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National and Regional Contests
Russia Contests
Soros Olympiad in Mathematics
IV Soros Olympiad 1997 - 98 (Russia)
IV Soros Olympiad 1997 - 98 (Russia)
Part of
Soros Olympiad in Mathematics
Subcontests
(39)
11.11
1
Hide problems
2 lines intersect inside a regular triangular pyramid
An arbitrary point
M
M
M
is taken on the basis of a regular triangular pyramid. Let
K
K
K
,
L
L
L
,
N
N
N
be the projections of
M
M
M
onto the lateral faces of this pyramid, and
P
P
P
be the intersection point of the medians of the triangle
K
L
N
KLN
K
L
N
. Prove that the straight line passing through the points
M
M
M
and
P
P
P
intersects the height of the pyramid (or its extension). Let us denote this intersection point by
E
E
E
. Find
M
P
:
P
E
MP: PE
MP
:
PE
if the dihedral angles at the base of the pyramid are equal to
a
a
a
.
11.12
1
Hide problems
4x4 non liear parameter system (IV Soros Olympiad 1997-98 R3 11.12)
Find how many different solutions depending on
a
a
a
has the system of equations :
{
x
+
z
=
2
a
y
+
u
+
x
z
=
a
−
3
y
z
+
x
u
=
2
a
y
u
=
1
\begin{cases} x+z=2a \\ y+u+xz=a-3 \\ yz+xu=2a \\ yu=1 \end{cases}
⎩
⎨
⎧
x
+
z
=
2
a
y
+
u
+
x
z
=
a
−
3
yz
+
xu
=
2
a
y
u
=
1
10.12
1
Hide problems
min F(M) = ctg \gamma + ctg \phi, fixed p/q
Two straight lines are given on a plane, intersecting at point
O
O
O
at an angle
a
a
a
. Let
A
A
A
,
B
B
B
and
C
C
C
be three points on one of the lines, located on one side of
O
O
O
and following in the indicated order,
M
M
M
be an arbitrary point on another line, different from
O
O
O
, Let
∠
A
M
B
=
γ
\angle AMB=\gamma
∠
A
MB
=
γ
,
∠
B
M
C
=
ϕ
\angle BMC = \phi
∠
BMC
=
ϕ
. Consider the function
F
(
M
)
=
c
t
g
γ
+
c
t
g
ϕ
F(M) = ctg \gamma + ctg \phi
F
(
M
)
=
c
t
g
γ
+
c
t
g
ϕ
. Prove that
F
(
M
)
F(M)
F
(
M
)
takes the smallest value on each of the rays into which
O
O
O
divides the second straight line. (Each has its own.) Let us denote one of these smallest values by
q
q
q
, and the other by
p
p
p
. Prove that the exprseeion
p
q
\frac{p}{q}
q
p
is independent of choice of points
A
A
A
,
B
B
B
and
C
C
C
. Express this relationship in terms of
a
a
a
.
10.11
1
Hide problems
cross sectional area of a cube drawn by a plane
A plane intersecting a unit cube divides it into two polyhedra. It is known that for each polyhedron the distance between any two points of it does not exceeds
3
2
\frac32
2
3
m. What can be the cross-sectional area of a cube drawn by a plane?
9.12
1
Hide problems
equal semicircles in a circle with 4 times larger diameter
One day, Professor Umzar Azum decided to fry dumplings for dinner. He took out a frying pan, opened a pack of dumplings, but suddenly thought about the question: how many dumplings could he fit in his frying pan? Measuring the sizes of the frying pan and dumplings, the professor came to the conclusion that the dumplings have the shape of a semicircle, the diameter of which is four times smaller than the diameter of the frying pan. Show how on the frying pan it is possible to place (without overlap): a)
20
20
20
pieces of dumplings; b)
24
24
24
pieces of dumplings; . (The problem boils down to placing, without overlapping, the appropriate number of identical semicircles inside a circle with a diameter four times larger.)Note: We (the authors of the problem) do not know the answer to the question whether it is possible to place 25 semicircles in a circle with a diameter four times smaller, and even more so we do not know what the largest number of such semicircles is. We will welcome any progress in solving the problem and evaluate it accordingly.
9.11
1
Hide problems
fixed point for a line, statring with intersecting circles
Given two circles intersecting at points
A
A
A
and
B
B
B
. A certain circle touches the first at point
A
A
A
, intersects the second at point
M
M
M
and intersects the straight line
A
B
AB
A
B
at point
P
P
P
(
M
M
M
and
P
P
P
are different from
B
B
B
). Prove that the straight line
M
P
MP
MP
passes through a fixed point of the plane (for any change in the third circle).
11.10
2
Hide problems
sum sin (4n-3)a_n/ (1-cos (4n-3)a_n) =kn
Let
a
n
=
π
2
n
a_n = \frac{\pi}{2n}
a
n
=
2
n
π
, where
n
n
n
is a natural number. Prove that for any
k
=
1
k = 1
k
=
1
,
2
2
2
,
.
.
.
...
...
,
n
n
n
holds the equality
sin
k
a
n
1
−
cos
a
n
+
sin
5
k
a
n
1
−
cos
5
a
n
+
sin
9
k
a
n
1
−
cos
9
a
n
+
.
.
.
+
sin
(
4
n
−
3
)
a
n
1
−
cos
(
4
n
−
3
)
a
n
=
k
n
\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn
1
−
cos
a
n
sin
k
a
n
+
1
−
cos
5
a
n
sin
5
k
a
n
+
1
−
cos
9
a
n
sin
9
k
a
n
+
...
+
1
−
cos
(
4
n
−
3
)
a
n
sin
(
4
n
−
3
)
a
n
=
kn
perimeter of ABC = k (BC)
The perimeter of triangle
A
B
C
ABC
A
BC
is
k
k
k
times larger than side
B
C
BC
BC
,
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
. In what ratio does the median to side
B
C
BC
BC
divide the diameter of the circle inscribed in this triangle, perpendicular to this side?
11.8
2
Hide problems
approx. \sqrt{5,44...4} (IV Soros Olympiad 1997-98 Correspondence 11.8)
Calculate
5
,
44...4
\sqrt{5,44...4}
5
,
44...4
(the decimal point is followed by
100
100
100
fours) with approximation to: a)
1
0
−
100
10^{-100}
1
0
−
100
, b)
1
0
−
200
10^{-200}
1
0
−
200
cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0
Sum of all roots of the equation
c
o
s
100
x
+
a
1
c
o
s
99
x
+
a
2
c
o
s
98
x
+
.
.
.
+
a
9
9
c
o
s
x
+
a
100
=
0
cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0
co
s
100
x
+
a
1
co
s
99
x
+
a
2
co
s
98
x
+
...
