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Problems
Contests
International Contests
Tournament Of Towns
2015 Tournament of Towns
2015 Tournament of Towns
Part of
Tournament Of Towns
Subcontests
(7)
1
2
Hide problems
First digit of Powers of an Integer
(a) The integers
x
x
x
,
x
2
x^2
x
2
and
x
3
x^3
x
3
begin with the same digit. Does it imply that this digit is
1
1
1
? (
2
2
2
points) (b) The same question for the integers
x
,
x
2
,
x
3
,
⋯
,
x
2015
x, x^2, x^3, \cdots, x^{2015}
x
,
x
2
,
x
3
,
⋯
,
x
2015
(
3
3
3
points) .
Relatively terms of a Geometric Series
A geometrical progression consists of
37
37
37
positive integers. The first and the last terms are relatively prime numbers. Prove that the
1
9
t
h
19^{th}
1
9
t
h
term of the progression is the
1
8
t
h
18^{th}
1
8
t
h
power of some positive integer. (
3
3
3
points)
2
3
Hide problems
Reflection of a Point lying on Circumcircle
A point
X
X
X
is marked on the base
B
C
BC
BC
of an isosceles
△
A
B
C
\triangle ABC
△
A
BC
, and points
P
P
P
and
Q
Q
Q
are marked on the sides
A
B
AB
A
B
and
A
C
AC
A
C
so that
A
P
X
Q
APXQ
A
PXQ
is a parallelogram. Prove that the point
Y
Y
Y
symmetrical to
X
X
X
with respect to line
P
Q
PQ
PQ
lies on the circumcircle of the
△
A
B
C
\triangle ABC
△
A
BC
. (
5
5
5
points)
Junior A-Level Question Two (Fall 2015)
From a set of integers
{
1
,
.
.
.
,
100
}
\{1,...,100\}
{
1
,
...
,
100
}
,
k
k
k
integers were deleted. Is it always possible to choose
k
k
k
distinct integers from the remaining set such that their sum is
100
100
100
if(a)
k
=
9
k=9
k
=
9
? (b)
k
=
8
k=8
k
=
8
?
Dividing a Grid into Polygons
A
10
×
10
10 \times 10
10
×
10
square on a grid is split by
80
80
80
unit grid segments into
20
20
20
polygons of equal area (no one of these segments belongs to the boundary of the square). Prove that all polygons are congruent. (
6
6
6
points)
3
2
Hide problems
Filling and permuting Numbers in Tables
(a) A
2
×
n
2 \times n
2
×
n
-table (with
n
>
2
n > 2
n
>
2
) is filled with numbers so that the sums in all the columns are different. Prove that it is possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different. (
2
2
2
points) (b) A
100
×
100
100 \times 100
100
×
100
-table is filled with numbers such that the sums in all the columns are different. Is it always possible to permute the numbers in the table so that the sums in the columns would still be different and the sums in the rows would also be different? (
6
6
6
points)
Coefficients and roots of a Polynomial
Each coefficient of a polynomial is an integer with absolute value not exceeding
2015
2015
2015
. Prove that every positive root of this polynomial exceeds
1
2016
\frac{1}{2016}
2016
1
. (
6
6
6
points)
4
2
Hide problems
Painting segments of a polygon Blue
A convex
N
−
N-
N
−
gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new
2
N
2N
2
N
segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. (
8
8
8
points)
Midpoints of diagonals of a Cyclic Quadrilateral
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral,
K
K
K
and
N
N
N
be the midpoints of the diagonals and
P
P
P
and
Q
Q
Q
be points of intersection of the extensions of the opposite sides. Prove that
∠
P
K
Q
+
∠
P
N
Q
=
180
\angle PKQ + \angle PNQ = 180
∠
P
K
Q
+
∠
PNQ
=
180
. (
7
7
7
points) .
