MathDB

2

1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)

a_1, a_2, a_3

be three distinct real numbers. Define

\begin{cases} b_1=\left(1+\dfrac{a_1a_2}{a_1-a_2}\right)\left(1+\dfrac{a_1a_3}{a_1-a_3}\right) \\ \\ b_2=\left(1+\dfrac{a_2a_3}{a_2-a_3}\right)\left(1+\dfrac{a_2a_1}{a_2-a_1}\right) \\ \\ b_3=\left(1+\dfrac{a_3a_1}{a_3-a_1}\right)\left(1+\dfrac{a_3a_2}{a_3-a_2}\right) \end {cases}

Prove that

1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)

When does equality hold?

a/b+b/c+c/a and a/c+b/a+c/b are both integers

Let

a, b, c

be nonzero integers such that

M = \frac{a}{b}+\frac{b}{c}+\frac{c}{a}

and

N =\frac{a}{c}+\frac{b}{a}+\frac{c}{b}

are both integers. Find

M

and

N

.

6

2

distance between two 5-digit numbers

Define the distance between two

5

-digit numbers

\overline{a_1a_2a_3a_4a_5}

and

\overline{b_1b_2b_3b_4b_5}

to be the largest integer

j

such that

a_j \ne b_j

. (Example: the distance between

16523

and

16452

is

5

.) Suppose all

5

-digit numbers are written in a line in some order. What is the minimal possible sum of the distances of adjacent numbers in that written order?

no 2 of them have the property that one divides other, in 7-digit numbers

Consider all the

7

-digit numbers formed by the digits

1,2 , 3,...,7

each digit being used exactly once in all the

7!

numbers. Prove that no two of them have the property that one divides the other.

5

2

N points, total number of pairwise distances of these N points is at most n

There are

N

points in the plane such that the total number of pairwise distances of these

N

points is at most

n

. Prove that

N \le (n + 1)^2

.

concurrent Euler lines

Let

P

be an interior point of triangle

ABC

such that

\angle BPC = \angle CPA =\angle APB = 120^o

. Prove that the Euler lines of triangles

APB,BPC,CPA

are concurrent.

4

2

circumradius of MIN is twice that of ABC

Let

BE

and

CF

be the bisectors of

\angle B

and

\angle C

of a triangle

ABC

whose incentre is

I

. Suppose

EF

, extended, meets the circumcircle of

ABC

in

M,N

. Show that the circumradius of

MIN

is twice that of

ABC

.

A_1 \cup A_2 \cup ... \cup A_n may be colored with 2 colors

Let

A_1,A_2,...,A_n

be

n

finite subsets of a set

X, n \ge 2

, such that (i)

|A_i| \ge 2, 1 \le i \le n

, (ii)

|A_i \cap A_j | \ne 1, j \le i < j \le n

. Prove that the elements of

A_1 \cup A_2 \cup ... \cup A_n

may be colored with

2

colors so that no

A_i

is colored by the same color.

3

2

if 4^n + 2^n + 1 is a prime then n = 3^k

Suppose

n

is a natural number such that

4^n + 2^n + 1

is a prime. Prove that

n = 3^k

for some nonnegative integer

k

.

a^{2007} = a + 1 and b^{4014} = b + 3a

Let

a

and

b

be two positive real numbers such that

a^{2007} = a + 1

and

b^{4014} = b + 3a

. Determine whether

a > b

or

b > a

.

1

2

sum |PA|^n + |PB|^n + |PC|^n + |PD|^n is constant wrt point P

Let

P

be a point on the circumcircle of a square

ABCD

. Find all integers

n > 0

such that the sum

S_n(P) = |PA|^n + |PB|^n + |PC|^n + |PD|^n

is constant with respect to the point

P

.

<APM + <BPC = 180^o , isosceles, < PAB =<PBC

Let

ABC

be an isosceles triangle with

AC = BC

, and let

M

be the midpoint of

AB

. Let

P

be a point inside the triangle such that

\angle PAB =\angle PBC

. Prove that

\angle APM + \angle BPC = 180^o

.

2007 Postal Coaching

Subcontests

1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)

a/b+b/c+c/a and a/c+b/a+c/b are both integers

distance between two 5-digit numbers

no 2 of them have the property that one divides other, in 7-digit numbers

N points, total number of pairwise distances of these N points is at most n

concurrent Euler lines

circumradius of MIN is twice that of ABC

A_1 \cup A_2 \cup ... \cup A_n may be colored with 2 colors

if 4^n + 2^n + 1 is a prime then n = 3^k

a^{2007} = a + 1 and b^{4014} = b + 3a

sum |PA|^n + |PB|^n + |PC|^n + |PD|^n is constant wrt point P

<APM + <BPC = 180^o , isosceles, < PAB =<PBC

2007 Postal Coaching

Subcontests

1 + |a_1b_1+a_2b_2+a_3b_3| \le (1+|a_1|) (1+|a_2|)(1+|a_3|)

a/b+b/c+c/a and a/c+b/a+c/b are both integers

distance between two 5-digit numbers

no 2 of them have the property that one divides other, in 7-digit numbers

N points, total number of pairwise distances of these N points is at most n

concurrent Euler lines

circumradius of MIN is twice that of ABC

A_1 \cup A_2 \cup ... \cup A_n may be colored with 2 colors

if 4^n + 2^n + 1 is a prime then n = 3^k

a^{2007} = a + 1 and b^{4014} = b + 3a

sum |PA|^n + |PB|^n + |PC|^n + |PD|^n is constant wrt point P

&lt;APM + &lt;BPC = 180^o , isosceles, &lt; PAB =&lt;PBC

<APM + <BPC = 180^o , isosceles, < PAB =<PBC