+
a
9
9
cos
x
+
a
100
=
0
, in interval
[
π
,
3
π
2
]
\left[\pi, \frac{3\pi}{2} \right]
[
π
,
2
3
π
]
, is equal to
21
π
21\pi
21
π
, and the sum of all roots of the equation
s
i
n
100
x
+
a
1
s
i
n
99
x
+
a
2
s
i
n
98
x
+
.
.
.
+
a
9
9
s
i
n
x
+
a
100
=
0
sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0
s
i
n
100
x
+
a
1
s
i
n
99
x
+
a
2
s
i
n
98
x
+
...
+
a
9
9
s
in
x
+
a
100
=
0
, in the same interval, is equal to
24
π
24\pi
24
π
. How many roots does the first equation have on the segment
[
π
2
,
π
]
\left[ \frac{\pi}{2}, \pi\right]
[
2
π
,
π
]
?
11.3
3
Hide problems
\sqrt{(x-2)^2(x-x^2)}<.. (IV Soros Olympiad 1997-98 Correspondence 11.3)
Solve the inequality
(
x
−
2
)
2
(
x
−
x
2
)
<
4
x
−
1
−
(
x
2
−
3
x
)
2
\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}
(
x
−
2
)
2
(
x
−
x
2
)
<
4
x
−
1
−
(
x
2
−
3
x
)
2
sinx cos^2 y +sin y cos^2 x =0 (IV Soros Olympiad 1997-98 R2 11.3)
Draw on the coordinate plane the set of points whose coordinates satisfy the equation
sin
x
cos
2
y
+
sin
y
cos
2
x
=
0
\sin x \cos^2 y +\sin y \cos^2 x =0
sin
x
cos
2
y
+
sin
y
cos
2
x
=
0
| ... |||x^2-x| -1|-1 |...-1|=x^2-2x-14 (IV Soros Olympiad 1997-98 R3 11.3)
Solve the equation
∣
.
.
.
∣
∣
∣
x
2
−
x
∣
−
1
∣
−
1
∣
.
.
.
−
1
∣
=
x
2
−
2
x
−
14.
\left| ... \left|\left||x^2-x| -1\right|-1 \right|...-1\right|=x^2-2x-14.
...
∣
x
2
−
x
∣
−
1
−
1
...
−
1
=
x
2
−
2
x
−
14.
(There are
11
11
11
units on the left side.)
10.2
3
Hide problems
sum k!/((x+k)..(x+1))=1 (IV Soros Olympiad 1997-98 Correspondence 10.2)
Solve the equation
10
x
+
10
+
10
⋅
9
(
x
+
10
)
(
x
+
9
)
+
10
⋅
9
⋅
8
(
x
+
10
)
(
x
+
9
)
(
x
+
8
)
+
.
.
.
+
10
⋅
9
⋅
.
.
.
⋅
2
⋅
1
(
x
+
10
)
(
x
+
9
)
⋅
.
.
.
⋅
(
x
+
1
)
=
11
\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11
x
+
10
10
+
(
x
+
10
)
(
x
+
9
)
10
⋅
9
+
(
x
+
10
)
(
x
+
9
)
(
x
+
8
)
10
⋅
9
⋅
8
+
...
+
(
x
+
10
)
(
x
+
9
)
⋅
...
⋅
(
x
+
1
)
10
⋅
9
⋅
...
⋅
2
⋅
1
=
11
line passes through arc midpoint
Let
M
M
M
be the point of intersection of the diagonals of the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
. Prove that if
A
B
=
A
M
,
AB = AM,
A
B
=
A
M
,
then a line passing through
M
M
M
perpendicular to
A
D
AD
A
D
passes through the midpoint of the arc
B
C
BC
BC
.
\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1} (IV Soros Olympiad 1997-98 R2 10.2)
Solve the equation
x
3
+
6
x
2
−
6
x
−
1
3
=
x
2
+
4
x
+
1
\sqrt[3]{x^3+6x^2-6x-1}=\sqrt{x^2+4x+1}
3
x
3
+
6
x
2
−
6
x
−
1
=
x
2
+
4
x
+
1
10.7
2
Hide problems
(\sqrt2+\sqrt3+\sqrt5)^1997 (IV Soros Olympiad 1997-98 Correspondence 10.7)
Prove that the number
(
2
+
3
+
5
)
1997
\left(\sqrt2+\sqrt3+\sqrt5\right)^{1997}
(
2
+
3
+
5
)
1997
can be represented as
A
2
+
B
3
+
C
5
+
D
30
A\sqrt2+B\sqrt3+C\sqrt5+D\sqrt{30}
A
2
+
B
3
+
C
5
+
D
30
where
A
A
A
,
B
B
B
,
C
C
C
,
D
D
D
are integers. Find with approximation to
1
0
−
10
10^{-10}
1
0
−
10
the ratio
D
A
\frac{D}{A}
A
D
6\sqrt2 sinx tgx - 2\sqrt2 tgx +3sinx -1=0 (IV Soros Olympiad 1997-98 R3 10.7)
How many different solutions on the interval
[
0
,
π
]
[0, \pi]
[
0
,
π
]
does the equation
6
2
sin
x
⋅
t
g
x
−
2
2
t
g
x
+
3
sin
x
−
1
=
0
6\sqrt2 \sin x \cdot tgx - 2\sqrt2 tgx +3\sin x -1=0
6
2
sin
x
⋅
t
gx
−
2
2
t
gx
+
3
sin
x
−
1
=
0
have?
10.6
3
Hide problems
numbers 0-2 in nxn board (IV Soros Olympiad 1997-98 Correspondence 10.6)
Is it possible to arrange
n
×
n
n \times n
n
×
n
in the cells of a square table the numbers
0
0
0
,
1
1
1
or
2
2
2
so that the sums of the numbers in rows and columns took on all different values from
1
1
1
to
2
n
2n
2
n
? Consider two cases: a)
n
n
n
is an odd number; b)
n
n
n
is an even number.
man gets lost in a large forest,
A man gets lost in a large forest, the boundary of which is a straight line. (We can assume that the forest fills the half-plane.) It is known that the distance from a person to Granina forest does not exceed
2
2
2
km. a) Suggest a path along which he will certainly be able to get out of the forest after walking no more than
14
14
14
km. (Of course, a person does not know in which direction the border of the forest is, BUT he has the opportunity to move along any pre-selected curve. It is believed that a person left the forest as soon as he reached its border, while the border of the forest is invisible to him, no matter how close he would have approached it.)b) Find a path with the same property and length no more than
13
13
13
km.
a fire at a speed of 1 km per hour
A fire that starts in the steppe spreads in all directions at a speed of
1
1
1
km per hour. A grader with a plow arrived on the fire line at the moment when the fire engulfed a circle with a radius of
1
1
1
km. The grader moves at a speed of
14
14
14
km per hour and cuts a strip with a plow that cuts off the fire. Indicate the path along which the grader should move so that the total area of the burnt steppe does not exceed: a)
4
π
4 \pi
4
π
km
2
^2
2
; b)
3
π
3 \pi
3
π
km
2
^2
2
. (We can assume that the grader’s path consists of straight segments and circular arcs.)
