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National and Regional Contests
Mathlinks Contests.
MathLinks Contest 6th
MathLinks Contest 6th
Part of
Mathlinks Contests.
Subcontests
(21)
1.3
1
Hide problems
0613 vector problem 6th edition Round 1 p3
Introductory part We call an
n
n
n
-tuple
x
=
(
x
1
,
x
2
,
.
.
.
,
x
n
)
x = (x_1, x_2, ... , x_n)
x
=
(
x
1
,
x
2
,
...
,
x
n
)
, with
x
k
∈
R
x_k \in R
x
k
∈
R
(or respectively with all
x
k
∈
Z
x_k \in Z
x
k
∈
Z
) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by
R
n
R^n
R
n
(respectively
Z
n
Z^n
Z
n
). A vector
x
x
x
is null if and only if
x
k
=
0
x_k = 0
x
k
=
0
, for all
k
∈
{
1
,
2
,
.
.
.
,
n
}
k \in \{1, 2,... , n\}
k
∈
{
1
,
2
,
...
,
n
}
. Also let
U
n
U_n
U
n
be the set of all real vectors
x
=
(
x
1
,
x
2
,
.
.
.
,
x
n
)
x = (x_1, x_2, ... , x_n)
x
=
(
x
1
,
x
2
,
...
,
x
n
)
, such that
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
1
x^2_1 + x^2_2 + ...+ x^2_n = 1
x
1
2
+
x
2
2
+
...
+
x
n
2
=
1
. For two vectors
x
=
(
x
1
,
.
.
.
,
x
n
)
,
y
=
(
y
1
,
.
.
.
,
y
n
)
x = (x_1, ... , x_n), y = (y_1, ..., y_n)
x
=
(
x
1
,
...
,
x
n
)
,
y
=
(
y
1
,
...
,
y
n
)
we define the scalar product as the real number
x
⋅
y
=
x
1
y
1
+
x
2
y
2
+
.
.
.
+
x
n
y
n
x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n
x
⋅
y
=
x
1
y
1
+
x
2
y
2
+
...
+
x
n
y
n
. We define the norm of the vector
x
x
x
as
∣
∣
x
∣
∣
=
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}
∣∣
x
∣∣
=
x
1
2
+
x
2
2
+
...
+
x
n
2
The problem Let
A
(
k
,
r
)
=
{
x
∈
U
n
∣
A(k, r) = \{x \in U_n |
A
(
k
,
r
)
=
{
x
∈
U
n
∣
for all
z
∈
Z
n
z \in Z^n
z
∈
Z
n
we have either
∣
x
⋅
z
∣
≥
k
∣
∣
z
∣
∣
r
|x \cdot z| \ge \frac{k}{||z||^r}
∣
x
⋅
z
∣
≥
∣∣
z
∣
∣
r
k
or
z
z
z
is null
}
\}
}
. Prove that if
r
>
n
−
1
r > n - 1
r
>
n
−
1
the we can find a positive number
k
k
k
such that
A
(
k
,
r
)
A(k, r)
A
(
k
,
r
)
is not empty, and if
r
<
n
−
1
r < n - 1
r
<
n
−
1
we cannot find such a positive number
k
k
k
.
6.1
1
Hide problems
0661 inequality 6th edition Round 6 p1
Let
p
>
1
p > 1
p
>
1
and let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive numbers such that
(
a
+
b
+
c
+
d
)
(
1
a
+
1
b
+
1
c
+
1
d
)
=
16
p
2
.
(a + b + c + d) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\right)= 16p^2.
(
a
+
b
+
c
+
d
)
(
a
1
+
b
1
+
c
1
+
d
1
)
=
16
p
2
.
Find all values of the ratio
R
=
max
{
a
,
b
,
c
,
d
}
min
{
a
,
b
,
c
,
d
}
R =\frac{\max \{a, b, c, d\}}{\min \{a, b, c, d\}}
R
=
m
i
n
{
a
,
b
,
c
,
d
}
m
a
x
{
a
,
b
,
c
,
d
}
(depending on the parameter
p
p
p
)
7.3
1
Hide problems
0673 lattice points 6th edition Round 7 p3
A lattice point in the Carthesian plane is a point with both coordinates integers. A rectangle minor (respectively a square minor) is a set of lattice points lying inside or on the boundaries of a rectangle (respectively square) with vertices lattice points. We assign to each lattice point a real number, such that the sum of all the numbers in any square minor is less than
1
1
1
in absolute value. Prove that the sum of all the numbers in any rectangle minor is less than
4
4
4
in absolute value.
