MathDB

1

silk road special polynomials, one subset with no real root, one with real roots

f_1, f_2, \ldots, f_n

with real coefficients is called special , if for any different

i,j,k \in \{ 1,2, \ldots, n\}

polynomial

\dfrac{2}{3}f_i + f_j + f_k

has no real roots, but for any different

p,q,r,s \in \{ 1,2, \ldots, n\}

of a polynomial

f_p + f_q + f_r + f_s

there is a real root. a) Give an example of a special set of four polynomials whose sum is not a zero polynomial. b) Is there a special set of five polynomials?

1

replacing primes a,b from [2,2007] with max prime <= \sqrt{a^2-ab+b^2}

On the board are written

2 , 3 , 5 ,... , 2003

, that is, all the prime numbers of the interval

[2,2007]

. The operation of simplification is the replacement of two numbers

a , b

by a maximal prime number not exceeding

\sqrt{a^2-a b+b^2}

. First, the student erases the number

q, 2<q<2003

, then applies the simplification operation to the remaining numbers until one number remains. Find the maximum possible and minimum possible values of the number obtained in the end. How do these values depend on the number

q

?

2

1

3 circles, incircle, circle tangent to incircle passing through BC, A- excircle

Let

\omega

be the incircle of triangle

ABC

touches

BC

at point

K

. Draw a circle passing through points

B

and

C

, and touching

\omega

at the point

S

. Prove that

S K

passes through the center of the exscribed circle of triangle

A B C

, tangent to side

B C

.

3

1

Inequality

Find the max. value of

M

,such that for all

a,b,c>0

:

a^{3}+b^{3}+c^{3}-3abc\geq M(|a-b|^{3}+|a-c|^{3}+|c-b|^{3})

2007 Silk Road

Subcontests

silk road special polynomials, one subset with no real root, one with real roots

replacing primes a,b from [2,2007] with max prime <= \sqrt{a^2-ab+b^2}

3 circles, incircle, circle tangent to incircle passing through BC, A- excircle

Inequality

2007 Silk Road

Subcontests

silk road special polynomials, one subset with no real root, one with real roots

replacing primes a,b from [2,2007] with max prime &lt;= \sqrt{a^2-ab+b^2}

3 circles, incircle, circle tangent to incircle passing through BC, A- excircle

Inequality

replacing primes a,b from [2,2007] with max prime <= \sqrt{a^2-ab+b^2}