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Contests
National and Regional Contests
Thailand Contests
Mathcenter Contest
2008 Mathcenter Contest
2008 Mathcenter Contest
Part of
Mathcenter Contest
Subcontests
(10)
1
2
Hide problems
x^2+xy+y^2=57, y^2+yz+z^2=84, z^2+zx+x^2=111
Given
x
,
y
,
z
∈
R
+
x,y,z\in \mathbb{R} ^+
x
,
y
,
z
∈
R
+
, that are the solutions to the system of equations :
x
2
+
x
y
+
y
2
=
57
x^2+xy+y^2=57
x
2
+
x
y
+
y
2
=
57
y
2
+
y
z
+
z
2
=
84
y^2+yz+z^2=84
y
2
+
yz
+
z
2
=
84
z
2
+
z
x
+
x
2
=
111
z^2+zx+x^2=111
z
2
+
z
x
+
x
2
=
111
What is the value of
x
y
+
3
y
z
+
5
z
x
xy+3yz+5zx
x
y
+
3
yz
+
5
z
x
?(maphybich)
sumx/\sqrt {x + y} >= \sqrt [4]{\frac {27(yz + zx + xy)}{4}}
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be a positive real numbers. Prove that
x
x
+
y
+
y
y
+
z
+
z
z
+
x
≥
27
(
y
z
+
z
x
+
x
y
)
4
4
\frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac { z}{\sqrt {z + x}}\geq\sqrt [4]{\frac {27(yz + zx + xy)}{4}}
x
+
y
x
+
y
+
z
y
+
z
+
x
z
≥
4
4
27
(
yz
+
z
x
+
x
y
)
(dektep)
10
1
Hide problems
5 questions, each options each , 2000 candidates
One test is a multiple choice test with
5
5
5
questions, each with
4
4
4
options,
2000
2000
2000
candidates, each choosing only one answer for each item.Find the smallest possible integer
n
n
n
that gives a student's answer sheet the following properties: In the student's answer sheet
n
n
n
, there are four sheets in it. Any two of the four tiles have exactly the same three answers.(tatari/nightmare)
9
1
Hide problems
p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x)
Set
P
P
P
as a polynomial function by
p
n
(
x
)
=
∑
k
=
0
n
−
1
x
k
p_n(x)=\sum_{k=0}^{n-1} x^k
p
n
(
x
)
=
∑
k
=
0
n
−
1
x
k
. a) Prove that for
m
,
n
∈
N
m,n\in N
m
,
n
∈
N
, when dividing
p
n
(
x
)
p_n(x)
p
n
(
x
)
by
p
m
(
x
)
p_m(x)
p
m
(
x
)
, the remainder is
p
i
(
x
)
,
∀
i
=
0
,
1
,
.
.
.
,
m
−
1.
p_i(x),\forall i=0,1,...,m-1.
p
i
(
x
)
,
∀
i
=
0
,
1
,
...
,
m
−
1.
b) Find all the positive integers
i
,
j
,
k
i,j,k
i
,
j
,
k
that make
p
i
(
x
)
+
p
j
(
x
2
)
+
p
k
(
x
4
)
=
p
100
(
x
)
.
p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).
p
i
(
x
)
+
p
j
(
x
2
)
+
p
k
(
x
4
)
=
p
100
(
x
)
.
(square1zoa)
8
3
Hide problems
sum [(1+ab)/(1+a)]^ {2008}>=4 if a,b,c,d>0 with abcd=1
Let
a
,
b
,
c
,
d
∈
R
+
a,b,c,d \in R^+
a
,
b
,
c
,
d
∈
R
+
with
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
(
1
+
a
b
1
+
a
)
2008
+
(
1
+
b
c
1
+
b
)
2008
+
(
1
+
c
d
1
+
c
)
2008
+
(
1
+
d
a
1
+
d
)
2008
≥
4
\left(\frac{1+ab}{1+a}\right)^{2008}+\left(\frac{1+bc}{1+b}\right)^{2008}+\left(\frac{1+cd }{1+c}\right)^{2008}+\left(\frac{1+da}{1+d}\right)^{2008} \geq 4
(
1
+
a
1
+
ab
)
2008
+
(
1
+
b
1
+
b
c
)
2008
+
(
1
+
c
1
+
c
d
)
2008
+
(
1
+
d
1
+
d
a
)
2008
≥
4
(dektep)
Goblin Tribe game
Once upon a time, there was a tribe called the Goblin Tribe, and their regular game was ''The ATM Game (Level Giveaway)'' . The game stats with a number of Goblin standing in a circle. Then the Chieftain assigns a Level to each Goblin, which can be the same or different (Level is a number which is a non-negative integer). Start play by selecting a Goblin with Level
k
k
k
(
k
≠
)
.
