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Problems
Contests
International Contests
Tuymaada Olympiad
2019 Tuymaada Olympiad
2019 Tuymaada Olympiad
Part of
Tuymaada Olympiad
Subcontests
(8)
5
1
Hide problems
is it possible to draw in the plane the graph presented in the figure ?
Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)
4
2
Hide problems
calculator operations limited, if S > x^n + 1 then S > x^n + x - 1
A calculator can square a number or add
1
1
1
to it. It cannot add
1
1
1
two times in a row. By several operations it transformed a number
x
x
x
into a number
S
>
x
n
+
1
S > x^n + 1
S
>
x
n
+
1
(
x
,
n
,
S
x, n,S
x
,
n
,
S
are positive integers). Prove that
S
>
x
n
+
x
−
1
S > x^n + x - 1
S
>
x
n
+
x
−
1
.
diplomas at the All-Russian Olympiad and percentages problem
A quota of diplomas at the All-Russian Olympiad should be strictly less than
45
%
45\%
45%
. More than
20
20
20
students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than
45
%
45\%
45%
. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to
45
%
45\%
45%
. Prove that the number of participants should be at least doubled.
3
2
Hide problems
custodians in rooms of gallery, where the plan is like a chessboard (juniors)
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of
n
n
n
rooms (
n
>
1
n > 1
n
>
1
)?
custodians in rooms of gallery, where the plan is like a chessboard
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of
n
n
n
rooms (
n
>
2
n > 2
n
>
2
)?
2
2
Hide problems
prove that point lies on a common external tangent of 2 circles
A triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB < AC
A
B
<
A
C
is inscribed in a circle
ω
\omega
ω
. Circles
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
touch the lines
A
B
AB
A
B
and
A
C
AC
A
C
, and their centres lie on the circumference of
ω
\omega
ω
. Prove that
C
C
C
lies on a common external tangent to
γ
1
\gamma_1
γ
1
and
γ
2
\gamma_2
γ
2
.
midpoint of segment circumcentres equidistant from 2 points
A trapezoid
A
B
C
D
ABCD
A
BC
D
with
B
C
/
/
A
D
BC // AD
BC
//
A
D
is given. The points
B
′
B'
B
′
and
C
′
C'
C
′
are symmetrical to
B
B
B
and
C
C
C
with respect to
C
D
CD
C
D
and
A
B
AB
A
B
, respectively. Prove that the midpoint of the segment joining the circumcentres of
A
B
C
′
ABC'
A
B
C
′
and
B
′
C
D
B'CD
B
′
C
D
is equidistant from
A
A
A
and
D
D
D
.
1
1
Hide problems
prove that |a_i| < 1 for some i, sequence where a_i +a_j has min abs. value
In a sequence
a
1
,
a
2
,
.
.
a_1, a_2, ..
a
1
,
a
2
,
..
of real numbers the product
a
1
a
2
a_1a_2
a
1
a
2
is negative, and to define
a
n
a_n
a
n
for
n
>
2
n > 2
n
>
2
one pair
(
i
,
j
)
(i, j)
(
i
,
j
)
is chosen among all the pairs
(
i
,
j
)
,
1
≤
i
<
j
<
n
(i, j), 1 \le i < j < n
(
i
,
j
)
,
1
≤
i
<
j
<
n
, not chosen before, so that
a
i
+
a
j
a_i +a_j
a
i
+
a
j
has minimum absolute value, and then
a
n
a_n
a
n
is set equal to
a
i
+
a
j
a_i + a_j
a
i
+
a
j
. Prove that
∣
a
i
∣
<
1
|a_i| < 1
∣
a
i
∣
<
1
for some
i
i
i
.
8
2
Hide problems
game with 2x1, 1x 2, 1x3, 3x1 rectangles in a 1000x1000 board
Andy, Bess, Charley and Dick play on a
1000
×
1000
1000 \times 1000
1000
×
1000
board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming
2
×
1
,
1
×
2
,
1
×
3
2 \times 1, 1 \times 2, 1 \times 3
2
×
1
,
1
×
2
,
1
×
3
, or
3
×
1
3 \times 1
3
×
1
rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.
Reflection of arc midpoint w.r.t side of triangle lies on some line
In
△
A
B
C
\triangle ABC
△
A
BC
∠
B
\angle B
∠
B
is obtuse and
A
B
≠
B
C
AB \ne BC
A
B
=
BC
. Let
O
O
O
is the circumcenter and
ω
\omega
ω
is the circumcircle of this triangle.
N
N
N
is the midpoint of arc
A
B
C
ABC
A
BC
. The circumcircle of
△
B
O
N
\triangle BON
△
BON
intersects
A
C
AC
A
C
on points
X
X
X
and
Y
Y
Y
. Let
B
X
∩
ω
=
P
≠
B
BX \cap \omega = P \ne B
BX
∩
ω
=
P
=
B
and
B
Y
∩
ω
=
Q
≠
B
BY \cap \omega = Q \ne B
B
Y
∩
ω
=
Q
=
B
. Prove that
P
,
Q
P, Q
P
,
Q
and reflection of
N
N
N
with respect to line
A
C
AC
A
C
are collinear.
7
2
Hide problems
computational for juniors in Tuymaada , tangent circle given
A circle
ω
\omega
ω
touches the sides
A
A
A
B and
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
and intersects its side
A
C
AC
A
C
at
K
K
K
. It is known that the tangent to
ω
\omega
ω
at
K
K
K
is symmetrical to the line
A
C
AC
A
C
with respect to the line
B
K
BK
B
K
. What can be the difference
A
K
−
C
K
AK -CK
A
K
−
C
K
if
A
B
=
9
AB = 9
A
B
=
9
and
B
C
=
11
BC = 11
BC
=
11
?
An inequality on grid cell
N
N
N
cells chosen on a rectangular grid. Let
a
i
a_i
a
i
is number of chosen cells in
i
i
i
-th row,
b
j
b_j
b
j
is number of chosen cells in
j
j
j
-th column. Prove that
∏
i
a
i
!
⋅
∏
j
b
j
!
≤
N
!
\prod_{i} a_i! \cdot \prod_{j} b_j! \leq N!
i
∏
a
i
!
⋅
j
∏
b
j
!
≤
N
!
6
2
Hide problems
Fun easy number theory
Let
S
\mathbb{S}
S
is the set of prime numbers that less or equal to 26. Is there any
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
N
a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}
a
1
,
a
2
,
a
3
,
a
4
,
a
5
,
a
6
∈
N
such that
g
c
d
(
a
i
,
a
j
)
∈
S
for
1
≤
i
≠
j
≤
6
gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for } 1\leq i \ne j \leq 6
g
c
d
(
a
i
,
a
j
)
∈
S
for
1
≤
i
=
j
≤
6
and for every element
p
p
p
of
S
\mathbb{S}
S
there exists a pair of
1
≤
k
≠
l
≤
6
1\leq k \ne l \leq 6
1
≤
k
=
l
≤
6
such that
s
=
g
c
d
(
a
k
,
a
l
)
?
s=gcd(a_k,a_l)?
s
=
g
c
d
(
a
k
,
a
l
)?
Prove that expression is not square of a natural number
Prove that the expression
(
1
4
+
1
2
+
1
)
(
2
4
+
2
2
+
1
)
…
(
n
4
+
n
2
+
1
)
(1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1)
(
1
4
+
1
2
+
1
)
(
2
4
+
2
2
+
1
)
…
(
n
4
+
n
2
+
1
)
is not square for all
n
∈
N
n \in \mathbb{N}
n
∈
N