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Contests
National and Regional Contests
Mathlinks Contests.
MathLinks Contest 7th
MathLinks Contest 7th
Part of
Mathlinks Contests.
Subcontests
(21)
7.2
1
Hide problems
0772
Prove that the set of all the points with both coordinates begin rational numbers can be written as a reunion of two disjoint sets
A
A
A
and
B
B
B
such that any line that that is parallel with
O
x
Ox
O
x
, and respectively
O
y
Oy
O
y
intersects
A
A
A
, and respectively
B
B
B
in a finite number of points.
7.3
1
Hide problems
0773
Let
n
n
n
be a positive integer, and let M \equal{} \{1,2,\ldots, 2n\}. Find the minimal positive integer
m
m
m
, such that no matter how we choose the subsets
A
i
⊂
M
A_i \subset M
A
i
⊂
M
,
1
≤
i
≤
m
1\leq i\leq m
1
≤
i
≤
m
, with the properties: (1) |A_i\minus{}A_j|\geq 1, for all
i
≠
j
i\neq j
i
=
j
, (2) \bigcup_{i\equal{}1}^m A_i \equal{} M, we can always find two subsets
A
k
A_k
A
k
and
A
l
A_l
A
l
such that A_k \cup A_l \equal{} M (here
∣
X
∣
|X|
∣
X
∣
represents the number of elements in the set
X
X
X
.)
7.1
1
Hide problems
0771
Find all pairs of positive integers
a
,
b
a,b
a
,
b
such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}
6.3
1
Hide problems
0763
Let
Ω
\Omega
Ω
be the circumcircle of triangle
A
B
C
ABC
A
BC
. Let
D
D
D
be the point at which the incircle of
A
B
C
ABC
A
BC
touches its side
B
C
BC
BC
. Let
M
M
M
be the point on
Ω
\Omega
Ω
such that the line
A
M
AM
A
M
is parallel to
B
C
BC
BC
. Also, let
P
P
P
be the point at which the circle tangent to the segments
A
B
AB
A
B
and
A
C
AC
A
C
and to the circle
Ω
\Omega
Ω
touches
Ω
\Omega
Ω
. Prove that the points
P
P
P
,
D
D
D
,
M
M
M
are collinear.
6.1
1
Hide problems
0761
Let
{
x
n
}
n
≥
1
\{x_n\}_{n\geq 1}
{
x
n
}
n
≥
1
be a sequences, given by x_1 \equal{} 1, x_2 \equal{} 2 and x_{n \plus{} 2} \equal{} \frac { x_{n \plus{} 1}^2 \plus{} 3 }{x_n} . Prove that
x
2008
x_{2008}
x
2008
is the sum of two perfect squares.
6.2
1
Hide problems
0762
Find all functions
f
,
g
:
Q
→
Q
f,g: \mathbb Q \to \mathbb Q
f
,
g
:
Q
→
Q
such that for all rational numbers
x
,
y
x,y
x
,
y
we have f(f(x) \plus{} g(y) ) \equal{} g(f(x)) \plus{} y .
5.3
1
Hide problems
0753
If
a
≥
b
≥
c
≥
d
>
0
a\geq b\geq c\geq d > 0
a
≥
b
≥
c
≥
d
>
0
such that abcd\equal{}1, then prove that \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.
5.2
1
Hide problems
0752
Let
A
′
A^{\prime}
A
′
be an arbitrary point on the side
B
C
BC
BC
of a triangle
A
B
C
ABC
A
BC
. Denote by
T
A
b
\mathcal{T}_{A}^{b}
T
A
b
,
T
A
c
\mathcal{T}_{A}^{c}
T
A
c
the circles simultanously tangent to
A
A
′
AA^{\prime}
A
A
′
,
A
′
B
A^{\prime}B
A
′
B
,
Γ
\Gamma
Γ
and
A
A
′
AA^{\prime}
A
A
′
,
A
′
C
A^{\prime}C
A
′
C
,
Γ
\Gamma
Γ
, respectively, where
Γ
\Gamma
Γ
is the circumcircle of
A
B
C
ABC
A
BC
. Prove that
T
A
b
\mathcal{T}_{A}^{b}
T
A
b
,
T
A
c
\mathcal{T}_{A}^{c}
T
A
c
are congruent if and only if
A
A
′
AA^{\prime}
A
A
′
passes through the Nagel point of triangle
A
B
C
ABC
A
BC
. (If
M
,
N
,
P
M,N,P
M
,
N
,
P
are the points of tangency of the excircles of the triangle
A
B
C
ABC
A
BC
with the sides of the triangle
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively, then the Nagel point of the triangle is the intersection point of the lines
A
M
AM
A
M
,
B
N
BN
BN
and
C
P
CP
CP
.)
