Give rectangle ABCD , connect C to any point E on the BD diagonal , on ray EC , choose F so that EF = EC. Draw FH and FK perpendicular to AB and AD . Prove that :

a) Quadrilateral AHFK is a rectangle 

b) AF parellel to BD and KH parellel to AC

c) Three points E,H,K are straight .

For \(a,x,y,z\) are positive real numbers satisfy \(xyz=1\) and \(a\ge 1\). Prove that \(\dfrac{x^a}{y+z}+\dfrac{y^a}{x+z}+\dfrac{z^a}{x+y}\ge\dfrac{3}{2}\)

-Like Nesbitt's inequality, i have a problem it's a general formula of Nesbitt's inequality,  if necessary please inbox to me :)

I have some good math problems in English :D

Number 1. Let a,b be the real numbers satisfy a + b = 2. Prove that \(a^4+b^4\ge a^2+b^2\)

Number 2. Let a,b,c be the positive numbers satisfy ab + bc + ca = 1.

Prove that  : \(\dfrac{a}{\sqrt{1+a^2}}+\dfrac{b}{\sqrt{1+b^2}}+\dfrac{c}{\sqrt{1+c^2}}\le\dfrac{3}{2}\)

The numbers 1 to 6 are placed in the circle so that each side of the triangle if 1 is placed in the circle shown .Fill the number in the circles so that each side of the triangle is equal to 10

Find a 9-digit number that satisfies these condition :

- It consists of 9 digits from 1 to 9.

- The first digit is smaller than the last one.

- The sum of the digits 1 and 2 and all the ones between them is 9

................................. 2 ..... 3 ................................................... 19

................................. 3 ..... 4 .................................................... 45

................................. 4 ..... 5 .................................................... 18