MathDB
Problems
Contests
National and Regional Contests
Thailand Contests
Mathcenter Contest
2009 Mathcenter Contest
2009 Mathcenter Contest
Part of
Mathcenter Contest
Subcontests
(5)
5
2
Hide problems
2^n can begin with any sequence of digits
For
n
∈
N
n\in\mathbb{N}
n
∈
N
, prove that
2
n
2^n
2
n
can begin with any sequence of digits.Hint:
log
2
\log 2
lo
g
2
is irrational number.
min P = (5a^{2}-3ab+2)/a^{2}(b-a)
Let
a
a
a
and
b
b
b
be real numbers, where
a
≠
0
a \not= 0
a
=
0
and
a
≠
b
a \not= b
a
=
b
and all the roots of the equation
a
x
3
−
x
2
+
b
x
−
1
=
0
ax^{3}-x^{2}+bx-1 = 0
a
x
3
−
x
2
+
b
x
−
1
=
0
is a real and positive number. Find the smallest possible value of
P
=
5
a
2
−
3
a
b
+
2
a
2
(
b
−
a
)
P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}
P
=
a
2
(
b
−
a
)
5
a
2
−
3
ab
+
2
.(Heir of Ramanujan)
4
2
Hide problems
sum 1/\sqrt{x+y} >=2+1/{\sqrt2} if xy+yz+zx=1
Let
x
,
y
,
z
∈
R
0
+
x,y,z\in \mathbb{R}^+_0
x
,
y
,
z
∈
R
0
+
such that
x
y
+
y
z
+
z
x
=
1
xy+yz+zx=1
x
y
+
yz
+
z
x
=
1
. Prove that
1
x
+
y
+
1
y
+
z
+
1
z
+
x
≥
2
+
1
2
.
\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{y+z}}+\frac{1}{\sqrt{z+x}}\ge 2+\frac{1}{\sqrt{2}}.
x
+
y
1
+
y
+
z
1
+
z
+
x
1
≥
2
+
2
1
.
(Anonymous314)
8(x+1/x) =15(y+1/y) = 17(z+1/z), xy + yz + zx=1
Find the values of the real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
that correspond to the system of equations.
8
(
x
+
1
x
)
=
15
(
y
+
1
y
)
=
17
(
z
+
1
z
)
8(x+\frac{1}{x}) =15(y+\frac{1}{y}) = 17(z+\frac{1}{z})
8
(
x
+
x
1
)
=
15
(
y
+
y
1
)
=
17
(
z
+
z
1
)
x
y
+
y
z
+
z
x
=
1
xy + yz + zx=1
x
y
+
yz
+
z
x
=
1
(Heir of Ramanujan)
3
2
Hide problems
infinite set of points with integral distances from each other
Prove that for each
k
k
k
points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear.(Anonymous314)PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source
sum (x^2+2)/\sqrt{z^2+xy} >=6
Let x,y,z>0 Prove that
x
2
+
2
z
2
+
x
y
+
y
2
+
2
x
2
+
y
z
+
z
2
+
2
y
2
+
z
x
≥
6
\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6
z
2
+
x
y
x
2
+
2
+
x
2
+
yz
y
2
+
2
+
y
2
+
z
x
z
2
+
2
≥
6
. (nooonuii)
2
2
Hide problems
locus of P, ABCD square such that max { PA, PC\}=1/2(PB+PD)
Find the locus of points
P
P
P
in the plane of a square
A
B
C
D
ABCD
A
BC
D
such that
max
{
P
A
,
P
C
}
=
1
2
(
P
B
+
P
D
)
.
\max\{ PA,\ PC\}=\frac12(PB+PD).
max
{
P
A
,
PC
}
=
2
1
(
PB
+
P
D
)
.
(Anonymous314)
4ab/(ab^2+1) is a natural
Find all natural numbers that can be written in the form
4
a
b
a
b
2
+
1
\frac{4ab}{ab^2+1}
a
b
2
+
1
4
ab
for some natural
a
,
b
a,b
a
,
b
.(nooonuii)
1
2
Hide problems
m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}
Let
m
,
n
m,n
m
,
n
be natural numbers. Prove that
m
m
m
m
+
n
n
n
n
≥
m
n
n
n
+
n
m
m
m
m^{m^{m^m}}+n^{n^{n^n}}\geq m^{n^{n^n}}+ n^{m^{m^m}}
m
m
m
m
+
n
n
n
n
≥
m
n
n
n
+
n
m
m
m
(nooonuii)
(a_1+a_2+...+a_n)!! / a_1!!a_2!!... a_n!! is an integer
For any natural
n
n
n
, define
n
!
!
=
(
n
!
)
!
n!!=(n!)!
n
!!
=
(
n
!)!
e.g.
3
!
!
=
(
3
!
)
!
=
6
!
=
720
3!!=(3!)!=6!=720
3
!!
=
(
3
!)!
=
6
!
=
720
. Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be a positive integer Prove that
(
a
1
+
a
2
+
⋯
+
a
n
)
!
!
a
1
!
!
a
2
!
!
⋯
a
n
!
!
\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}
a
1
!!
a
2
!!
⋯
a
n
!!
(
a
1
+
a
2
+
⋯
+
a
n
)!!
is an integer.(nooonuii)