=
18
0
o
2 \angle B_1 A_1 C_1 + \angle BAC = 180^o
2∠
B
1
A
1
C
1
+
∠
B
A
C
=
18
0
o
,
2
∠
A
1
C
1
B
1
+
∠
A
C
B
=
18
0
o
2 \angle A_1 C_1 B_1 + \angle ACB = 180^o
2∠
A
1
C
1
B
1
+
∠
A
CB
=
18
0
o
,
2
∠
C
1
B
1
A
1
+
∠
C
B
A
=
18
0
o
2 \angle C_1 B_1 A_1 + \angle CBA = 180^o
2∠
C
1
B
1
A
1
+
∠
CB
A
=
18
0
o
. Find the locus of the centers of the circles circumscribed about the triangles
A
1
B
1
C
1
A_1 B_1 C_1
A
1
B
1
C
1
(all possible such triangles are considered).
9.7
1
Hide problems
area chasing, 2 equal ratios in quadrilateral (VII Soros Olympiad R1 9.7)
Sides
A
B
AB
A
B
and
C
D
CD
C
D
of quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at point
E
E
E
. On the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
points
M
M
M
and
N
N
N
are taken, respectively, so that
A
M
/
A
C
=
B
N
/
B
D
=
k
AM / AC = BN / BD = k
A
M
/
A
C
=
BN
/
B
D
=
k
. Find the area of a triangle
E
M
N
EMN
EMN
if the area of
A
B
C
D
ABCD
A
BC
D
is
S
S
S
.
9.6
1
Hide problems
ratio wanted, rectangle inscribed in triangle (VII Soros Olympiad R1 9.6)
Two vertices of the rectangle are located on side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
, and the other two are on sides
A
B
AB
A
B
and
A
C
AC
A
C
. It is known that the midpoint of the altitude of this triangle, drawn on the side
B
C
BC
BC
, lies on one of the diagonals of the rectangle, and the side of the rectangle located on
B
C
BC
BC
is three times less than
B
C
BC
BC
. In what ratio does the altitude of the triangle divide the side
B
C
BC
BC
?
8.6
1
Hide problems
area of triangle on 3 // sides, by 3 cyclists (VII Soros Olympiad R1 8.6)
Three cyclists started simultaneously on three parallel straight paths (at the time of the start, the athletes were on the same straight line). Cyclists travel at constant speeds.
1
1
1
second after the start, the triangle formed by the cyclists had an area of
5
5
5
m
2
^2
2
. What area will such a triangle have in
10
10
10
seconds after the start?
10.5
1
Hide problems
locus of incenters of A_1B_1C_1 when <B_1A_1C_1 + 2 <BAC = 180^o ...
An acute-angled triangle
A
B
C
ABC
A
BC
is given. Points
A
1
,
B
1
A_1, B_1
A
1
,
B
1
and
C
1
C_1
C
1
are taken on its sides
B
C
,
C
A
BC, CA
BC
,
C
A
and
A
B
AB
A
B
, respectively, such that
∠
B
1
A
1
C
1
+
2
∠
B
A
C
=
18
0
o
\angle B_1A_1C_1 + 2 \angle BAC = 180^o
∠
B
1
A
1
C
1
+
2∠
B
A
C
=
18
0
o
,
∠
A
1
C
1
B
1
+
2
∠
A
C
B
=
18
0
o
\angle A_1C_1B_1 + 2 \angle ACB = 180^o
∠
A
1
C
1
B
1
+
2∠
A
CB
=
18
0
o
,
∠
C
1
B
1
A
1
+
2
∠
C
B
A
=
18
0
o
\angle C_1B_1A_1 + 2 \angle CBA = 180^o
∠
C
1
B
1
A
1
+
2∠
CB
A
=
18
0
o
. Find the locus of the centers of the circles inscribed in triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
(all kinds of such triangles are considered).
10.8
1
Hide problems
sin of smallest angle less tahn 2 \sin 18^o, constructing triangle from triangle
There is a set of triangles, in each of which the smallest angle does not exceed
3
6
o
36^o
3
6
o
. A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than
2
sin
1
8
o
2 \sin 18^o
2
sin
1
8
o
.
10.6
1
Hide problems
computational with circumcircle and incircle
A circle is inscribed in triangle
A
B
C
ABC
A
BC
.
M
M
M
and
N
N
N
are the points of its tangency with the sides
B
C
BC
BC
and
C
A
CA
C
A
, respectively. The segment
A
M
AM
A
M
intersects
B
N
BN
BN
at point
P
P
P
and the inscribed circle at point
Q
Q
Q
. It is known that
M
P
=
a
MP = a
MP
=
a
,
P
Q
=
b
PQ = b
PQ
=
b
. Find
A
Q
AQ
A
Q
.
10.4
1
Hide problems
inradius of triangle with vertices feet of altitudes, circumcenter, centroid
An acute-angled triangle is inscribed in a circle of radius
R
R
R
. The distance between the center of the circle and the point of intersection of the medians of the triangle is
d
d
d
. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.
11.8
1
Hide problems
min and max radius of smallest sphere of 3, tangent to plane, line, each other
Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is
1
1
1
. Within what limits can the radius of the smallest sphere vary?
11.6
1
Hide problems
sum of radii of 6 inner circles = diameter of large circle, tangent to all 6
A circle is tangent internally by
6
6
6
circles so that each one is tangent internally to two adjacent ones and the radii of opposite circles are pairwise equal. Prove that the sum of the radii of the inner circles is equal to the diameter of the given circle.
8.8
1
Hide problems
conguent quadr. by moving 1 point (1998-99 Savin Competition 6-8 p1)
Is there a quadrilateral, any vertex of which can be moved to another location so that the new quadrilateral is congruent to the original one?