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Contests
National and Regional Contests
Mathlinks Contests.
MathLinks Contest 4th
MathLinks Contest 4th
Part of
Mathlinks Contests.
Subcontests
(21)
7.1
1
Hide problems
0471 inequalites 4th edition Round 7 p1
Let
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
be positive reals such that
a
b
c
d
=
1
abcd = 1
ab
c
d
=
1
. Prove that
1
a
(
b
+
1
)
+
1
b
(
c
+
1
)
+
1
c
(
d
+
1
)
+
1
d
(
a
+
1
)
≥
2.
\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.
a
(
b
+
1
)
1
+
b
(
c
+
1
)
1
+
c
(
d
+
1
)
1
+
d
(
a
+
1
)
1
≥
2.
7.3
1
Hide problems
0473 Fibonacci 4th edition Round 7 p3
Let
{
f
n
}
n
≥
0
\{f_n\}_{n \ge 0}
{
f
n
}
n
≥
0
be the Fibonacci sequence, given by
f
0
=
f
1
=
1
f_0 = f_1 = 1
f
0
=
f
1
=
1
, and for all positive integers
n
n
n
the recurrence
f
n
+
1
=
f
n
+
f
n
−
1
f_{n+1} = f_n + f_{n-1}
f
n
+
1
=
f
n
+
f
n
−
1
. Let
a
n
=
f
n
+
1
f
n
a_n = f_{n+1}f_n
a
n
=
f
n
+
1
f
n
for any non-negative integer
n
n
n
, and let
P
n
(
X
)
=
X
n
+
a
n
−
1
X
n
−
1
+
.
.
.
+
a
1
X
+
a
0
.
P_n(X) = X^n + a_{n-1}X^{n-1} + ... + a_1X + a_0.
P
n
(
X
)
=
X
n
+
a
n
−
1
X
n
−
1
+
...
+
a
1
X
+
a
0
.
Prove that for all positive integers
n
≥
3
n \ge 3
n
≥
3
the polynomial
P
n
(
X
)
P_n(X)
P
n
(
X
)
is irreducible in
Z
[
X
]
Z[X]
Z
[
X
]
.
7.2
1
Hide problems
0472 geometry 4th edition Round 7 p2
Let
Ω
\Omega
Ω
be the incircle of a triangle
A
B
C
ABC
A
BC
. Suppose that there exists a circle passing through
B
B
B
and
C
C
C
and tangent to
Ω
\Omega
Ω
in
A
′
A'
A
′
. Suppose the similar points
B
′
B'
B
′
,
C
′
C'
C
′
exist. Prove that the lines
A
A
′
,
B
B
′
AA', BB'
A
A
′
,
B
B
′
and
C
C
′
CC'
C
C
′
are concurrent.
6.3
1
Hide problems
0463 tetahedron 4th edition Round 6 p3
If
n
>
2
n>2
n
>
2
is an integer and
x
1
,
…
,
x
n
x_1, \ldots ,x_n
x
1
,
…
,
x
n
are positive reals such that
1
x
1
+
1
x
2
+
⋯
+
1
x
n
=
n
\frac 1{x_1} + \frac 1{x_2} + \cdots + \frac 1{x_n} = n
x
1
1
+
x
2
1
+
⋯
+
x
n
1
=
n
then the following inequality takes place
x
2
2
+
⋯
+
x
n
2
n
−
1
⋅
x
1
2
+
x
3
2
+
⋯
+
x
n
2
n
−
1
⋯
x
1
2
+
⋯
+
x
n
−
1
2
n
−
1
≥
(
x
1
2
+
.
.
.
+
x
n
2
n
)
n
−
1
.
\frac{x_2^2+\cdots+x_n^2}{n-1}\cdot \frac {x_1^2+x_3^2+\cdots +x_n^2} {n-1} \cdots \frac{x_1^2+\cdots+x_{n-1}^2}{n-1}\geq \left(\frac{x_1^2+...+x_n^2}{n}\right)^{n-1}.
n
−
1
x
2
2
+
⋯
+
x
n
2
⋅
n
−
1
x
1
2
+
x
3
2
+
⋯
+
x
n
2
⋯
n
−
1
x
1
2
+
⋯
+
x
n
−
1
2
≥
(
n
x
1
2
+
...