10.4
3
Hide problems
2x4 radical system (IV Soros Olympiad 1997-98 Correspondence 10.4)
Solve the system of equations
{
x
+
y
+
z
+
t
=
6
1
−
x
2
+
4
−
y
2
+
9
−
z
2
+
16
−
t
2
=
8
\begin{cases} x+y+z+t=6 \\ \sqrt{1-x^2}+\sqrt{4-y^2}+\sqrt{9-z^2}+\sqrt{16-t^2}=8 \end{cases}
{
x
+
y
+
z
+
t
=
6
1
−
x
2
+
4
−
y
2
+
9
−
z
2
+
16
−
t
2
=
8
sin^2 x+psin x+q=0 (IV Soros Olympiad 1997-98 R2 10.4)
Draw on the plane
(
p
,
q
)
(p, q)
(
p
,
q
)
all points with coordinates
(
p
,
q
)
(p,q)
(
p
,
q
)
, for which the equation
sin
2
x
+
p
sin
x
+
q
=
0
\sin^2x+p\sin x+q=0
sin
2
x
+
p
sin
x
+
q
=
0
has solutions and all its positive solutions form an arithmetic progression.
\sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x (IV Soros Olympiad 1997-98 R3 10.4)
Solve the equation
2
x
2
+
x
−
3
+
2
x
2
−
3
=
x
.
\sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.
2
x
2
+
x
−
3
+
2
x
2
−
3
=
x
.
10.3
3
Hide problems
distance in 2D (IV Soros Olympiad 1997-98 Correspondence 10.3)
For any two points
A
(
x
1
,
y
1
)
A (x_1 , y_1)
A
(
x
1
,
y
1
)
and
B
(
x
2
,
y
2
)
B (x_2, y_2)
B
(
x
2
,
y
2
)
, the distance
r
(
A
,
B
)
r (A, B)
r
(
A
,
B
)
between them is determined by the equality
r
(
A
,
B
)
=
m
a
x
{
∣
x
1
−
x
2
∣
,
∣
y
1
−
y
2
∣
}
r(A, B) = max\{| x_1- x_2 | , | y_1 - y_2 |\}
r
(
A
,
B
)
=
ma
x
{
∣
x
1
−
x
2
∣
,
∣
y
1
−
y
2
∣
}
. Prove that the triangle inequality
r
(
A
,
C
)
+
r
(
C
,
B
)
≥
r
(
A
,
B
)
r(A, C) + r(C, B) \ge r(A, B)
r
(
A
,
C
)
+
r
(
C
,
B
)
≥
r
(
A
,
B
)
. holds for the distance introduced in this way .Let
A
A
A
and
B
B
B
be two points of the plane . Find the locus of points
C
C
C
for which a)
r
(
A
,
C
)
+
r
(
C
,
B
)
=
r
(
A
,
B
)
r(A, C) + r(C, B) = r(A, B)
r
(
A
,
C
)
+
r
(
C
,
B
)
=
r
(
A
,
B
)
b)
r
(
A
,
C
)
=
r
(
C
,
B
)
.
r(A, C) = r(C, B).
r
(
A
,
C
)
=
r
(
C
,
B
)
.
distance of feet of 2 altitudes = 1/2 R
What can angle
B
B
B
of triangle
A
B
C
ABC
A
BC
be equal to if it is known that the distance between the feet of the altitudes drawn from vertices
A
A
A
and
C
C
C
is equal to half the radius of the circle circumscribed around this triangle?
3 3digit numbers in artihm. progession (IV Soros Olympiad 1997-98 R3 10.3)
Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?
9.4
3
Hide problems
min 16 x^3/y+y^3/x-\sqrt{xy} (IV Soros Olympiad 1997-98 Correspondence 9.4)
Find the smallest value of the expression
16
⋅
x
3
y
+
y
3
x
−
x
y
16 \cdot \frac{x^3}{y}+\frac{y^3}{x}-\sqrt{xy}
16
⋅
y
x
3
+
x
y
3
−
x
y
min max , nested absolute values (IV Soros Olympiad 1997-98 R3 9.4)
Find the smallest and largest values of the expression
∣
.
.
.
∣
∣
x
−
1
∣
−
1
∣
.
.
.
−
1
∣
+
1
∣
∣
x
−
2
∣
−
1
∣
+
1
\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}
∣
∣
x
−
2∣
−
1
∣
+
1
∣
...
∣
∣
x
−
1∣
−
1
∣
...
−
1
∣
+
1
(The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)
(x^2-x-1)^2-x^3=5 (IV Soros Olympiad 1997-98 R2 9.4)
Solve the equation
(
x
2
−
x
−
1
)
2
−
x
3
=
5
(x^2-x-1)^2-x^3=5
(
x
2
−
x
−
1
)
2
−
x
3
=
5
9.2
3
Hide problems
nested radicals (IV Soros Olympiad 1997-98 Correspondence 9.2)
Solve the equation
2
1
+
x
1
+
(
x
+
1
)
1
+
(
x
+
2
)
1
+
(
x
+
3
)
(
x
+
5
)
=
x
2\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)(x+5)}}}}=x
2
1
+
x
1
+
(
x
+
1
)
1
+
(
x
+
2
)
1
+
(
x
+
3
)
(
x
+
5
)
=
x
7-digit number by 3 terms of A.P. (IV Soros Olympiad 1997-98 R3 9.2)
The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?
x^2+5\sqrt2 x+a<0 (IV Soros Olympiad 1997-98 R2 9.2)
Find all values of the parameter
a
a
a
for which there exist exactly two integer values of
x
x
x
that satisfy the inequality
x
2
+
5
2
x
+
a
<
0.
x^2+5\sqrt2 x+a<0.
x
2
+
5
2
x
+
a
<
0.