5.1
1
Hide problems
0751
Find all real polynomials
g
(
x
)
g(x)
g
(
x
)
of degree at most n \minus{} 3,
n
≥
3
n\geq 3
n
≥
3
, knowing that all the roots of the polynomial f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x) are real.
4.3
1
Hide problems
0743
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers such that ab\plus{}bc\plus{}ca\equal{}3. Prove that \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .
4.2
1
Hide problems
0742
Find the number of finite sequences \{a_1,a_2,\ldots,a_{2n\plus{}1}\}, formed with nonnegative integers, for which a_1\equal{}a_{2n\plus{}1}\equal{}0 and |a_k \minus{}a_{k\plus{}1}|\equal{}1, for all
k
∈
{
1
,
2
,
…
,
2
n
}
k\in\{1,2,\ldots,2n\}
k
∈
{
1
,
2
,
…
,
2
n
}
.
4.1
1
Hide problems
0741
Let
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
be five distinct points, such that no three of them lie on the same line. Prove that AB\plus{}BC\plus{}CA \plus{} DE < AD \plus{} AE \plus{} BD\plus{}BE \plus{} CD\plus{}CE .
3.3
1
Hide problems
0733
Find the greatest positive real number
k
k
k
such that the inequality below holds for any positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
: \frac ab \plus{} \frac bc \plus{} \frac ca \minus{} 3 \geq k \left( \frac a{b \plus{} c} \plus{} \frac b{c \plus{} a} \plus{} \frac c{a \plus{} b} \minus{} \frac 32 \right).
3.2
1
Hide problems
0732
Prove that for positive integers
x
,
y
,
z
x,y,z
x
,
y
,
z
the number x^2 \plus{} y^2 \plus{} z^2 is not divisible by 3(xy \plus{} yz \plus{} zx).
3.1
1
Hide problems
0731
Let
p
p
p
be a prime and let
d
∈
{
0
,
1
,
…
,
p
}
d \in \left\{0,\ 1,\ \ldots,\ p\right\}
d
∈
{
0
,
1
,
…
,
p
}
. Prove that \sum_{k \equal{} 0}^{p \minus{} 1}{\binom{2k}{k \plus{} d}}\equiv r \pmod{p}, where r \equiv p\minus{}d \pmod 3, r\in\{\minus{}1,0,1\}.
2.3
1
Hide problems
0723
Let
A
B
C
ABC
A
BC
be a given triangle with the incenter
I
I
I
, and denote by
X
X
X
,
Y
Y
Y
,
Z
Z
Z
the intersections of the lines
A
I
AI
A
I
,
B
I
BI
B
I
,
C
I
CI
C
I
with the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
, respectively. Consider
K
a
\mathcal{K}_{a}
K
a
the circle tangent simultanously to the sidelines
A
B
AB
A
B
,
A
C
AC
A
C
, and internally to the circumcircle
C
(
O
)
\mathcal{C}(O)
C
(
O
)
of
A
B
C
ABC
A
BC
, and let
A
′
A^{\prime}
A
′
be the tangency point of
K
a
\mathcal{K}_{a}
K
a
with
C
\mathcal{C}
C
. Similarly, define
B
′
B^{\prime}
B
′
, and
C
′
C^{\prime}
C
′
. Prove that the circumcircles of triangles
A
X
A
′
AXA^{\prime}
A
X
A
′
,
B
Y
B
′
BYB^{\prime}
B
Y
B
′
, and
C
Z
C
′
CZC^{\prime}
CZ
C
′
all pass through two distinct points.
2.2
1
Hide problems
0722
For a prime
p
p
p
an a positive integer
n
n
n
, denote by
ν
p
(
n
)
\nu_p(n)
ν
p
(
n
)
the exponent of
p
p
p
in the prime factorization of
n
!
n!
n
!