+
x
n
2
)
n
−
1
.
6.2
1
Hide problems
0462 f(P) is a square 4th edition Round 6 p2
Let
P
P
P
be the set of points in the plane, and let
f
:
P
→
P
f : P \to P
f
:
P
→
P
be a function such that the image through
f
f
f
of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that
f
(
P
)
f(P)
f
(
P
)
is a square.
6.1
1
Hide problems
0461 number theory 4th edition Round 6 p1
Find all positive integers
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
, such that the following equality takes place for an infinity of positive integers
n
n
n
(
1
a
+
2
a
+
.
.
.
+
n
a
)
b
=
(
1
c
+
2
c
+
.
.
.
+
n
c
)
d
(1^a + 2^a +...+ n^a)^b = (1^c + 2^c +...+ n^c)^d
(
1
a
+
2
a
+
...
+
n
a
)
b
=
(
1
c
+
2
c
+
...
+
n
c
)
d
5.3
1
Hide problems
0453 number theory 4th edition Round 5 p3
The sequence
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
is defined as follows:
x
1
=
0
x_1 = 0
x
1
=
0
, and for all
n
≥
1
n \ge 1
n
≥
1
(
n
+
1
)
3
x
n
+
1
=
2
n
2
(
2
n
+
1
)
x
n
+
2
(
3
n
+
1
)
.
(n + 1)^3 x_{n+1} = 2n^2 (2n + 1)x_n + 2(3n + 1).
(
n
+
1
)
3
x
n
+
1
=
2
n
2
(
2
n
+
1
)
x
n
+
2
(
3
n
+
1
)
.
Prove that
{
x
n
}
n
\{x_n\}_n
{
x
n
}
n
contains infinitely many integer numbers.
5.2
1
Hide problems
0452 geometry 4th edition Round 5 p2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and let
K
K
K
be a point on side
A
B
AB
A
B
such that
∠
K
D
A
=
∠
B
C
D
\angle KDA = \angle BCD
∠
KD
A
=
∠
BC
D
. Let
L
L
L
be a point on the diagonal
A
C
AC
A
C
such that
K
L
∥
B
C
KL \parallel BC
K
L
∥
BC
. Prove that
∠
K
D
B
=
∠
L
D
C
\angle KDB = \angle LDC
∠
KD
B
=
∠
L
D
C
.
5.1
1
Hide problems
0451 combinatorics 4th edition Round 5 p1
Let
n
n
n
be a positive integer and let
a
n
a_n
a
n
be the number of ways to write
n
n
n
as a sum of positive integers, such that any two summands differ by at least
2
2
2
. Also, let
b
n
b_n
b
n
be the number of ways to write
n
n
n
as a sum of positive integers of the form
5
k
±
1
5k\pm 1
5
k
±
1
,
k
∈
Z
k \in Z
k
∈
Z
. Prove that
a
n
b
n
\frac{a_n}{b_n}
b
n
a
n
is a constant for all positive integers
n
n
n
.
4.3
1
Hide problems
0443 graph theory 4th edition Round 4 p3
Given is a graph
G
G
G
. An empty subgraph of
G
G
G
is a subgraph of
G
G
G
with no edges between its vertices. An edge of
G
G
G
is called important if and only if the removal of this edge will increase the size of the maximal empty subgraph. Suppose that two important edges in
G
G
G
have a common endpoint. Prove there exists a cycle of odd length in
G
G
G
.