11.9
2
Hide problems
cut pyramid ABCD into 8 equal and similar pyramids
Cut pyramid
A
B
C
D
ABCD
A
BC
D
into
8
8
8
equal and similar pyramids, if: a)
A
B
=
B
C
=
C
D
AB = BC = CD
A
B
=
BC
=
C
D
,
∠
A
B
C
=
∠
B
C
D
=
9
0
o
\angle ABC =\angle BCD = 90^o
∠
A
BC
=
∠
BC
D
=
9
0
o
, dihedral angle at edge
B
C
BC
BC
is right b) all plane angles at vertex
B
B
B
are right and
A
B
=
B
C
=
B
D
2
AB = BC = BD\sqrt2
A
B
=
BC
=
B
D
2
.Note. Whether there are other types of triangular pyramids that can be cut into any number similar to the original pyramids (their number is not necessarily
8
8
8
and the pyramids are not necessarily equal to each other) is currently unknown
0< a <=b <= c, a+b+ c = 7, abc = 9. (IV Soros Olympiad 1997-98 R3 11.9)
The numbers
a
a
a
,
b
b
b
and
c
c
c
satisfy the conditions
0
<
a
≤
b
≤
c
,
a
+
b
+
c
=
7
,
a
b
c
=
9.
0 < a \le b \le c\,\,\,,\,\,\, a+b+ c = 7\,\,\,, \,\,\,abc = 9.
0
<
a
≤
b
≤
c
,
a
+
b
+
c
=
7
,
ab
c
=
9.
Within what limits can each of the numbers
a
a
a
,
b
b
b
and
c
c
c
vary?
11.7
2
Hide problems
4 collinear points, 3 circles, computational
On straight line
ℓ
\ell
ℓ
there are points
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
, following in the indicated order:
A
B
=
a
AB = a
A
B
=
a
,
B
C
=
b
BC = b
BC
=
b
,
C
D
=
c
CD = c
C
D
=
c
. Segments
A
D
AD
A
D
and
B
C
BC
BC
serve as chords of two circles, and the sum of the angular values of the arcs of these circles located on one side of
ℓ
\ell
ℓ
is equal to
36
0
o
360^o
36
0
o
. A third circle passes through
A
A
A
and
B
B
B
, intersecting the first two at points
K
K
K
and
M
M
M
. The straight line
K
M
KM
K
M
intersects
ℓ
\ell
ℓ
at point
E
E
E
. Find
A
E
AE
A
E
.
log_{\frac12} x\ ge 16^x (IV Soros Olympiad 1997-98 R3 11.7)
Solve the inequality
log
1
2
x
≥
1
6
x
\log_{\frac12} x\ge 16^x
lo
g
2
1
x
≥
1
6
x
11.6
3
Hide problems
all triangles DPQ by moving point M are similar to each other
It is known that the bisector of the angle
∠
A
D
C
\angle ADC
∠
A
D
C
of the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
passes through the center of the circle inscribed in the triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be an arbitrary point of the arc
A
B
C
ABC
A
BC
of the circle circumscribed around
A
B
C
D
ABCD
A
BC
D
. Denote by
P
P
P
and
Q
Q
Q
the centers of the circles inscribed in the triangles
A
B
M
ABM
A
BM
and
B
C
M
BCM
BCM
. Prove that all triangles
D
P
Q
DPQ
D
PQ
obtained by moving point
M
M
M
are similar to each other. Find the angle
∠
P
D
Q
\angle PDQ
∠
P
D
Q
and ratio
B
P
:
P
Q
BP : PQ
BP
:
PQ
if
∠
B
A
C
=
α
\angle BAC = \alpha
∠
B
A
C
=
α
,
∠
B
C
A
=
β
\angle BCA = \beta
∠
BC
A
=
β
max no of acute, 6 points (IV Soros Olympiad 1997-98 Correspondence 11.6)
There are
6
6
6
points marked on the plane. Find the greatest possible number of acute triangles with vertices at the marked points.
shortest path on surface of rectangular parallelepiped
On the planet Brick, which has the shape of a rectangular parallelepiped with edges of
1
1
1
km,
2
2
2
km and
4
4
4
km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)
11.5
3
Hide problems
log_{2n-2} (n^2 + 2) is rational (IV Soros Olympiad 1997-98 R2 11.5)
Find all integers
n
n
n
for which
log
2
n
−
2
(
n
2
+
2
)
\log_{2n-2} (n^2 + 2)
lo
g
2
n
−
2
(
n
2
+
2
)
is a rational number.
sides of parallelogram are diagonals of 4 squares
The sides of the parallelogram serve as the diagonals of the four squares. The vertices of the squares lying in the part of the plane external to the parallelogram (the sides of the squares emerging from these vertices do not have common points with the parallelogram) serve as the vertices of a quadrilateral of area
a
a
a
, the four vertices opposite to them form a quadrilateral of area
b
b
b
. Find the area of the parallelogram.
KM _|_ AD, cyclic ABCD, BK=CK, <KBC+< AMB= 90^o
Let
M
M
M
be the point of intersection of the diagonals of the inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
, and let the angle
∠
A
M
B
\angle AMB
∠
A
MB
be an acute angle. On the side
B
C
BC
BC
, as a base, an isosceles triangle
B
C
K
BCK
BC
K
is constructed in the direction external to the quadrilateral such that
∠
K
B
C
+
∠
A
M
B
=
9
0
o
\angle KBC+\angle AMB= 90^o
∠
K
BC
+
∠
A
MB
=
9
0
o
. Prove that line
K
M
KM
K
M
is perpendicular to
A
D
AD
A
D
.
11.4
3
Hide problems
king in 8x8 chessboard (IV Soros Olympiad 1997-98 Correspondence 11.4)
In the lower left corner of the
8
×
8
8 \times 8
8
×
8
chessboard there is a king. He can move one cell either to the right, or up, or diagonally - to the right and up. How many ways can the king go to the upper right corner of the board if his route does not contain cells located on opposite sides of the diagonal going from the lower left to the upper right corner of the board?
max area of projection of cylinder on a plane
Find the largest value of the area of the projection of the cylinder onto the plane if its radius is
r
r
r
and its height is
h
h
h
(orthogonal projection).
set of 1998 different naturals (IV Soros Olympiad 1997-98 R3 11.4)
There is a set of
1998
1998
1998
different natural numbers. It is known that none of them can be represented as the sum of several other numbers in this set. What is the smallest value that the largest of these numbers can take?
11.2
3
Hide problems
x^2 -\pi x+a < 0, 1998 integer solutions (IV Soros Olympiad 1997-98 R2 11.2)
Find all values of the parameter
a
a
a
for which there are exactly
1998
1998
1998
integers
x
x
x
satisfying the inequality
x
2
−
π
x
+
a
<
0.
x^2 -\pi x +a < 0.
x
2
−
π
x
+
a
<
0.
(y^3-arcsinx)(x^3+arcsin y)>=0 (IV Soros Olympiad 1997-98 Correspondence 11.2)
Find the area of a figure consisting of points whose coordinates satisfy the inequality
(
y
3
−
a
r
c
s
i
n
x
)
(
x
3
+
a
r
c
s
i
n
y
)
≥
0.
(y^3 - arcsin x)(x^3 + arcsin y) \ge 0.