. Given a positive integer
d
d
d
and a finite set
{
p
1
,
p
2
,
…
,
p
k
}
\{p_1,p_2,\ldots, p_k\}
{
p
1
,
p
2
,
…
,
p
k
}
of primes, show that there are infinitely many positive integers
n
n
n
such that
ν
p
i
(
n
)
≡
0
(
m
o
d
d
)
\nu_{p_i}(n) \equiv 0 \pmod d
ν
p
i
(
n
)
≡
0
(
mod
d
)
, for all
1
≤
i
≤
k
1\leq i \leq k
1
≤
i
≤
k
.
2.1
1
Hide problems
0721
Let
k
k
k
be an integer,
k
≥
2
k \geq 2
k
≥
2
, and let
p
1
,
p
2
,
…
,
p
k
p_{1},\ p_{2},\ \ldots,\ p_{k}
p
1
,
p
2
,
…
,
p
k
be positive reals with p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1. Suppose we have a collection
(
A
1
,
1
,
A
1
,
2
,
…
,
A
1
,
k
)
\left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)
(
A
1
,
1
,
A
1
,
2
,
…
,
A
1
,
k
)
,
(
A
2
,
1
,
A
2
,
2
,
…
,
A
2
,
k
)
\left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)
(
A
2
,
1
,
A
2
,
2
,
…
,
A
2
,
k
)
,
…
\ldots
…
,
(
A
m
,
1
,
A
1
,
2
,
…
,
A
m
,
k
)
\left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)
(
A
m
,
1
,
A
1
,
2
,
…
,
A
m
,
k
)
of
k
k
k
-tuples of finite sets satisfying the following two properties: (i) for every
i
i
i
and every
j
≠
j
′
j \neq j^{\prime}
j
=
j
′
, A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset, and (ii) for every
i
≠
i
′
i\neq i^{\prime}
i
=
i
′
there exist
j
≠
j
′
j\neq j^{\prime}
j
=
j
′
for which
A
i
,
j
∩
A
i
′
,
j
′
≠
∅
A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset
A
i
,
j
∩
A
i
′
,
j
′
=
∅
. Prove that \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1.
1.3
1
Hide problems
0713
We are given the finite sets
X
X
X
,
A
1
A_1
A
1
,
A
2
A_2
A
2
,
…
\dots
…
, A_{n \minus{} 1} and the functions
f
i
:
X
→
A
i
f_i: \ X\rightarrow A_i
f
i
:
X
→
A
i
. A vector
(
x
1
,
x
2
,
…
,
x
n
)
∈
X
n
(x_1,x_2,\dots,x_n)\in X^n
(
x
1
,
x
2
,
…
,
x
n
)
∈
X
n
is called nice, if f_i(x_i) \equal{} f_i(x_{i \plus{} 1}), for each i \equal{} 1,2,\dots,n \minus{} 1. Prove that the number of nice vectors is at least \frac {|X|^n}{\prod\limits_{i \equal{} 1}^{n \minus{} 1} |A_i|}.
1.2
1
Hide problems
0712
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be four distinct positive integers in arithmetic progression. Prove that
a
b
c
d
abcd
ab
c
d
is not a perfect square.
1.1
1
Hide problems
0711
Given is an acute triangle
A
B
C
ABC
A
BC
and the points
A
1
,
B
1
,
C
1
A_1,B_1,C_1
A
1
,
B
1
,
C
1
, that are the feet of its altitudes from
A
,
B
,
C
A,B,C
A
,
B
,
C
respectively. A circle passes through
A
1
A_1
A
1
and
B
1
B_1
B
1
and touches the smaller arc
A
B
AB
A
B
of the circumcircle of
A
B
C
ABC
A
BC
in point
C
2
C_2
C
2
. Points
A
2
A_2
A
2
and
B
2
B_2
B
2
are defined analogously. Prove that the lines
A
1
A
2
A_1A_2
A
1
A
2
,
B
1
B
2
B_1B_2
B
1
B
2
,
C
1
C
2
C_1C_2
C
1
C
2
have a common point, which lies on the Euler line of
A
B
C
ABC
A
BC
.