MathDB
Problems
Contests
International Contests
Tuymaada Olympiad
2018 Tuymaada Olympiad
2018 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
6
1
Hide problems
Numbers on blackboard replaced by difference
The numbers
1
,
2
,
3
,
…
,
1024
1, 2, 3, \dots, 1024
1
,
2
,
3
,
…
,
1024
are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead.
512
512
512
numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.Proposed by A. Golovanov
8
1
Hide problems
Circumcircles of two triangles are tangent
Quadrilateral
A
B
C
D
ABCD
A
BC
D
with perpendicular diagonals is inscribed in a circle with centre
O
O
O
. The tangents to this circle at
A
A
A
and
C
C
C
together with line
B
D
BD
B
D
form the triangle
Δ
\Delta
Δ
. Prove that the circumcircles of
B
O
D
BOD
BO
D
and
Δ
\Delta
Δ
are tangent.[hide=Additional information for Junior League]Show that this point lies belongs to
ω
\omega
ω
, the circumcircle of
O
A
C
OAC
O
A
C
Proposed by A. Kuznetsov
7
2
Hide problems
Inequality with x, y, z >= 1
Prove the inequality
(
x
3
+
2
y
2
+
3
z
)
(
4
y
3
+
5
z
2
+
6
x
)
(
7
z
3
+
8
x
2
+
9
y
)
≥
720
(
x
y
+
y
z
+
x
z
)
(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)
(
x
3
+
2
y
2
+
3
z
)
(
4
y
3
+
5
z
2
+
6
x
)
(
7
z
3
+
8
x
2
+
9
y
)
≥
720
(
x
y
+
yz
+
x
z
)
for
x
,
y
,
z
≥
1
x, y, z \geq 1
x
,
y
,
z
≥
1
.Proposed by K. Kokhas
Sending non-intersecting teams to an olympiad
A school has three senior classes of
M
M
M
students each. Every student knows at least
3
4
M
\frac{3}{4}M
4
3
M
people in each of the other two classes. Prove that the school can send
M
M
M
non-intersecting teams to the olympiad so that each team consists of
3
3
3
students from different classes who know each other.Proposed by C. Magyar, R. Martin
5
2
Hide problems
Product divisble by p^3
A prime
p
p
p
and a positive integer
n
n
n
are given. The product
(
1
3
+
1
)
(
2
3
+
1
)
.
.
.
(
(
n
−
1
)
3
+
1
)
(
n
3
+
1
)
(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)
(
1
3
+
1
)
(
2
3
+
1
)
...
((
n
−
1
)
3
+
1
)
(
n
3
+
1
)
is divisible by
p
3
p^3
p
3
. Prove that
p
≤
n
+
1
p \leq n+1
p
≤
n
+
1
.Proposed by Z. Luria
Spectrometer test tells if a ball is zinc or copper
99
99
99
identical balls lie on a table.
50
50
50
balls are made of copper, and
49
49
49
balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to
2
2
2
balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?Proposed by N. Vlasova, S. Berlov
4
1
Hide problems
Greatest Common Divisor of Sequence 2^2^n+d
Prove that for every positive integer
d
>
1
d > 1
d
>
1
and
m
m
m
the sequence
a
n
=
2
2
n
+
d
a_n=2^{2^n}+d
a
n
=
2
2
n
+
d
contains two terms
a
k
a_k
a
k
and
a
l
a_l
a
l
(
k
≠
l
k \neq l
k
=
l
) such that their greatest common divisor is greater than
m
m
m
.Proposed by T. Hakobyan
2
1
Hide problems
Equal segments, circle tangent to sides of triangle
A circle touches the side
A
B
AB
A
B
of the triangle
A
B
C
ABC
A
BC
at
A
A
A
, touches the side
B
C
BC
BC
at
P
P
P
and intersects the side
A
C
AC
A
C
at
Q
Q
Q
. The line symmetrical to
P
Q
PQ
PQ
with respect to
A
C
AC
A
C
meets the line
A
P
AP
A
P
at
X
X
X
. Prove that
P
C
=
C
X
PC=CX
PC
=
CX
.Proposed by S. Berlov
3
2
Hide problems
Rooks and pawns
n
n
n
rooks and
k
k
k
pawns are arranged on a
100
×
100
100 \times 100
100
×
100
board. The rooks cannot leap over pawns. For which minimum
k
k
k
is it possible that no rook can capture any other rook?Junior League:
n
=
2551
n=2551
n
=
2551
(Proposed by A. Kuznetsov) Senior League:
n
=
2550
n=2550
n
=
2550
(Proposed by N. Vlasova)
Line bisects segment
A point
P
P
P
on the side
A
B
AB
A
B
of a triangle
A
B
C
ABC
A
BC
and points
S
S
S
and
T
T
T
on the sides
A
C
AC
A
C
and
B
C
BC
BC
are such that
A
P
=
A
S
AP=AS
A
P
=
A
S
and
B
P
=
B
T
BP=BT
BP
=
BT
. The circumcircle of
P
S
T
PST
PST
meets the sides
A
B
AB
A
B
and
B
C
BC
BC
again at
Q
Q
Q
and
R
R
R
, respectively. The lines
P
S
PS
PS
and
Q
R
QR
QR
meet at
L
L
L
. Prove that the line
C
L
CL
C
L
bisects the segment
P
Q
PQ
PQ
.Proposed by A. Antropov
1
2
Hide problems
Polynomial divisible by x^2+1
Real numbers
a
≠
0
,
b
,
c
a \neq 0, b, c
a
=
0
,
b
,
c
are given. Prove that there is a polynomial
P
(
x
)
P(x)
P
(
x
)
with real coefficients such that the polynomial
x
2
+
1
x^2+1
x
2
+
1
divides the polynomial
a
P
(
x
)
2
+
b
P
(
x
)
+
c
aP(x)^2+bP(x)+c
a
P
(
x
)
2
+
b
P
(
x
)
+
c
.Proposed by A. Golovanov
Roots of quadratic trinomials
Do there exist three different quadratic trinomials
f
(
x
)
,
g
(
x
)
,
h
(
x
)
f(x), g(x), h(x)
f
(
x
)
,
g
(
x
)
,
h
(
x
)
such that the roots of the equation
f
(
x
)
=
g
(
x
)
f(x)=g(x)
f
(
x
)
=
g
(
x
)
are
1
1
1
and
4
4
4
, the roots of the equation
g
(
x
)
=
h
(
x
)
g(x)=h(x)
g
(
x
)
=
h
(
x
)
are
2
2
2
and
5
5
5
, and the roots of the equation
h
(
x
)
=
f
(
x
)
h(x)=f(x)
h
(
x
)
=
f
(
x
)
are
3
3
3
and
6
6
6
?Proposed by A. Golovanov