(
y
3
−
a
rcs
in
x
)
(
x
3
+
a
rcs
in
y
)
≥
0.
3-digit with max no of different divisors (IV Soros Olympiad 1997-98 R3 11.2)
Find the three-digit number that has the greatest number of different divisors.
11.1
3
Hide problems
diophantine xy =1997(x + y) (IV Soros Olympiad 1997-98 Correspondence 11.1)
Solve the equation
x
y
=
1997
(
x
+
y
)
xy =1997(x + y)
x
y
=
1997
(
x
+
y
)
in integers.
dig together the garden bed (IV Soros Olympiad 1997-98 R2 11.1)
Petya digs the garden bed alone for
a
a
a
minutes longer than he does with Vasya. Vasya digs up the same bed for
b
b
b
minutes longer than he would have done with Petya. How many minutes does it take Vasya and Petya to dig up the same bed together? orthogonal).
2 arc cos x >= arc cos y (IV Soros Olympiad 1997-98 R3 11.1)
On the coordinate plane, draw a set of points whose coordinates
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfy the inequality
2
a
r
c
cos
x
≥
a
r
c
cos
y
2 arc \cos x \ge arc \cos y
2
a
rc
cos
x
≥
a
rc
cos
y
10.10
2
Hide problems
equal circles
A circle touches the extensions of sides
C
A
CA
C
A
and
C
B
CB
CB
of triangle
A
B
C
ABC
A
BC
, and also touches side
A
B
AB
A
B
of this triangle at point
P
P
P
. Prove that the radius of the circle tangent to segments
A
P
AP
A
P
,
C
P
CP
CP
and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.
15 teams in football tournament (IV Soros Olympiad 1997-98 R3 10.10)
The football tournament, held in one round, involved
16
16
16
teams, each two of which scored a different number of points. (
3
3
3
points were given for a victory,
1
1
1
point for a draw,
0
0
0
points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?
10.9
2
Hide problems
max no of acute, 16 points (IV Soros Olympiad 1997-98 Correspondence 10.9)
There are
16
16
16
points marked on the circle. Find the greatest possible number of acute triangles with vertices at the marked points.
comp. geo with circumcircle and another circle
In triangle
A
B
C
ABC
A
BC
, side
B
C
BC
BC
is equal to
a
a
a
, and the angles of the triangle adjacent to it are equal to
α
\alpha
α
and
β
\beta
β
. A circle passing through points
A
A
A
and
B
B
B
intersects lines
C
A
CA
C
A
and
C
B
CB
CB
for second time at points
P
P
P
and
M
M
M
. It is known that straight line
R
M
RM
RM
passes through the center of the circle circumscribed around
A
B
C
ABC
A
BC
. Find the length of the segment
P
M
PM
PM
10.8
2
Hide problems
ratio, lines symmetric wrt line (IV Soros Olympiad 1997-98 Correspondence 10.8)
In triangle
A
B
C
ABC
A
BC
, angle
B
B
B
is different from a right angle,
A
B
:
B
C
=
k
AB : BC = k
A
B
:
BC
=
k
. Let
M
M
M
be the midpoint of
A
C
AC
A
C
. Lines symmetric to
B
M
BM
BM
wrt
A
B
AB
A
B
and
B
C
BC
BC
intersect line
A
C
AC
A
C
at points
D
D
D
and
E
E
E
. Find
B
D
:
B
E
BD : BE
B
D
:
BE
.
a root of x^3-x-1=0 (IV Soros Olympiad 1997-98 R3 10.8)
Let
a
a
a
be the root of the equation
x
3
−
x
−
1
=
0
x^3-x-1=0
x
3
−
x
−
1
=
0
. Find an equation of the third degree with integer coefficients whose root is
a
3
a^3
a
3
.
10.5
3
Hide problems
king in 7x7 chessboard (IV Soros Olympiad 1997-98 Correspondence 10.5)
In the lower left corner of the square
7
×
7
7 \times 7
7
×
7
board there is a king. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different ways can the king get to the upper right corner of the board if he is prohibited from visiting the central square?
triangle connstruction, 3 rays with common origin
Three rays with a common origin are drawn on the plane, dividing the plane into three angles. One point is marked inside each corner. Using one ruler, construct a triangle whose vertices lie on the given rays and whose sides contain the given points.
all plane angles at vertex A are equal, triangular pyramid
At the base of the triangular pyramid
A
B
C
D
ABCD
A
BC
D
lies a regular triangle
A
B
C
ABC
A
BC
such that
A
D
=
B
C
AD = BC
A
D
=
BC
. All plane angles at vertex
B
B
B
are equal to each other. What might these angles be equal to?
10.1
3
Hide problems
cyclic quad. comp. geo (IV Soros Olympiad 1997-98 Correspondence 10.1)
Two sides of the cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
are known:
A
B
=
a
AB = a
A
B
=
a
,
B
C
=
b
BC = b
BC
=
b
. A point
K
K
K
is taken on the side
C
D
CD
C
D
so that
C
K
=
m
CK = m
C
K
=
m
. A circle passing through
B
B
B
,
K
K
K
and
D
D
D
intersects line
D
A
DA
D
A
at a point
M
M
M
, different from
D
D
D
. Find
A
M
AM
A
M
.
minutes vs hours angle is 1^o (IV Soros Olympiad 1997-98 R2 10.1)
Indicate the moment in time when for the first time after midnight the angle between the minute and hour hands will be equal to
1
o
1^o
1
o
, despite the fact that the minute hand shows an integer number of minutes.
y=x+|y-3x-2x^2| (IV Soros Olympiad 1997-98 R3 10.1)
On the coordinate plane, draw a set of points whose coordinates
(
x
,
y
)
(x, y)
(
x
,
y
)
satisfy the equation
y
=
x
+
∣
y
−
3
x
−
2
x
2
∣
y=x+|y-3x-2x^2|
y
=
x
+
∣
y
−
3
x
−
2
x
2
∣
.
9.10
2
Hide problems
construct bisector using ruler (IV Soros Olympiad 1997-98 Correspondence 9.10)
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.
incenter lies on a chord of a circle passing through A and B
A circle is drawn through vertices
A
A
A
and
B
B
B
of triangle
A
B
C
ABC
A
BC
, intersecting sides
A
C
AC
A
C
and
B
C
BC
BC
at points
M
M
M
and
P
P
P
. It is known that the segment
M
P
MP
MP
contains the center of the circle inscribed in
A
B
C
ABC
A
BC
. Find
M
P
MP
MP
if
A
B
=
c
AB = c
A
B
=
c
,
B
C
=
a
BC = a
BC
=
a
,
C
A
=
b
CA=b
C
A
=
b
.