7.2
1
Hide problems
0672 geometry 6th edition Round 7 p2
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Let
M
,
N
M, N
M
,
N
be the midpoints of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
and let
P
P
P
be the midpoint of
M
N
MN
MN
. Let
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
be the intersections of the rays
A
P
AP
A
P
,
B
P
BP
BP
,
C
P
CP
CP
and
D
P
DP
D
P
respectively with the circumcircle of the quadrilateral
A
B
C
D
ABCD
A
BC
D
.Find, with proof, the value of the sum
σ
=
A
P
P
A
′
+
B
P
P
B
′
+
C
P
P
C
′
+
D
P
P
D
′
.
\sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} .
σ
=
P
A
′
A
P
+
P
B
′
BP
+
P
C
′
CP
+
P
D
′
D
P
.
7.1
1
Hide problems
0671 product of irreducible polynomials 6th edition Round 7 p1
Write the following polynomial as a product of irreducible polynomials in
Z
[
X
]
\mathbb{Z}[X]
Z
[
X
]
f
(
X
)
=
X
2005
−
2005
X
+
2004.
f(X) = X^{2005} - 2005 X + 2004 .
f
(
X
)
=
X
2005
−
2005
X
+
2004.
Justify your answer.
6.2
1
Hide problems
0662 non-negative real numbers in nxn matrix 6th edition Round 6 p2
A
n
×
n
n \times n
n
×
n
matrix is filled with non-negative real numbers such that on each line and column the sum of the elements is
1
1
1
. Prove that one can choose n positive entries from the matrix, such that each of them lies on a different line and different column.
6.3
1
Hide problems
0663 geometry 6th edition Round 6 p3
Let
C
1
,
C
2
C_1, C_2
C
1
,
C
2
and
C
3
C_3
C
3
be three circles, of radii
2
,
4
2, 4
2
,
4
and
6
6
6
respectively. It is known that each of them are tangent exteriorly with the other two circles. Let
Ω
1
\Omega_1
Ω
1
and
Ω
2
\Omega_2
Ω
2
be two more circles, each of them tangent to all of the
3
3
3
circles above, of radius
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
respectively. Prove that
ω
1
+
ω
2
=
2
ω
1
ω
2
\omega_1 + \omega_2 = 2\omega_1\omega_2
ω
1
+
ω
2
=
2
ω
1
ω
2
.
5.3
1
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0653 geometric inequality with areas 6th edition Round 5 p3
Let
A
B
C
ABC
A
BC
be a triangle, and let
A
B
B
2
A
3
ABB_2A_3
A
B
B
2
A
3
,
B
C
C
3
B
1
BCC_3B_1
BC
C
3
B
1
and
C
A
A
1
C
2
CAA_1C_2
C
A
A
1
C
2
be squares constructed outside the triangle. Denote with
S
S
S
the area of the triangle
A
B
C
ABC
A
BC
and with s the area of the triangle formed by the intersection of the lines
A
1
B
1
A_1B_1
A
1
B
1
,
B
2
C
2
B_2C_2
B
2
C
2
and
C
3
A
3
C_3A_3
C
3
A
3
. Prove that
s
≤
(
4
−
2
3
)
S
s \le (4 - 2\sqrt3)S
s
≤
(
4
−
2
3
)
S
.
5.2
1
Hide problems
0652 algebra 6th edition Round 5 p2
Let
n
≥
5
n \ge 5
n
≥
5
be an integer and let
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
be
n
n
n
distinct integer numbers such that no
3
3
3
of them can be in arithmetic progression. Prove that if for all
1
≤
i
,
j
≤
n
1 \le i, j \le n
1
≤
i
,
j
≤
n
we have
2
∣
x
i
−
x
j
∣
≤
n
(
n
−
1
)
2|x_i - x_j | \le n(n - 1)
2∣
x
i
−
x
j
∣
≤
n
(
n
−
1
)
then there exist
4
4
4
distinct indices
i
,
j
,
k
,
l
∈
{
1
,
2
,
.
.
.
,
n
}
i, j, k, l \in \{1, 2, ... , n\}
i
,
j
,
k
,
l
∈
{
1
,
2
,
...
,
n
}
such that
x
i
+
x
j
=
x
k
+
x
l
.
x_i + x_j = x_k + x_l.
x
i
+
x
j
=
x
k
+
x
l
.