4.2
1
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0442 triangle inside triangle 4th edition Round 4 p2
We say that two triangles
T
1
T_1
T
1
and
T
2
T_2
T
2
are contained one in each other, and we write
T
1
⊂
T
2
T_1 \subset T_2
T
1
⊂
T
2
, if and only if all the points of the triangle
T
1
T_1
T
1
lie on the sides or in the interior of the triangle
T
2
T_2
T
2
. Let
Δ
\Delta
Δ
be a triangle of area
S
S
S
, and let
Δ
1
⊂
Δ
\Delta_1 \subset \Delta
Δ
1
⊂
Δ
be the largest equilateral triangle with this property, and let
Δ
⊂
Δ
2
\Delta \subset \Delta_2
Δ
⊂
Δ
2
be the smallest equilateral triangle with this property (in terms of areas). Let
S
1
,
S
2
S_1, S_2
S
1
,
S
2
be the areas of
Δ
1
,
Δ
2
\Delta_1, \Delta_2
Δ
1
,
Δ
2
respectively. Prove that
S
1
S
2
=
S
2
S_1S_2 = S^2
S
1
S
2
=
S
2
.Bonus question: : Does this statement hold for quadrilaterals (and squares)?
4.1
1
Hide problems
0441 inequality 4th edition Round 4 p1
Let
N
0
N_0
N
0
be the set of all non-negative integers and let
f
:
N
0
×
N
0
→
[
0
,
+
∞
)
f : N_0 \times N_0 \to [0, +\infty)
f
:
N
0
×
N
0
→
[
0
,
+
∞
)
be a function such that
f
(
a
,
b
)
=
f
(
b
,
a
)
f(a, b) = f(b, a)
f
(
a
,
b
)
=
f
(
b
,
a
)
and
f
(
a
,
b
)
=
f
(
a
+
1
,
b
)
+
f
(
a
,
b
+
1
)
,
f(a, b) = f(a + 1, b) + f(a, b + 1),
f
(
a
,
b
)
=
f
(
a
+
1
,
b
)
+
f
(
a
,
b
+
1
)
,
for all
a
,
b
∈
N
0
a, b \in N_0
a
,
b
∈
N
0
. Denote by
x
n
=
f
(
n
,
0
)
x_n = f(n, 0)
x
n
=
f
(
n
,
0
)
for all
n
∈
N
0
n \in N_0
n
∈
N
0
. Prove that for all
n
∈
N
0
n \in N_0
n
∈
N
0
the following inequality takes place
2
n
x
n
≥
x
0
.
2^n x_n \ge x_0.
2
n
x
n
≥
x
0
.
3.1
1
Hide problems
0431 Fibonacci inequality 4th edition Round 3 p1
Let
{
f
n
}
n
≥
1
\{f_n\}_{n\ge 1}
{
f
n
}
n
≥
1
be the Fibonacci sequence, defined by
f
1
=
f
2
=
1
f_1 = f_2 = 1
f
1
=
f
2
=
1
, and for all positive integers
n
n
n
,
f
n
+
2
=
f
n
+
1
+
f
n
f_{n+2} = f_{n+1} + f_n
f
n
+
2
=
f
n
+
1
+
f
n
. Prove that the following inequality takes place for all positive integers
n
n
n
:
(
n
1
)
f
1
+
(
n
2
)
f
2
+
.
.
.
+
(
n
n
)
f
n
<
(
2
n
+
2
)
n
n
!
{n \choose 1}f_1 +{n \choose 2}f_2+... +{n \choose n}f_n < \frac{(2n + 2)^n}{n!}
(
1
n
)
f
1
+
(
2
n
)
f
2
+
...
+
(
n
n
)
f
n
<
n
!
(
2
n
+
2
)
n
.
3.3
1
Hide problems
0433 geometry 4th edition Round 3 p3
Let
A
B
C
ABC
A
BC
be a triangle, and let
C
C
C
be its circumcircle. Let
T
T
T
be the circle tangent to
A
B
,
A
C
AB, AC
A
B
,
A
C
and
C
C
C
internally in the points
F
,
E
F, E
F
,
E
and
D
D
D
respectively. Let
P
,
Q
P, Q
P
,
Q
be the intersection points between the line
E
F
EF
EF
and the lines
D
B
DB
D
B
and
D
C
DC
D
C
respectively. Prove that if
D
P
=
D
Q
DP = DQ
D
P
=
D
Q
then the triangle
A
B
C
ABC
A
BC
is isosceles.
