2011 Silk Road

Part of Silk Road

Subcontests

(4)

1

infinite primes of form m^2+mn+n^2 whereas m,n integers

Prove that there are infinitely many primes representable in the form

m^2+mn+n^2

for some integers

m,n

1

equal angles in an isosceles triangle, line passes thought midpoint of base

Given an isosceles triangle

ABC

AB

K

is taken on the extension of the side

AC

(beyond the point

C

\angle KBC = \angle ABC

S

the intersection point of angle - bisectors of

\angle BKC

\angle ACB

AB

KS

intersect at point

L

BS

CL

intersect at point

M

. Prove that line

KM

passes through the midpoint of the segment

BC

1

SRMC 2011 P1

Determine the smallest possible value of

| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |

A_{1}, A_{2}, A_{3}, A_{4}, A_{5}

sets simultaneously satisfying the following conditions:

(i)

| A_{i}\cap A_{j} | = 1

1\leq i < j\leq 5

, i.e. any two distinct sets contain exactly one element in common;

(ii)

A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing

1\leq i<j<k<l\leq 5

, i.e. any four different sets contain no common element. Where

| S |

means the number of elements of

S

1

Ineq

For all a,b,c\in \bb{R}^+ such that

a+b+c=1

( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc

M