5.1
1
Hide problems
0651 diophantine equation 6th edition Round 5 p1
Find all solutions in integers of the equation
x
2
+
2
2
=
y
3
+
3
3
.
x^2 + 2^2 = y^3 + 3^3.
x
2
+
2
2
=
y
3
+
3
3
.
4.3
1
Hide problems
0643 radical inequality 6th edition Round 4 p3
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
c
=
1
abc = 1
ab
c
=
1
. Prove that
a
+
b
b
+
1
+
b
+
c
c
+
1
+
c
+
a
a
+
1
≥
3
\sqrt{\frac{a+b}{b+1}}+\sqrt{\frac{b+c}{c+1}}+\sqrt{\frac{c+a}{a+1}} \ge 3
b
+
1
a
+
b
+
c
+
1
b
+
c
+
a
+
1
c
+
a
≥
3
4.2
1
Hide problems
0642 infinite multiples of n without 9 6th edition Round 4 p2
Let
n
n
n
be a positive integer. Prove that there exist an infinity of multiples of
n
n
n
which do not contain the digit “
9
9
9
” in their decimal representation
4.1
1
Hide problems
0641 n subsets of a set with 5 elements 6th edition Round 4 p1
Let
F
F
F
be a family of n subsets of a set
K
K
K
with
5
5
5
elements, such that any two subsets in
F
F
F
have a common element. Find the minimal value of
n
n
n
such that no matter how we choose
F
F
F
with the properties above, there exists exactly one element of
K
K
K
which belongs to all the sets in
F
F
F
.
3.3
1
Hide problems
0633 subsets in disjoint convex sets 6th edition Round 3 p3
We say that a set of points
M
M
M
in the plane is convex if for any two points
A
,
B
∈
M
A, B \in M
A
,
B
∈
M
, all the points from the segment
(
A
B
)
(AB)
(
A
B
)
also belong to
M
M
M
. Let
n
≥
2
n \ge 2
n
≥
2
be an integer and let
F
F
F
be a family of
n
n
n
disjoint convex sets in the plane. Prove that there exists a line
ℓ
\ell
ℓ
in the plane, a set
M
∈
F
M \in F
M
∈
F
and a subset
S
⊂
F
S \subset F
S
⊂
F
with at least
⌈
n
12
⌉
\lceil \frac{n}{12} \rceil
⌈
12
n
⌉
elements such that
M
M
M
is contained in one closed half-plane determined by
ℓ
\ell
ℓ
, and all the sets
N
∈
S
N \in S
N
∈
S
are contained in the complementary closed half-plane determined by
ℓ
\ell
ℓ
.
3.2
1
Hide problems
0632 geometry with tangential quadr. 6th edition Round 3 p2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and the points
A
1
∈
(
C
D
)
A_1 \in (CD)
A
1
∈
(
C
D
)
,
A
2
∈
(
B
C
)
A_2 \in (BC)
A
2
∈
(
BC
)
,
C
1
∈
(
A
B
)
C_1 \in (AB)
C
1
∈
(
A
B
)
,
C
2
∈
(
A
D
)
C_2 \in (AD)
C
2
∈
(
A
D
)
. Let
M
,
N
M, N
M
,
N
be the intersection points between the lines
A
A
2
,
C
C
1
AA_2, CC_1
A
A
2
,
C
C
1
and
A
A
1
,
C
C
2
AA_1, CC_2
A
A
1
,
C
C
2
respectively. Prove that if three of the quadrilaterals
A
B
C
D
ABCD
A
BC
D
,
A
2
B
C
1
M
A_2BC_1M
A
2
B
C
1
M
,
A
M
C
N
AMCN
A
MCN
,
A
1
N
C
2
D
A_1NC_2D
A
1
N
C
2
D
are circumscriptive (i.e. there exists an incircle tangent to all the sides of the quadrilateral) then the forth quadrilateral is also circumscriptive.
3.1
1
Hide problems
0631 number theory 6th edition Round 3 p1
For each positive integer
n
n
n
let
τ
(
n
)
\tau (n)
τ
(
n
)
be the sum of divisors of
n
n
n
. Find all positive integers
k
k
k
for which
τ
(
k
n
−
1
)
≡
0
\tau (kn - 1) \equiv 0
τ
(
kn
−
1
)
≡
0
(mod
k
k
k
) for all positive integers
n
n
n
.
2.3
1
Hide problems
0623 bijective mappings 6th edition Round 2 p3
Let
σ
:
{
1
,
2
,
.
.