9.9
2
Hide problems
touchpoints of incircle (IV Soros Olympiad 1997-98 Correspondence 9.9)
In triangle
A
B
C
ABC
A
BC
, angle
A
A
A
is equal to
a
a
a
and the altitude drawn to side
B
C
BC
BC
is equal to
h
h
h
. The inscribed circle of the triangle touches the sides of the triangle at points
K
K
K
,
M
M
M
and
P
P
P
, where
P
P
P
lies on side
B
C
BC
BC
. Find the distance from
P
P
P
to
K
M
KM
K
M
.
sum of digits of divisors = 33 (IV Soros Olympiad 1997-98 R3 9.9)
Find an odd natural number not exceeding
1000
1000
1000
if you know that the sum of the last digits of all its divisors (including
1
1
1
and the number itself) is
33
33
33
.
9.8
2
Hide problems
king in 6x6 chessboard (IV Soros Olympiad 1997-98 Correspondence 9.8)
There is a king in the lower left corner of a chessboard of dimensions
6
6
6
and
6
6
6
. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?
P(P(P(x))) = 0 if P(x) = x^2+bx+c (IV Soros Olympiad 1997-98 R3 9.8)
The equation
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
, where
P
(
x
)
=
x
2
+
b
x
+
c
P(x) = x^2+bx+c
P
(
x
)
=
x
2
+
b
x
+
c
, has a single root, and the equation
P
(
P
(
P
(
x
)
)
)
=
0
P(P(P(x))) = 0
P
(
P
(
P
(
x
)))
=
0
has exactly three different roots. Solve the equation
P
(
P
(
P
(
x
)
)
)
=
0.
P(P(P(x))) = 0.
P
(
P
(
P
(
x
)))
=
0.
9.7
2
Hide problems
different distance in 2D (IV Soros Olympiad 1997-98 Correspondence 9.7)
For any two points
A
(
x
1
,
y
1
)
A (x_1 , y_1)
A
(
x
1
,
y
1
)
and
B
(
x
2
,
y
2
)
B (x_2, y_2)
B
(
x
2
,
y
2
)
, the distance
r
(
A
,
B
)
r (A, B)
r
(
A
,
B
)
between them is determined by the equality
r
(
A
,
B
)
=
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
r(A, B) = | x_1- x_2 | + | y_1 - y_2 |
r
(
A
,
B
)
=
∣
x
1
−
x
2
∣
+
∣
y
1
−
y
2
∣
. Prove that the triangle inequality
r
(
A
,
C
)
+
r
(
C
,
B
)
≥
r
(
A
,
B
)
r(A, C) + r(C, B) \ge r(A, B)
r
(
A
,
C
)
+
r
(
C
,
B
)
≥
r
(
A
,
B
)
. holds for the distance introduced in this way .Let
A
A
A
and
B
B
B
be two points of the plane (you can take
A
(
1
,
3
)
A(1, 3)
A
(
1
,
3
)
,
B
(
3
,
7
)
B(3, 7)
B
(
3
,
7
)
). Find the locus of points
C
C
C
for which a)
r
(
A
,
C
)
+
r
(
C
,
B
)
=
r
(
A
,
B
)
r(A, C) + r(C, B) = r(A, B)
r
(
A
,
C
)
+
r
(
C
,
B
)
=
r
(
A
,
B
)
b)
r
(
A
,
C
)
=
r
(
C
,
B
)
.
r(A, C) = r(C, B).
r
(
A
,
C
)
=
r
(
C
,
B
)
.
3 solutions with different % (IV Soros Olympiad 1997-98 R3 9.7)
There are three solutions with different percentages of alcohol. If you mix them in a ratio of
1
:
2
:
3
1:2:3
1
:
2
:
3
, you get a
20
%
20\%
20%
solution. If you mix them in a ratio of
5
:
4
:
3
,
5: 4: 3,
5
:
4
:
3
,
you will get a solution with
50
%
50\%
50%
alcohol content. What percentage of alcohol will the solution contain if equal amounts of the original solutions are mixed?
9.6
3
Hide problems
square inscribed in rhombus (IV Soros Olympiad 1997-98 Correspondence 9.6)
A rhombus is circumscribed around a square with side
1997
1997
1997
. Find its diagonals if it is known that they are equal to different integers.
cut acute triangle into 4 parts (IV Soros Olympiad 1997-98 R3 9.6)
Cut an acute triangle, one of whose sides is equal to the altitude drawn, by two straight cuts, into four parts, from which you can fold a square.
ratio on a chord of a cyclic quad
A chord is drawn through the intersection point of the diagonals of an inscribed quadrilateral. It is known that the parts of this chord located outside the quadrilateral have lengths equal to
1
3
\frac13
3
1
and
1
4
\frac14
4
1
of this chord. In what ratio is this chord divided by the intersection point of the diagonals of the quadrilateral?
9.5
3
Hide problems
numbers 0-2 in nxn board (IV Soros Olympiad 1997-98 Correspondence 9.5)
There is a square table with side
n
n
n
. Is it possible to enter the numbers
0
0
0
,
1
1
1
or
2
2
2
into the cells of this table so that all sums of numbers in rows and columns are different and take values from
1
1
1
to
2
n
2n
2
n
, if: a)
n
=
7
n = 7
n
=
7
? b)
n
=
8
n = 8
n
=
8
?
locus of center of rotations
Given triangle
A
B
C
ABC
A
BC
. Find the locus of points
M
M
M
such that there is a rotation with center at
M
M
M
that takes
C
C
C
to a certain point on side
A
B
AB
A
B
.
...<5/8< ... (IV Soros Olympiad 1997-98 R2 9.5)
All ordinary proper irreducible fractions whose numerators are two-digit numbers were ordered in ascending order. Between what two consecutive fractions is the number
5
8
\frac58
8
5
located?
9.3
3
Hide problems
machines in the workshop (IV Soros Olympiad 1997-98 Correspondence 9.3)
Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?
OM + ON >= R
Through point
O
O
O
- the center of a circle circumscribed around an acute triangle - a straight line is drawn, perpendicular to one of its sides and intersecting the other two sides of the triangle (or their extensions) at points
M
M
M
and
N
N
N
. Prove that
O
M
+
O
N
≥
R
OM+ON \ge R
OM
+
ON
≥
R
, where
R
R
R
is the radius of the circumscribed circle around the triangle.
angle, ratio 2:1, 1:1 by orhocenter (IV Soros Olympiad 1997-98 R2 9.3)
What is angle
B
B
B
of triangle
A
B
C
ABC
A
BC
, if it is known that the altitudes drawn from
A
A
A
and
C
C
C
intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of
2
:
1
2: 1
2
:
1
, counting from the vertex?