3.2
1
Hide problems
0432 functional 4th edition Round 3 p2
Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that
f
(
x
)
≥
0
f(x) \ge 0
f
(
x
)
≥
0
for all positive reals
x
x
x
,
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
and for all reals
x
,
y
x, y
x
,
y
f
(
x
+
y
−
x
y
)
=
f
(
x
)
+
f
(
y
)
−
f
(
x
y
)
.
f(x + y -xy) = f(x) + f(y) - f(xy).
f
(
x
+
y
−
x
y
)
=
f
(
x
)
+
f
(
y
)
−
f
(
x
y
)
.
2.3
1
Hide problems
0423 combinations sum 4th edition Round 2 p3
Let
m
≥
2
n
m \ge 2n
m
≥
2
n
be two positive integers. Find a closed form for the following expression:
E
(
m
,
n
)
=
∑
k
=
0
n
(
−
1
)
k
(
m
−
k
n
)
(
n
k
)
E(m, n) = \sum_{k=0}^{n} (-1)^k {{m- k} \choose n} { n \choose k}
E
(
m
,
n
)
=
k
=
0
∑
n
(
−
1
)
k
(
n
m
−
k
)
(
k
n
)
2.2
1
Hide problems
0422 tetrahedron 4th edition Round 2 p2
Prove that the six sides of any tetrahedron can be the sides of a convex hexagon.
2.1
1
Hide problems
0421 number theory 4th edition Round 2 p1
For a positive integer
n
n
n
let
σ
(
n
)
\sigma (n)
σ
(
n
)
be the sum of all its positive divisors. Find all positive integers
n
n
n
such that the number
σ
(
n
)
n
+
1
\frac{\sigma (n)}{n + 1}
n
+
1
σ
(
n
)
is an integer.
1.3
1
Hide problems
0413 geometry 4th edition Round 1 p3
Let
Ω
1
(
O
1
,
r
1
)
\Omega_1(O_1, r_1)
Ω
1
(
O
1
,
r
1
)
and
Ω
2
(
O
2
,
r
2
)
\Omega_2(O_2, r_2)
Ω
2
(
O
2
,
r
2
)
be two circles that intersect in two points
X
,
Y
X, Y
X
,
Y
. Let
A
,
C
A, C
A
,
C
be the points in which the line
O
1
O
2
O_1O_2
O
1
O
2
cuts the circle
Ω
1
\Omega_1
Ω
1
, and let
B
B
B
be the point in which the circle
Ω
2
\Omega_2
Ω
2
itnersect the interior of the segment
A
C
AC
A
C
, and let
M
M
M
be the intersection of the lines
A
X
AX
A
X
and
B
Y
BY
B
Y
. Prove that
M
M
M
is the midpoint of the segment
A
X
AX
A
X
if and only if
O
1
O
2
=
1
2
(
r
1
+
r
2
)
O_1O_2 =\frac12 (r_1 + r_2)
O
1
O
2
=
2
1
(
r
1
+
r
2
)
.
1.2
1
Hide problems
0412 number theory 4th edition Round 1 p2
Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.
1.1
1
Hide problems
0411 polynomials 4th edition Round 1 p1
Let
a
≥
2
a \ge 2
a
≥
2
be an integer. Find all polynomials
f
f
f
with real coefficients such that
A
=
{
a
n
2
∣
n
≥
1
,
n
∈
Z
}
⊂
{
f
(
n
)
∣
n
≥
1
,
n
∈
Z
}
=
B
.
A = \{a^{n^2} | n \ge 1, n \in Z\} \subset \{f(n) | n \ge 1, n \in Z\} = B.
A
=
{
a
n
2
∣
n
≥
1
,
n
∈
Z
}
⊂
{
f
(
n
)
∣
n
≥
1
,
n
∈
Z
}
=
B
.