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National and Regional Contests
Thailand Contests
Mathcenter Contest
2012 Mathcenter Contest + Longlist
2012 Mathcenter Contest + Longlist
Part of
Mathcenter Contest
Subcontests
(14)
6
1
Hide problems
sum a/b^2(c+a)(a+b) >=3/4 if a,b,c>0 with abc=1
Let a,b,c>0 and
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
a
b
2
(
c
+
a
)
(
a
+
b
)
+
b
c
2
(
a
+
b
)
(
b
+
c
)
+
c
a
2
(
c
+
a
)
(
a
+
b
)
≥
3
4
.
\frac{a}{b^2(c+a)(a+b)}+\frac{b}{c^2(a+b)(b+c)}+\frac{c}{a^2(c+a)(a+b)}\ge \frac{3}{4}.
b
2
(
c
+
a
)
(
a
+
b
)
a
+
c
2
(
a
+
b
)
(
b
+
c
)
b
+
a
2
(
c
+
a
)
(
a
+
b
)
c
≥
4
3
.
(Zhuge Liang)
5
1
Hide problems
sum a/ \sqrt{b+c} >= (a+b+c) /\sqrt{2} if a,b,c>0, with a+b+c+abc=4
Let a,b,c>0 and
a
+
b
+
c
+
a
b
c
=
4
a+b+c+abc=4
a
+
b
+
c
+
ab
c
=
4
. Prove that
a
b
+
c
+
b
c
+
a
+
c
a
+
b
≥
1
2
(
a
+
b
+
c
)
.
\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{c+a}}+\frac{c}{\sqrt{a+b} }\ge \frac{1}{\sqrt{2}}(a+b+c).
b
+
c
a
+
c
+
a
b
+
a
+
b
c
≥
2
1
(
a
+
b
+
c
)
.
(Zhuge Liang)
4
1
Hide problems
sum a/ \sqrt{2b^2+2c^2-a^2} >= \sqrt3 for sidelengths a,b,c
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be the side lengths of any triangle. Prove that
a
2
b
2
+
2
c
2
−
a
2
+
b
2
c
2
+
2
a
2
−
b
2
+
c
2
a
2
+
2
b
2
−
c
2
≥
3
.
\frac{a}{\sqrt{2b^2+2c^2-a^2}}+\frac{b}{\sqrt{2c^2+2a^2-b^2 }}+\frac{c}{\sqrt{2a^2+2b^2-c^2}}\ge \sqrt{3}.
2
b
2
+
2
c
2
−
a
2
a
+
2
c
2
+
2
a
2
−
b
2
b
+
2
a
2
+
2
b
2
−
c
2
c
≥
3
.
(Zhuge Liang)
7
1
Hide problems
\tau (n^m) = \sum_{d|n} m^{\nu (d)}
The arithmetic function
ν
\nu
ν
is defined by
ν
(
n
)
=
{
0
,
n
=
1
k
,
n
=
p
1
a
1
p
2
a
2
.
.
.
p
k
a
k
\nu (n) = \begin{cases}0, \,\,\,\,\, n=1 \\ k, \,\,\,\,\, n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}\end{cases}
ν
(
n
)
=
{
0
,
n
=
1
k
,
n
=
p
1
a
1
p
2
a
2
...
p
k
a
k
, where
n
=
p
1
a
1
p
2
a
2
.
.
.
p
k
a
k
n=p_1^{a_1} p_2^{a_2} ... p_k^{a_k}
n
=
p
1
a
1
p
2
a
2
...
p
k
a
k
represents the prime factorization of the number. Prove that for any naturals
m
,
n
m,n
m
,
n
,
τ
(
n
m
)
=
∑
d
∣
n
m
ν
(
d
)
.
\tau (n^m) = \sum_{d | n} m^{\nu (d)}.
τ
(
n
m
)
=
d
∣
n
∑
m
ν
(
d
)
.
(PP-nine)
3
1
Hide problems
no of divisors of (p^5+2p^2) if p,p^2+2 are both primes
If
p
,
p
2
+
2
p,p^2+2
p
,
p
2
+
2
are both primes, how many divisors does
p
5
+
2
p
2
p^5+2p^2
p
5
+
2
p
2
have?(Zhuge Liang)
10
1
Hide problems
any 2 adjacent numbers in 8x8 table are relative primes , numbers 1-8
The table size
8
×
8
8 \times 8
8
×
8
contains the numbers
1
,
2
,
.
.
.
,
8
1,2,...,8
1
,
2
,
...