9.1
3
Hide problems
unit square and a circle (IV Soros Olympiad 1997-98 Correspondence 9.1)
Through vertices
A
A
A
and
B
B
B
of the unit square
A
B
C
D
ABCD
A
BC
D
, passes a circle intersecting lines
A
D
AD
A
D
and
A
C
AC
A
C
at points
K
K
K
and
M
M
M
, other than
A
A
A
. Find the length of the projection
K
M
KM
K
M
onto
A
C
AC
A
C
.
2(x-6)=x^2/ (1+\sqrt{x+1})^2 (IV Soros Olympiad 1997-98 R3 9.1)
Solve the equation
2
(
x
−
6
)
=
x
2
(
1
+
x
+
1
)
2
2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}
2
(
x
−
6
)
=
(
1
+
x
+
1
)
2
x
2
football tournament (IV Soros Olympiad 1997-98 R2 9.1)
The football tournament was played in one round.
3
3
3
points were given for a win,
1
1
1
point for a draw, and
0
0
0
points for a loss. Could it be that the first place team under the old scoring system (win -
2
2
2
points, draw -
1
1
1
point, loss -
0
0
0
) would be last?
grade8
2
Hide problems
grade 8 problems (IV Soros Olympiad 1997-98 Correspondence Round)
p1. What is the maximum amount of a
12
%
12\%
12%
acid solution that can be obtained from
1
1
1
liter of
5
%
5\%
5%
,
10
%
10\%
10%
and
15
%
15\%
15%
solutions? p2. Which number is greater:
199
,
719
,
971
,
99
7
2
199,719,971,997^2
199
,
719
,
971
,
99
7
2
or
199
,
719
,
971
,
996
∗
19
,
9719
,
971
,
998
199,719,971,996 * 19,9719,971,998
199
,
719
,
971
,
996
∗
19
,
9719
,
971
,
998
? p3. Is there a convex
1998
1998
1998
-gon whose angles are all integer degrees? p4. Is there a ten-digit number divisible by
11
11
11
that uses all the digits from
0
0
0
to
9
9
9
? p5. There are
20
20
20
numbers written in a circle, each of which is equal to the sum of its two neighbors. Prove that the sum of all numbers is
0
0
0
. p6. Is there a convex polygon that has neither an axis of symmetry nor a center of symmetry, but which transforms into itself when rotated around some point through some angle less than
180
180
180
degrees? p7. In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. p8. Give an example of a natural number that is divisible by
30
30
30
and has exactly
105
105
105
different natural factors, including
1
1
1
and the number itself. p9. In the writing of the antipodes, numbers are also written with the digits
0
,
.
.
.
,
9
0, ..., 9
0
,
...
,
9
, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes
5
∗
8
+
7
+
1
=
48
5 * 8 + 7 + 1 = 48
5
∗
8
+
7
+
1
=
48
2
∗
2
∗
6
=
24
2 * 2 * 6 = 24
2
∗
2
∗
6
=
24
5
∗
6
=
30
5* 6 = 30
5
∗
6
=
30
a) How will the equality
2
3
=
.
.
.
2^3 = ...
2
3
=
...
in the writing of the antipodes be continued? b) What does the number
9
9
9
mean among the Antipodes?Clarifications: a) It asks to convert
2
3
2^3
2
3
in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit
9
9
9
mean among the antipodes, i.e. with which digit is it equal in our number system? p10. Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts?PS.1. There was typo in problem
9
9
9
, it asks for
2
3
2^3
2
3
and not
23
23
23
. PS.2. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade 8 problems (IV Soros Olympiad 1997-98 Round 2)
p1. a) There are barrels weighing
1
,
2
,
3
,
4
,
.
.
.
,
19
,
20
1, 2, 3, 4, ..., 19, 20
1
,
2
,
3
,
4
,
...
,
19
,
20
pounds. Is it possible to distribute them equally (by weight) into three trucks?b) The same question for barrels weighing
1
,
2
,
3
,
4
,
.
.
.
,
9
,
10
1, 2, 3, 4, ..., 9, 10
1
,
2
,
3
,
4
,
...
,
9
,
10
pounds. p2. There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now? p3. What is the smallest number of integers from
1000
1000
1000
to
1500
1500
1500
that must be marked so that any number
x
x
x
from
1000
1000
1000
to
1500
1500
1500
differs from one of the marked numbers by no more than
10
%
10\%
10%
of the value of
x
x
x
? p4. Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”). p5. There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png p6. The natural number
a
a
a
is less than the natural number
b
b
b
. In this case, the sum of the digits of number
a
a
a
is
100
100
100
less than the sum of the digits of number
b
b
b
. Prove that between the numbers
a
a
a
and
b
b
b
there is a number whose sum of digits is
43
43
43
more than the sum of the digits of
a
a
a
. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade7
2
Hide problems
grade 7 problems (IV Soros Olympiad 1997-98 Correspondence Round)
p1. The oil pipeline passes by three villages
A
A
A
,
B
B
B
,
C
C
C
. In the first village,
30
%
30\%
30%
of the initial amount of oil is drained, in the second -
40
%
40\%
40%
of the amount that will reach village
B
B
B
, and in the third -
50
%
50\%
50%
of the amount that will reach village
C
C
C
What percentage of the initial amount of oil reaches the end of the pipeline? p2. There are several ordinary irreducible fractions (not necessarily proper) with natural numerators and denominators (and the denominators are greater than
1
1
1
). The product of all fractions is equal to
10
10
10
. All numerators and denominators are increased by
1
1
1
. Can the product of the resulting fractions be greater than
10
10
10
? p3. The garland consists of
10
10
10
light bulbs connected in series. Exactly one of the light bulbs has burned out, but it is not known which one. There is a suitable light bulb available to replace a burnt out one. To unscrew a light bulb, you need
10
10
10
seconds, to screw it in - also
10
10
10
seconds (the time for other actions can be neglected). Is it possible to be guaranteed to find a burnt out light bulb: a) in
10
10
10
minutes, b) in
5
5
5
minutes? p4. When fast and slow athletes run across the stadium in one direction, the fast one overtakes the slow one every
15
15
15
minutes, and when they run towards each other, they meet once every
5
5
5
minutes. How many times is the speed of a fast runner greater than the speed of a slow runner? p5. Petya was
35
35
35
minutes late for school. Then he decided to run to the kiosk for ice cream. But when he returned, the second lesson had already begun. He immediately ran for ice cream a second time and was gone for the same amount of time. When he returned, it turned out that he was late again, and he had to wait
50
50
50
minutes before the start of the fourth lesson. How long does it take to run from school to the ice cream stand and back if each lesson, including recess after it, lasts
55
55
55
minutes? p6. In a convex heptagon, draw as many diagonals as possible so that no three of them are sides of the same triangle, the vertices of which are at the vertices of the original heptagon. p7. In the writing of the antipodes, numbers are also written with the digits
0
,
.
