MathDB

1

BB' < CC' wanted when <IBA > <ICA and <IBC > <ICB

ABC

be a triangle and a point

I

lies inside the triangle. Assume

\angle IBA > \angle ICA

and

\angle IBC >\angle ICB

. Prove that, if extensions of

BI

,

CI

intersect

AC

,

AB

at

B'

,

C'

respectively, then

BB' < CC'

. b) Let

ABC

be a triangle with

AB < AC

and angle bisector

AD

. Prove that for every point

I, J

on the segment

[AD]

and

I \ne J

, we always have

\angle JBI > \angle JCI

.
c) Let

ABC

be a triangle with

AB < AC

and angle bisector

AD

. Choose

M, N

on segments

CD

and

BD

, respectively, such that

AD

is the bisector of angle

\angle MAN

. On the segment

[AD]

take an arbitrary point

I

(other than

D

). The lines

BI

,

CI

intersect

AM

,

AN

at

B', C'

. Prove that

BB' < CC'

.

9

1

9 queens and 1 king on a 8x8 board, no queen may capture another

On a board with

64

(

8 \times 8

) squares, find a way to arrange

9

queens and

1

king so that every queen cannot capture another queen.

10

1

every rational x=(a^3_1 + a^3_2 + ... + a^3_m)/(b^3_1 + b^3_2 + ... + b^3_m)

a) Prove that for any positive integer

m > 2

, the equation

y^3 = x^3_1 + x^3_2 + ... + x^3_m

always has a positive integer solution. b) Given a positive integer

n > 1

and suppose

n \ne 3

. Prove that every rational number

x > 0

can be expressed as

x =\frac{a^3_1 + a^3_2 + ... + a^3_n}{b^3_1 + b^3_2 + ... + b^3_n}

where

a_i, b_i

(i = 1, . . . , n)

are positive integers.

11

1

f(P(x) + f(y)) = y + Q(f(x)), f(P(x_0)) = Q(f(x_0))

Given

P

a real polynomial with degree greater than

1

. Find all pairs

(f,Q)

with function

f : R \to R

and the real polynomial

Q

satisfying the following two conditions: i) for all

x, y \in R

, we have

f(P(x) + f(y)) = y + Q(f(x))

. ii) there exists

x_0 \in R

such that

f(P(x_0)) = Q(f(x_0))

.

3

1

a_1 = a_2 = ...= a_m if a_1^n+... +a_m^n has finite no of prime divisors

Given positive integers

a_1

,

a_2

,

...

,

a_m

(

m \ge 1

). Consider the sequence

\{u_n\}_{n=1}^{\infty}

, with

u_n = a_1^n + a_2^n + ... + a_m^n.

We know that this sequence has a finite number of prime divisors. Prove that

a_1 = a_2 = ...= a_m

.

2

1

sum and product of 2 whites is white, coloring positive integers bw

Positive integers are colored in black and white. We know that the sum of two numbers of different colors is always black, and that there are infinitely many numbers that are white. Prove that the sum and product of two white numbers are also white numbers.

1

x + y + z >= 2/d \sqrt{(a + d)(b + d)(c + d)} if ax + by + cz = xyz

Let

a, b, c

be three positive real numbers. a) Prove that there exists a unique positive real number

d

that satisfies

\frac{1}{a + d}+ \frac{1}{b + d}+\frac{1}{c + d}=\frac{2}{d} .

b) With

x, y, z

being positive real numbers such that

ax + by + cz = xyz

, prove the inequality

x + y + z \ge \frac{2}{d}\sqrt{(a + d)(b + d)(c + d)}.

12

1

(x^2 + 1)(y^2 + 1)(z^2 + 1) >= c_k (x + y + z)^k

a) Find all real numbers

k

such that there exists a positive constant

c_k

satisfying

(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k

for all positive real numbers.
b) Given the numbers

k

found, determine the largest number

c_k

.

6

1

<BIE =<DCF wanted, IE//AB, IF// AD, cyclic ABCD

For a given cyclic quadrilateral

ABCD

, let

I

be a variable point on the diagonal

AC

such that

I

and

A

are on the same side of the diagonal

BD

. Assume

E,F

lie on the diagonal

BD

such that

IE\parallel AB

and

IF\parallel AD

. Show that

\angle BIE =\angle DCF

5

1

no of positive integers in form ap+bq with p+q<=2005

Let

a,b

be positive integers . How many integers numbers can be written in the form

ap+bq

, where

p,q

are nonnegative integers and

p+q\le{2005}

?

7

1

A hard functional equation

Find all function

f:[0,\infty )\to\mathbb{R}

such that

f

is monotonic and

[f(x)+f(y)]^2=f(x^2-y^2)+f(2xy)

for all

x\geq y\geq 0

8

1

Ineq

If a,b,c>0, prove that:

\frac{1}{a\sqrt{(a+b)}}+\frac{1}{b\sqrt{(b+c)}}+\frac{1}{c\sqrt{(c+a)}} \geq \frac{3}{\sqrt{2abc}}

thank u for ur help :oops:

VMEO II 2005

Subcontests

BB' < CC' wanted when <IBA > <ICA and <IBC > <ICB

9 queens and 1 king on a 8x8 board, no queen may capture another

every rational x=(a^3_1 + a^3_2 + ... + a^3_m)/(b^3_1 + b^3_2 + ... + b^3_m)

f(P(x) + f(y)) = y + Q(f(x)), f(P(x_0)) = Q(f(x_0))

a_1 = a_2 = ...= a_m if a_1^n+... +a_m^n has finite no of prime divisors

sum and product of 2 whites is white, coloring positive integers bw

x + y + z >= 2/d \sqrt{(a + d)(b + d)(c + d)} if ax + by + cz = xyz

(x^2 + 1)(y^2 + 1)(z^2 + 1) >= c_k (x + y + z)^k

<BIE =<DCF wanted, IE//AB, IF// AD, cyclic ABCD

no of positive integers in form ap+bq with p+q<=2005

A hard functional equation

Ineq

VMEO II 2005

Subcontests

BB' &lt; CC' wanted when &lt;IBA &gt; &lt;ICA and &lt;IBC &gt; &lt;ICB

9 queens and 1 king on a 8x8 board, no queen may capture another

every rational x=(a^3_1 + a^3_2 + ... + a^3_m)/(b^3_1 + b^3_2 + ... + b^3_m)

f(P(x) + f(y)) = y + Q(f(x)), f(P(x_0)) = Q(f(x_0))

a_1 = a_2 = ...= a_m if a_1^n+... +a_m^n has finite no of prime divisors

sum and product of 2 whites is white, coloring positive integers bw

x + y + z &gt;= 2/d \sqrt{(a + d)(b + d)(c + d)} if ax + by + cz = xyz

(x^2 + 1)(y^2 + 1)(z^2 + 1) &gt;= c_k (x + y + z)^k

&lt;BIE =&lt;DCF wanted, IE//AB, IF// AD, cyclic ABCD

no of positive integers in form ap+bq with p+q&lt;=2005

A hard functional equation

Ineq

BB' < CC' wanted when <IBA > <ICA and <IBC > <ICB

x + y + z >= 2/d \sqrt{(a + d)(b + d)(c + d)} if ax + by + cz = xyz

(x^2 + 1)(y^2 + 1)(z^2 + 1) >= c_k (x + y + z)^k

<BIE =<DCF wanted, IE//AB, IF// AD, cyclic ABCD

no of positive integers in form ap+bq with p+q<=2005