0
k \not=). 0
k
=
)
.0
) comes up. Let's assume Goblin
A
A
A
. Goblin
A
A
A
explodes itself. Goblin A's Level becomes
0
0
0
. After that, Level of Goblin
k
k
k
next to Goblin
A
A
A
clockwise gets Level
1
1
1
. Prove that: 1.) If after that Goblin
k
k
k
next to Goblin
A
A
A
explodes itself and keep doing this,
k
′
k'
k
′
next to that Goblin clockwise explodes itself. Prove that the level of each Goblin will be the same again. 2) 2.) If after that we can choose any Goblin whose level is not
0
0
0
to explode itself. And keep doing this. Prove that no matter what the initial level is, we can make each level the way we want. But there is a condition that the sum of all Goblin's levels must be equal to the beginning.(gools)
\sum d(A_i,A_j)<10^{-2008}
Prove that there are different points
A
0
,
A
1
,
⋯
A
2550
A_0 \,\, ,A_1 \,\, , \cdots A_{2550}
A
0
,
A
1
,
⋯
A
2550
on the
X
Y
XY
X
Y
plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If
d
(
A
i
,
A
j
)
d(A_i,A_j)
d
(
A
i
,
A
j
)
represents the distance between
A
i
,
A
j
A_i\,\, , A_j
A
i
,
A
j
then
∑
0
≤
i
<
j
≤
2550
{
d
(
A
i
,
A
j
)
}
<
1
0
−
2008
\sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}
0
≤
i
<
j
≤
2550
∑
{
d
(
A
i
,
A
j
)}
<
1
0
−
2008
Note :
{
x
}
\{x \}
{
x
}
represents the decimal part of x e.g.
{
3.16
}
=
0.16
\{ 3.16\} = 0.16
{
3.16
}
=
0.16
. (passer-by)
7
3
Hide problems
(DEF) <=\frac{EF^2}{4 AD^2} if AFDE is cyclic and (ABC)=1
A
B
C
ABC
A
BC
is a triangle with an area of
1
1
1
square meter. Given the point
D
D
D
on
B
C
BC
BC
, point
E
E
E
on
C
A
CA
C
A
, point
F
F
F
on
A
B
AB
A
B
, such that quadrilateral
A
F
D
E
AFDE
A
F
D
E
is cyclic. Prove that the area of
D
E
F
≤
E
F
2
4
A
D
2
DEF \le \frac{EF^2}{4 AD^2}
D
EF
≤
4
A
D
2
E
F
2
.(holmes)
\sigma(p^2)=\sigma(q^b)
For every positive integer
n
n
n
,
σ
(
n
)
\sigma(n)
σ
(
n
)
is equal to the sum of all the positive divisors of
n
n
n
(for example,
σ
(
6
)
=
1
+
2
+
3
+
6
=
12
\sigma(6)=1+2+3+6=12
σ
(
6
)
=
1
+
2
+
3
+
6
=
12
) . Find the solution of the equation
σ
(
p
2
)
=
σ
(
q
b
)
\sigma(p^2)=\sigma(q^b)
σ
(
p
2
)
=
σ
(
q
b
)
where
p
p
p
and
q
q
q
are primes where p>q and
b
b
b
are positive integers.(gools)
arithmetic sequence a_n with a_1 prime factor >=i
Let
n
,
d
n,d
n
,
d
be natural numbers. Prove that there is an arithmetic sequence of positive integers.
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
with common difference of
d
d
d
and
a
i
a_i
a
i
with prime factor greater than or equal to
i
i
i
for all values
i
=
1
,
2
,
.
.
.
,
n
i=1,2,...,n
i
=
1
,
2
,
...