.
.
,
9
0, ..., 9
0
,
...
,
9
, but each of the numbers has different meanings for them and for us. It turned out that the equalities are also true for the antipodes
5
∗
8
+
7
+
1
=
48
5 * 8 + 7 + 1 = 48
5
∗
8
+
7
+
1
=
48
2
∗
2
∗
6
=
24
2 * 2 * 6 = 24
2
∗
2
∗
6
=
24
5
∗
6
=
30
5* 6 = 30
5
∗
6
=
30
a) How will the equality
2
3
=
.
.
.
2^3 = ...
2
3
=
...
in the writing of the antipodes be continued? b) What does the number 9 mean among the Antipodes?Clarifications: a) It asks to convert
2
3
2^3
2
3
in antipodes language, and write with what number it is equal and find a valid equality in both numerical systems. b) What does the digit
9
9
9
mean among the antipodes, i.e. with which digit is it equal in our number system? p8. They wrote the numbers
1
,
2
,
3
,
4
,
.
.
.
,
1996
,
1997
1, 2, 3, 4, ..., 1996, 1997
1
,
2
,
3
,
4
,
...
,
1996
,
1997
in a row. Which digits were used more when writing these numbers - ones or twos? How long? p9. On the number axis there lives a grasshopper who can jump
1
1
1
and
4
4
4
to the right and left. Can he get from point
1
1
1
to point
2
2
2
of the numerical axis
i
n
1996
in 1996
in
1996
jumps if he must not get to points with coordinates divisible by
4
4
4
(points
0
0
0
,
±
4
\pm 4
±
4
,
±
8
\pm 8
±
8
, etc.)? p10. Is there a convex quadrilateral that can be cut along a straight line into two parts of the same size and shape, but neither the diagonal nor the straight line passing through the midpoints of opposite sides divides it into two equal parts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade 7 problems (IV Soros Olympiad 1997-98 Round 2)
p1. In the correct identity
(
x
2
−
1
)
(
x
+
.
.
.
)
=
(
x
+
3
)
(
x
−
1
)
(
x
+
.
.
.
)
(x^2 - 1)(x + ...) = (x + 3)(x- 1)(x +...)
(
x
2
−
1
)
(
x
+
...
)
=
(
x
+
3
)
(
x
−
1
)
(
x
+
...
)
two numbers were replaced with dots. What were these numbers? p2. A merchant is carrying money from point A to point B. There are robbers on the roads who rob travelers: on one road the robbers take
10
%
10\%
10%
of the amount currently available, on the other -
20
%
20\%
20%
, etc. . How should the merchant travel to bring as much of the money as possible to B? What part of the original amount will he bring to B? https://cdn.artofproblemsolving.com/attachments/f/5/ab62ce8fce3d482bc52b89463c953f4271b45e.png p3. Find the angle between the hour and minute hands at
7
7
7
hours
38
38
38
minutes. p4. The lottery game is played as follows. A random number from
1
1
1
to
1000
1000
1000
is selected. If it is divisible by
2
2
2
, they pay a ruble, if it is divisible by
10
10
10
- two rubles, by
12
12
12
- four rubles, by
20
20
20
- eight, if it is divisible by several of these numbers, then they pay the sum. How much can you win (at one time) in such a game? List all options. p5.The sum of the digits of a positive integer
x
x
x
is equal to
n
n
n
. Prove that between
x
x
x
and
10
x
10x
10
x
you can find an integer whose sum of digits is
n
+
5
n + 5
n
+
5
. p6.
9
9
9
people took part in the campaign, which lasted
12
12
12
days. There were
3
3
3
people on duty every day. At the same time, the duty officers quarreled with each other and no two of them wanted to be on duty together ever again. Nevertheless, the participants of the campaign claim that for all
12
12
12
days they were able to appoint three people on duty, taking into account this requirement. Could this be so? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade6
2
Hide problems
grade 6 problems (IV Soros Olympiad 1997-98 Correspondence Round)
p1. For
25
25
25
bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? p2. Cut the square into the figure into
4
4
4
parts of the same shape and size so that each part contains exactly one shaded square. https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png p3. The numerator and denominator of the fraction are positive numbers. The numerator is increased by
1
1
1
, and the denominator is increased by
10
10
10
. Can this increase the fraction? p4. The brother left the house
5
5
5
minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? p5. Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) p6. Give an example of a natural number divisible by
6
6
6
and having exactly
15
15
15
different natural divisors (counting
1
1
1
and the number itself). p7. In a round dance,
30
30
30
children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? p8. A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got
4
4
4
holes. The positions of three of them are marked in figure Where might the fourth hole be? https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png p9 The numbers 1
,
2
,
3
,
4
,
5
,
,
2000
, 2, 3, 4, 5, _, 2000
,
2
,
3
,
4
,
5
,
,
2000
are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining
1000
1000
1000
numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? p10. On the number axis there lives a grasshopper who can jump
1
1
1
and
4
4
4
to the right and left. Can he get from point
1
1
1
to point
2
2
2
of the numerical axis in
1996
1996
1996
jumps if he must not get to points with coordinates divisible by
4
4
4
(points
0
0
0
,
±
4
\pm 4
±
4
,
±
8
\pm 8
±
8
etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.
grade 6 problems (IV Soros Olympiad 1997-98 Round 2)
p1. The numerator of the fraction was increased by 20%. By what percentage should its denominator be reduced so that the resulting fraction doubles? p2. From point
O
O
O
on the plane there are four rays
O
A
OA
O
A
,
O
B
OB
OB
,
O
C
OC
OC
and
O
D
OD
O
D
(not necessarily in that order). It is known that
∠
A
O
B
=
4
0
o
\angle AOB =40^o
∠
A
OB
=
4
0
o
,
∠
B
O
C
=
7
0
o
\angle BOC = 70^o
∠
BOC
=
7
0
o
,
∠
C
O
D
=
8
0
o
\angle COD = 80^o
∠
CO
D
=
8
0
o
. What values can the angle between rays
O
A
OA
O
A
and
O
D
OD
O
D
take? (The angle between the rays is from
0
o
0^o
0
o
to
18
0
o
180^o
18
0
o
.) p3. Three equal circles have a common interior point. Prove that there is a circle of the same radius containing the centers of these three circles. p4. Two non-leap years are consecutive. The first one has more Mondays than Wednesdays. Which of the seven days of the week will occur most often in the second year? p5. The difference between two four-digit numbers is
7
7
7
. How much can the sums of their digits differ? p6. The numbers
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
1, 2, 3, 4, 5, 6, 7, 8, 9
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
are written on the board. In one move you can increase any of the numbers by
3
3
3
or
5
5
5
. What is the minimum number of moves you need to make for all the numbers to become equal? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.