.
,
n
}
→
{
1
,
2
,
.
.
.
,
n
}
\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}
σ
:
{
1
,
2
,
...
,
n
}
→
{
1
,
2
,
...
,
n
}
be a bijective mapping. Let
S
n
S_n
S
n
be the set of all such mappings and let
d
k
(
σ
)
=
∣
σ
(
k
)
−
σ
(
k
+
1
)
∣
d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|
d
k
(
σ
)
=
∣
σ
(
k
)
−
σ
(
k
+
1
)
∣
, for all
k
∈
{
1
,
2
,
.
.
.
,
n
}
k \in \{1, 2, ..., n\}
k
∈
{
1
,
2
,
...
,
n
}
, where
σ
(
n
+
1
)
=
σ
(
1
)
\sigma (n + 1) = \sigma (1)
σ
(
n
+
1
)
=
σ
(
1
)
. Also let
d
(
σ
)
=
min
{
d
k
(
σ
)
∣
1
≤
k
≤
n
}
d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}
d
(
σ
)
=
min
{
d
k
(
σ
)
∣1
≤
k
≤
n
}
. Find
max
σ
∈
S
n
d
(
σ
)
\max_{\sigma \in S_n} d(\sigma)
max
σ
∈
S
n
d
(
σ
)
.
2.2
1
Hide problems
0622 number theory with combinations 6th edition Round 2 p2
Let
a
1
,
a
2
,
.
.
.
,
a
n
−
1
a_1, a_2, ..., a_{n-1}
a
1
,
a
2
,
...
,
a
n
−
1
be
n
−
1
n - 1
n
−
1
consecutive positive integers in increasing order such that
k
k
k
(
n
k
)
{n \choose k}
(
k
n
)
≡
0
\equiv 0
≡
0
(mod
a
k
a_k
a
k
), for all
k
∈
{
1
,
2
,
.
.
.
,
n
−
1
}
k \in \{1, 2, ... , n - 1\}
k
∈
{
1
,
2
,
...
,
n
−
1
}
. Find the possible values of
a
1
a_1
a
1
.
2.1
1
Hide problems
0621 expontential equation 6th edition Round 2 p1
Solve in positive real numbers the following equation
x
−
y
+
y
−
x
=
4
−
x
−
y
x^{-y} + y^{-x} = 4 - x - y
x
−
y
+
y
−
x
=
4
−
x
−
y
.
1.2
1
Hide problems
0612 5 points lie on sphere 6th edition Round 1 p2
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle of center
O
O
O
in the plane
α
\alpha
α
, and let
V
∉
α
V \notin\alpha
V
∈
/
α
be a point in space such that
V
O
⊥
α
V O \perp \alpha
V
O
⊥
α
. Let
A
′
∈
(
V
A
)
A' \in (V A)
A
′
∈
(
V
A
)
,
B
′
∈
(
V
B
)
B'\in (V B)
B
′
∈
(
V
B
)
,
C
′
∈
(
V
C
)
C'\in (V C)
C
′
∈
(
V
C
)
,
D
′
∈
(
V
D
)
D'\in (V D)
D
′
∈
(
V
D
)
be four points, and let
M
M
M
and
N
N
N
be the midpoints of the segments
A
′
C
′
A'C'
A
′
C
′
and
B
′
D
′
B'D'
B
′
D
′
. .Prove that
M
N
∥
α
MN \parallel \alpha
MN
∥
α
if and only if
V
,
A
′
,
B
′
,
C
′
,
D
′
V , A', B', C', D'
V
,
A
′
,
B
′
,
C
′
,
D
′
all lie on a sphere.
1.1
1
Hide problems
0611 inequality 6th edition Round 1 p1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
b
c
+
c
a
+
b
=
1
,
bc +ca +b = 1,
b
c
+
c
a
+
b
=
1
,
. Prove that
1
+
b
2
c
2
(
b
+
c
)
2
+
1
+
c
2
a
2
(
c
+
a
)
2
+
1
+
a
2
b
2
(
a
+
b
)
2
≥
5
2
.
\frac {1 +b^2c^2}{(b +c)^2} + \frac {1+ c^2a^2}{(c + a)^2} +\frac {1 +a^2b^2}{(a +b)^2} \geq \frac {5}{2}.
(
b
+
c
)
2
1
+
b
2
c
2
+
(
c
+
a
)
2
1
+
c
2
a
2
+
(
a
+
b
)
2
1
+
a
2
b
2
≥
2
5
.