,
n
.(nooonuii)
6
2
Hide problems
x^5-y^2=4 diophantine
Find the total number of integer solutions of the equation
x
5
−
y
2
=
4
x^5-y^2=4
x
5
−
y
2
=
4
(Erken)
for even a, exist infinite n such that n|a^n+1
For even positive integers
a
>
1
a>1
a
>
1
. Prove that there are infinite positive integers
n
n
n
that makes
n
∣
a
n
+
1
n | a^n+1
n
∣
a
n
+
1
.(tomoyo-jung)
5
3
Hide problems
P_{2008}(2008) =? P_n(x)=P_{n-1}(x)+P_{n-1}(x-1), P_1(x)=2/x
Let
P
1
(
x
)
=
1
x
P_1(x)=\frac{1}{x}
P
1
(
x
)
=
x
1
and
P
n
(
x
)
=
P
n
−
1
(
x
)
+
P
n
−
1
(
x
−
1
)
P_n(x)=P_{n-1}(x)+P_{n-1}(x-1)
P
n
(
x
)
=
P
n
−
1
(
x
)
+
P
n
−
1
(
x
−
1
)
for every natural
n
n
n
greater than
1
1
1
. Find the value of
P
2008
(
2008
)
P_{2008}(2008)
P
2008
(
2008
)
.(Mathophile)
a,b,c of 6 irrational exists usch that a+b,b+c,c+a are irrationals
There are
6
6
6
irrational numbers. Prove that there are always three of them, suppose
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
+
b
a+b
a
+
b
,
b
+
c
b+c
b
+
c
,
c
+
a
c+a
c
+
a
are irrational numbers.(Erken)
sum 1/(a^2+1) >= 3/2 if ab+bc+ca = 3
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers where
a
b
+
b
c
+
c
a
=
3
ab+bc+ca = 3
ab
+
b
c
+
c
a
=
3
. Prove that
1
a
2
+
1
+
1
b
2
+
1
+
1
c
2
+
1
≥
3
2
.
\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\geq\dfrac{3} {2}.
a
2
+
1
1
+
b
2
+
1
1
+
c
2
+
1
1
≥
2
3
.
(dektep)
4
3
Hide problems
(a^2+b^2+c^2)/(a+b+ c) is an integer if (a\sqrt{3}+b)/(b\sqrt3+c) is rational
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive integers that
a
3
+
b
b
3
+
c
\frac{a\sqrt{3}+b}{b\sqrt3+c}
b
3
+
c
a
3
+
b
is a rational number, show that
a
2
+
b
2
+
c
2
a
+
b
+
c
\frac{a^2+b^2+c^2}{a+b+ c}
a
+
b
+
c
a
2
+
b
2
+
c
2
is an integer.(Anonymous314)
AD + DX - (BC + CX) = 8 , trapezoid ABCD
The trapezoid
A
B
C
D
ABCD
A
BC
D
has sides
A
B
AB
A
B
and
C
D
CD
C
D
that are parallel
D
A
B
^
=
6
∘
\hat{DAB} = 6^{\circ}
D
A
B
^
=
6
∘
and
A
B
C
^
=
4
2
∘
\hat{ABC} = 42^{\circ}
A
BC
^
=
4
2
∘
. Point
X
X
X
lies on the side
A
B
AB
A
B
, such that
A
X
D
^
=
7
8
∘
\hat{AXD} = 78^{\circ}
A
X
D
^
=
7
8
∘
and
C
X
B
^
=
6
6
∘
\hat{CXB} = 66^{\circ}
CXB
^
=
6
6
∘
. The distance between
A
B
AB
A
B
and
C
D
CD
C
D
is
1
1
1
unit . Prove that
A
D
+
D
X
−
(
B
C
+
C
X
)
=
8
AD + DX - (BC + CX) = 8
A
D
+
D
X
−
(
BC
+
CX
)
=
8
units.(Heir of Ramanujan)
1/(p^n+q^n+1) + 1/(q^n+r^n+1)+ 1/(r^n+p^n+ 1) <=1 if pqr=1
Let
p
,
q
,
r
∈
R
+
p,q,r \in \mathbb{R}^+
p
,
q
,
r
∈
R
+
and for every
n
∈
N
n \in \mathbb{N}
n
∈
N
where
p
q
r
=
1
pqr=1
pq
r
=
1
, denote
1
p
n
+
q
n
+
1
+
1
q
n
+
r
n
+
1
+
1
r
n
+
p
n
+
1
≤
1
\frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+ 1} \leq 1
p
n
+
q
n
+
1
1
+
q
n
+
r
n
+
1
1
+
r
n
+
p
n
+
1
1
≤
1
(Art-Ninja)
3
2
Hide problems
remainder from division sum_{i=1}^{2006} x_i^2 with 2551
Set
M
=
{
1
,
2
,
⋯
,
2550
}
M= \{1,2,\cdots,2550\}
M
=
{
1
,
2
,
⋯
,
2550
}
and
min
A
,
max
A
\min A ,\ \max A
min
A
,
max
A
represents the minimum and maximum values of the elements in the set
A
A
A
. For
k
∈
{
1
,
2
,
⋯
2006
}
k \in \{1,2,\cdots 2006\}
k
∈
{
1
,
2
,
⋯
2006
}
define
x
k
=
1
2008
(
∑
A
⊂
M
:
n
(
A
)
=
k
(
m
i
n
A
+
max
A
)
)
x_k = \frac{1}{2008} \bigg (\sum_{A \subset M : n(A)= k} (\ min A + \max A) \, \bigg)
x
k
=
2008
1
(
A
⊂
M
:
n
(
A
)
=
k
∑
(
min
A
+
max
A
)
)
. Find remainder from division
∑
i
=
1
2006
x
i
2
\sum_{i=1}^{2006} x_i^2
∑
i
=
1
2006
x
i
2
with
2551
2551
2551
.(passer-by)
(absin2C+bcsin2A+casin2B){ab+bc+ca)<=\sqrt3/2
Let
A
B
C
ABC
A
BC
be a triangle whose side lengths are opposite the angle
A
,
B
,
C
A,B,C
A
,
B
,
C
are
a
,
b
,
c
a,b,c
a
,
b
,
c
respectively. Prove that
a
b
sin
2
C
+
b
c
sin
2
A
+
c
a
sin
2
B
a
b
+
b
c
+
c
a
≤
3
2
\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}
ab
+
b
c
+
c
a
ab
sin
2
C
+
b
c
sin
2
A
+
c
a
sin
2
B
≤
2
3
.(nooonuii)
2
3
Hide problems
f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y), f(2008) =f(-2008)
Find all the functions
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
which satisfy the functional equation
f
(
x
y
2
)
+
f
(
x
2
y
)
=
y
2
f
(
x
)
+
x
2
f
(
y
)
f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)
f
(
x
y
2
)
+
f
(
x
2
y
)
=
y
2
f
(
x
)
+
x
2
f
(
y
)
for every
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
and
f
(
2008
)
=
f
(
−
2008
)
f(2008) =f(-2008)
f
(
2008
)
=
f
(
−
2008
)
(nooonuii)
polynomial wanted, If P(a)=0 then P(a|a|)=0
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
which have the properties: 1)
P
(
x
)
P(x)
P
(
x
)
is not a constant polynomial and is a mononic polynomial. 2)
P
(
x
)
P(x)
P
(
x
)
has all real roots and no duplicate roots. 3) If
P
(
a
)
=
0
P(a)=0
P
(
a
)
=
0
then
P
(
a
∣
a
∣
)
=
0
P(a|a|)=0
P
(
a
∣
a
∣
)
=
0
(nooonui)
AX=AY wanted, touchpoints of incircle related
In triangle
A
B
C
ABC
A
BC
(
A
B
≠
A
C
AB\not= AC
A
B
=
A
C
), the incircle is tangent to the sides of
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
respectively. Let
A
D
AD
A
D
meet the incircle again at point
P
P
P
, let
E
F
EF
EF
and the line passing through the point
P
P
P
and perpendicular to
A
D
AD
A
D
intersect at
Q
Q
Q
. Let
A
Q
AQ
A
Q
intersect
D
E
DE
D
E
at
X
X
X
and
D
F
DF
D
F
at
Y
Y
Y
. Prove that
A
X
=
A
Y
AX=AY
A
X
=
A
Y
.(tatari/nightmare)