,
8
in each amount as much as you want provided that two numbers that are adjacent vertically, horizontally, diagonally are relative primes. Prove that some number appears in the table at least
12
12
12
times.(PP-nine)
2
1
Hide problems
p=2^n+1 is prime if 3^{(p-1)/2}+1\equiv 0 \pmod p
Let
p
=
2
n
+
1
p=2^n+1
p
=
2
n
+
1
and
3
(
p
−
1
)
/
2
+
1
≡
0
(
m
o
d
p
)
3^{(p-1)/2}+1\equiv 0 \pmod p
3
(
p
−
1
)
/2
+
1
≡
0
(
mod
p
)
. Show that
p
p
p
is a prime. (Zhuge Liang)
1
1
Hide problems
a^2+b^2-8c = 6
Prove without using modulo that there are no integers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
2
+
b
2
−
8
c
=
6
a^2+b^2-8c = 6
a
2
+
b
2
−
8
c
=
6
(Metamorphosis)
6 sl14
1
Hide problems
max \frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2} if bc-ca-ab=1
For a real number a,b,c>0 where
b
c
−
c
a
−
a
b
=
1
bc-ca-ab=1
b
c
−
c
a
−
ab
=
1
find the maximum value of
P
=
4024
1
+
a
2
−
4024
1
+
b
2
−
2555
1
+
c
2
P=\frac{4024}{1+a^2}-\frac{4024}{1+b^2}-\frac{2555}{1+c^2}
P
=
1
+
a
2
4024
−
1
+
b
2
4024
−
1
+
c
2
2555
and find out when that holds . (PP-nine)
5 sl13
1
Hide problems
a>0, f(a)<0 for f(\sqrt{xy})=\frac{f(x)+f(y)}{2}
Define
f
:
R
+
→
R
f : \mathbb{R}^+ \rightarrow \mathbb{R}
f
:
R
+
→
R
as the strictly increasing function such that
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
2
f(\sqrt{xy})=\frac{f(x)+f(y)}{2}
f
(
x
y
)
=
2
f
(
x
)
+
f
(
y
)
for all positive real numbers
x
,
y
x,y
x
,
y
. Prove that there are some positive real numbers
a
a
a
where f(a)<0. (PP-nine)
4 sl12
1
Hide problems
n|a if sum 1/a_(\phi (k)})=a/b
Given a natural n>2, let
{
a
1
,
a
2
,
.
.
.
,
a
ϕ
(
n
)
}
⊂
Z
\{ a_1,a_2,...,a_{\phi (n)} \} \subset \mathbb{Z}
{
a
1
,
a
2
,
...
,
a
ϕ
(
n
)
}
⊂
Z
is the Reduced Residue System (RRS) set of modulo
n
n
n
(also known as the set of integers
k
k
k
where
(
k
,
n
)
=
1
(k,n)=1
(
k
,
n
)
=
1
and no pairs are congruent in modulo
n
n
n
). if write
1
a
1
+
1
a
2
+
⋯
+
1
a
ϕ
(
n
)
=
a
b
\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_{\phi (n)}}=\frac{a}{b}
a
1
1
+
a
2
1
+
⋯
+
a
ϕ
(
n
)
1
=
b
a
where
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
and
(
a
,
b
)
=
1
(a,b)=1
(
a
,
b
)
=
1
, then prove that
n
∣
a
n|a
n
∣
a
.(PP-nine)
2 sl11
1
Hide problems
infinite set of whose prime divisor does not exceed p_n, perfect squares
Define the sequence of positive prime numbers.
p
1
,
p
2
,
p
3
,
.
.
.
p_1,p_2,p_3,...
p
1
,
p
2
,
p
3
,
...
. Let set
A
A
A
be the infinite set of positive integers whose prime divisor does not exceed
p
n
p_n
p
n
. How many at least members must be selected from the set
A
A
A
, such that we ensures that there are
2
2
2
numbers whose products are perfect squares? (PP-nine)
2 sl9
1
Hide problems
min of a+b+c+ {3/(ab+bc+ca) if a,b,c>0 with a^2+b^2+c^2=1
Let
a
,
b
,
c
∈
R
+
a,b,c \in \mathbb{R}^+
a
,
b
,
c
∈
R
+
where
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
. Find the minimum value of .
a
+
b
+
c
+
3
a
b
+
b
c
+
c
a
a+b+c+\frac{3}{ab+bc+ca}
a
+
b
+
c
+
ab
+
b
c
+
c
a
3
(PP-nine)
1 sl8
1
Hide problems
matrices with integers
For matrices
A
=
[
a
i
j
]
m
×
m
A=[a_{ij}]_{m \times m}
A
=
[
a
ij
]
m
×
m
and
B
=
[
b
i
j
]
m
×
m
B=[b_{ij}]_{m \times m}
B
=
[
b
ij
]
m
×
m
where
A
,
B
∈
Z
m
×
m
A,B \in \mathbb{Z} ^{m \times m}
A
,
B
∈
Z
m
×
m
let
A
≡
B
(
m
o
d
n
)
A \equiv B \pmod{n}
A
≡
B
(
mod
n
)
only if
a
i
j
≡
b
i
j
(
m
o
d
n
)
a_{ij} \equiv b_{ij} \pmod{n}
a
ij
≡
b
ij
(
mod
n
)
for every
i
,
j
∈
{
1
,
2
,
.
.
.
,
m
}
i,j \in \{ 1,2,...,m \}
i
,
j
∈
{
1
,
2
,
...
,
m
}
, that's
A
−
B
=
n
Z
A-B=nZ
A
−
B
=
n
Z
for some
Z
∈
Z
m
×
m
Z \in \mathbb{Z}^{m \times m}
Z
∈
Z
m
×
m
. (The symbol
A
∈
Z
m
×
m
A \in \mathbb{Z} ^{m \times m}
A
∈
Z
m
×
m
means that every element in
A
A
A
is an integer.) Prove that for
A
∈
Z
m
×
m
A \in \mathbb{Z} ^{m \times m}
A
∈
Z
m
×
m
there is
B
∈
Z
m
×
m
B \in \mathbb{Z} ^{m \times m}
B
∈
Z
m
×
m
, where
A
B
≡
I
(
m
o
d
n
)
AB \equiv I \pmod{n }
A
B
≡
I
(
mod
n
)
only if
(
det
(
A
)
,
n
)
=
1
(\det (A),n)=1
(
det
(
A
)
,
n
)
=
1
and find the value of
B
B
B
in the form of
A
A
A
where
I
I
I
represents the dimensional identity matrix
m
×
m
m \times m
m
×
m
.(PP-nine)