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International Contests
Tournament Of Towns
2016 Tournament Of Towns
2016 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(7)
7
3
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Minimizing connections of batteries
a.) There are
2
n
+
1
2n+1
2
n
+
1
(
n
>
2
n>2
n
>
2
) batteries. We don't know which batteries are good and which are bad but we know that the number of good batteries is greater by
1
1
1
than the number of bad batteries. A lamp uses two batteries, and it works only if both of them are good. What is the least number of attempts sufficient to make the lamp work?b.) The same problem but the total number of batteries is
2
n
2n
2
n
(
n
>
2
n>2
n
>
2
) and the numbers of good and bad batteries are equal.Proposed by Alexander Shapovalov
Trains on a Planet
A spherical planet has the equator of length
1
1
1
. On this planet,
N
N
N
circular roads of length
1
1
1
each are to be built and used for several trains each. The trains must have the same constant positive speed and never stop or collide. What is the greatest possible sum of lengths of all the trains? The trains are arcs of zero width with endpoints removed (so that if only endpoints of two arcs have coincided then it is not a collision). Solve the problem for : (a)
N
=
3
N=3
N
=
3
(4 points) (b)
N
=
4
N=4
N
=
4
(6 points)Alexandr Berdnikov
Equal number of left and right jumps
Several frogs are sitting on the real line at distinct integer points. In each move, one of them can take a
1
1
1
-jump towards the right as long as they are still in on distinct points. We calculate the number of ways they can make
N
N
N
moves in this way for a positive integer
N
N
N
. Prove that if the jumps were all towards the left, we will still get the same number of ways.(F. Petrov)(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
6
4
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5
4
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4
4
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3
4
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2
3
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No roots but roots
Do there exist integers
a
a
a
and
b
b
b
such that : (a) the equation
x
2
+
a
x
+
b
=
0
x^2 + ax + b = 0
x
2
+
a
x
+
b
=
0
has no real roots, and the equation
⌊
x
2
⌋
+
a
x
+
b
=
0
\lfloor x^2 \rfloor + ax + b = 0
⌊
x
2
⌋
+
a
x
+
b
=
0
has at least one real root? (2 points) (b) the equation
x
2
+
2
a
x
+
b
x^2 + 2ax + b
x
2
+
2
a
x
+
b
= 0 has no real roots, and the equation
⌊
x
2
⌋
+
2
a
x
+
b
=
0
\lfloor x^2 \rfloor + 2ax + b = 0
⌊
x
2
⌋
+
2
a
x
+
b
=
0
has at least one real root? 3 points (By
⌊
k
⌋
\lfloor k \rfloor
⌊
k
⌋
we denote the integer part of
k
k
k
, that is, the greatest integer not exceeding
k
k
k
.) Alexandr Khrabrov
All lines touch same circle
On plane there is fixed ray
s
s
s
with vertex
A
A
A
and a point
P
P
P
not on the line which contains
s
s
s
. We choose a random point
K
K
K
which lies on ray. Let
N
N
N
be a point on a ray outside
A
K
AK
A
K
such that
N
K
=
1
NK=1
N
K
=
1
. Let
M
M
M
be a point such that
N
M
=
1
,
M
∈
P
K
NM=1,M \in PK
NM
=
1
,
M
∈
P
K
and
M
!
=
K
.
M!=K.
M
!
=
K
.
Prove that all lines
N
M
NM
NM
, provided by some point
K
K
K
, touch some fixed circle.
Tiling with dominoes
A natural number is written in each cell of an
8
×
8
8 \times 8
8
×
8
board. It turned out that for any tiling of the board with dominoes, the sum of numbers in the cells of each domino is different. Can it happen that the largest number on the board is no greater than
32
32
32
?(N. Chernyatyev)(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)
1
3
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Donald and cards
On a blackboard the product
l
o
g
(
)
[
]
×
⋯
×
l
o
g
(
)
[
]
log_{( )}[ ]\times\dots\times log_{( )}[ ]
l
o
g
(
)
[
]
×
⋯
×
l
o
g
(
)
[
]
is written (there are 50 logarithms in the product). Donald has
100
100
100
cards:
[
2
]
,
[
3
]
,
…
,
[
51
]
[2], [3],\dots, [51]
[
2
]
,
[
3
]
,
…
,
[
51
]
and
(
52
)
,
…
,
(
101
)
(52),\dots,(101)
(
52
)
,
…
,
(
101
)
. He is replacing each
(
)
()
(
)
with some card of form
(
x
)
(x)
(
x
)
and each
[
]
[]
[
]
with some card of form
[
y
]
[y]
[
y
]
. Find the difference between largest and smallest values Donald can achieve.
Writing Numbers on a Tape
All integers from
1
1
1
to one million are written on a tape in some arbitrary order. Then the tape is cut into pieces containing two consecutive digits each. Prove that these pieces contain all two-digit integers for sure, regardless of the initial order of integers.(4 points) Alexey Tolpygo
Children avoiding candies
100
100
100
children stand in a line each having
100
100
100
candies. In one move, one of them may take some of their candies and distribute them to a non-empty set of the remaining children. After what least number of moves can it happen that no two children have the same number of candies?(N. Chernyatevya)(Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.)