5
2
Hide problems
Coefficients of Products of 2 Polynomials
Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding
2015
2015
2015
but all coefficients of their product have absolute values not exceeding
1
1
1
? (
10
10
10
points)
Symbol Combinations on Real Numbers
Several distinct real numbers are written on a blackboard. Peter wants to create an algebraic expression such that among its values there would be these and only these numbers. He may use any real numbers, brackets, signs
+
,
−
,
×
+, -, \times
+
,
−
,
×
and a special sign
±
\pm
±
. Usage of
±
\pm
±
is equivalent to usage of
+
+
+
and
−
-
−
in all possible combinations. For instance, the expression
5
±
1
5 \pm 1
5
±
1
results in
{
4
,
6
}
\{4, 6\}
{
4
,
6
}
, while
(
2
±
0.5
)
±
0.5
(2 \pm 0.5) \pm 0.5
(
2
±
0.5
)
±
0.5
results in
{
1
,
2
,
3
}
\{1, 2, 3\}
{
1
,
2
,
3
}
. Can Peter construct an expression if the numbers on the blackboard are : (a)
1
,
2
,
4
1, 2, 4
1
,
2
,
4
? (
2
2
2
points) (b) any
100
100
100
distinct real numbers ? (
6
6
6
points)
6
3
Hide problems
Sequence with + , - and x
Several distinct real numbers are written on a blackboard. Peter wants to make an expression such that its values are exactly these numbers. To make such an expression, he may use any real numbers, brackets, and usual signs
+
+
+
,
−
-
−
and
×
\times
×
. He may also use a special sign
±
\pm
±
: computing the values of the resulting expression, he chooses values
+
+
+
or
−
-
−
for every
±
\pm
±
in all possible combinations. For instance, the expression
5
±
1
5 \pm 1
5
±
1
results in
{
4
,
6
}
\{4, 6 \}
{
4
,
6
}
, and
(
2
±
0.5
)
±
0.5
(2 \pm 0.5) \pm 0.5
(
2
±
0.5
)
±
0.5
results in
{
1
,
2
,
3
}
\{1, 2, 3 \}
{
1
,
2
,
3
}
. Can Pete construct such an expression:
a
)
a)
a
)
If the numbers on the blackboard are
1
,
2
,
4
1, 2, 4
1
,
2
,
4
;
b
)
b)
b
)
For any collection of
100
100
100
distinct real numbers on a blackboard?
Expelling Binary Wizards
An Emperor invited
2015
2015
2015
wizards to a festival. Each of the wizards knows who of them is good and who is evil, however the Emperor doesn’t know this. A good wizard always tells the truth, while an evil wizard can tell the truth or lie at any moment. The Emperor gives each wizard a card with a single question, maybe different for different wizards, and after that listens to the answers of all wizards which are either “yes” or “no”. Having listened to all the answers, the Emperor expels a single wizard through a magic door which shows if this wizard is good or evil. Then the Emperor makes new cards with questions and repeats the procedure with the remaining wizards, and so on. The Emperor may stop at any moment, and after this the Emperor may expel or not expel a wizard. Prove that the Emperor can expel all the evil wizards having expelled at most one good wizard. (
10
10
10
points)
Cutting a Melon
Basil has a melon in a shape of a ball,
20
20
20
in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut,
h
h
h
deep (
h
h
h
is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if (a)
h
=
17
h = 17
h
=
17
? (6 points) (b)
h
=
18
h = 18
h
=
18
? (6 points)
7
3
Hide problems
Circumcircle-Incircle Analogy for 3-D Cuboid
It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to
3
3
3
edges.) (
10
10
10
points)
Santa Clause Problem
Santa Clause had
n
n
n
sorts of candies,
k
k
k
candies of each sort. He distributed them at random between
k
k
k
gift bags,
n
n
n
candies per a bag and gave a bag to everyone of
k
k
k
children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.
Heights in Descending order
N
N
N
children no two of the same height stand in a line. The following two-step procedure is applied: first, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of their heights (a group may consist of a single child). Second, the order of children in each group is reversed, so now in each group the children stand in descending order of their heights. Prove that in result of applying this procedure
N
−
1
N - 1
N
−
1
times the children in the line would stand from the left to the right in descending order of their heights. (12 points)