Give a,b,c,d are positive integer numbers so that \(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+d}+\dfrac{d}{d+a}=2\)

Prove that :

abcd is the square number 

Find x know :

|x - 1| + |2x - 4| = 3

With a,b,c are positive real numbers so that a + b + c = 1 .

Find Max_S know :

\(S=6\left(ab+bc+ca\right)+a\left(a-b\right)^2+b\left(b-c\right)^2+c\left(c-a\right)^2\)

Polynomial analysis into factor multiplication :

x - x² + x³ - x⁴ 

Give x + y = 1. 

Find smallest value of F know :

F = x² + y² + xy 

For triangle ABC (AB <AC). Let d be the centroid of BC. Draw D symmetric to A through d. Call O is the intersection of d with AC.

Prove that : d is the symmetry axis of the quadrilateral ABCD

Find x if :

a) \(\left(x-1\right)\left(3-x\right)>0\)

b) xy = x + y 

Give a,b,c,d are real numbers and abcd = 1

Prove that : \(a^3+b^3+c^3+d^3\ge a+b+c+d\)

Give a,b,c are positive real number and a² + b² + c² = 3

Find Max_P know :

P = a + b + c - abc 

Give a,b,c \(\ge\) 0 

Prove that : \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{9}{a+b+c}\)

Prove Nesbitt inequality :

Give a,b,c are positive real numbers , then , we have :

\(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\ge\dfrac{3}{2}\)

P/s : Use AM - GM inequality

Prove that : 

\(2^{2x+1}+3^{2x+1}⋮5\)

\(\forall x\in N\)

Compact :

A = 5 + 5³ + 5⁵ + ....... + 5²⁰¹⁵ + 5²⁰¹⁷    

Give \(\Delta ABC\) with AM is the median line .

Prove that : \(AM\le\dfrac{AB+AC}{2}\)

Give a,b,c > 0

Prove that : \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)

Give a,b,c > 0 and abc = 1 .

Prove that :  \(\Sigma\dfrac{a\left(3a+1\right)}{\left(a+1\right)^2}\ge3\)

Give x,y > 0 , x² + y² = 1

Prove that : \(P=\left(1+x\right)\left(1+\dfrac{1}{y}\right)+\left(1+y\right)\left(1+\dfrac{1}{x}\right)\ge4+3\sqrt{2}\)

P/s : By inequality AM-GM 

Give x,y,z,a,b,c are positive real number .

Prove that : \(\dfrac{x^2+1}{yz+b}+\dfrac{y^2+b}{xz+c}+\dfrac{z^2+c}{xy+a}\ge3\)

Prove this :

For every natural number (n) greater than 4, fraction 4 / n is equal to the sum of 3 different Egyptian fractions.

Give \(\Delta ABC\) have \(\widehat{ABC}=30^0\) and \(\widehat{BAC}=130^0\). Call Ax is the opposite ray of AB , the bisectrix of \(\widehat{ABC}\)cut bisectrix of \(\widehat{CAx}\)at D, Ray BA cut CD at E. Compare the long of AC and CE