Find pairs of integers (x,y) so that x² + 102 = y²

Find the residual polynomial of division :

(x¹⁰⁵ + x⁹⁰ + x⁷⁵ + ..... + x¹⁵ + 1) : (x² - 1)

Solve the equation :

(x + 3)⁴ - (x - 3)⁴ - 24x³ = 108 

Compact : 

P = 12(5² + 1)(5⁴ + 1)(5⁸ + 1)(5¹⁶ + 1)

Given x,y satisfy:  x² + 2xy + 6x + 6y + 2y² + 8 = 0 

Find minimum and maximum value of B = x + y + 2016 

Polynomial analysis into multiplication

a) x⁴ + 4

b) 4x⁴ + 81 

c) 4x⁴ - 9x² + 81 

Polynomial analysis into multiplication

a) x⁵ + x⁴ + 1

b) x⁸ + x⁴ + 1

Given rhombus ABCD , know AB = BC = CD = DA = 4 (centimeter) and 1 of 4 angles , there is a angle that it equals to 30 degrees 

Find the area of ABCD 

Given Parallelogram ABCD. M,N are midpoints of AB,CD. Call E is the intersection of AN and DM , F is the intersection of BN and CM

Show that :

a) \(S_{\Delta MCD}=\dfrac{1}{2}.S_{ABCD}\)

b) \(S_{\Delta MCD}=S_{\Delta NAB}\)

c) \(S_{\Delta EDN}+S_{\Delta FCN}=S_{\Delta EAM}+S_{\Delta FBM}\)

Given Parallelogram ABCD, O is the intersection of AC and BD .

Show that :

a) \(S_{\Delta ABD}=S_{\Delta ABC}=S_{\Delta ACD}=S_{\Delta BCD}\)

b) \(S_{\Delta OAD}=S_{\Delta OBC}=S_{\Delta OCD}=S_{\Delta OAB}\)

Given a,b are positive number satisfy a + b \(\le\) 1

Find minimum value of S know :

S = ab + \(\dfrac{1}{ab}\)

Given x,y satisfy 16x² - 9y² \(\ge\) 144 .

Show that : |2x - y + 1| \(\ge\) \(2\sqrt{5}-1\)

Given quadrilateral ABCD , E is the midpoint of BD

a) Show the ratio of : \(\dfrac{S_{ABE}}{S_{ADE}};\dfrac{S_{ABE}}{S_{ABD}}\)

b) Show that : \(S_{ABCE}=\dfrac{1}{2}.S_{ABCD}\)

Can or can't if a triangle be divided into 5 equal triangles ?

Prove that :

a) If n is the sum of two square numbers so that 2n is the sum of two square numbers too.

b) If 2n is the sum of two square numbers so that n is the sum of two square numbers too. 

Prove that :

a) a² + 2(a + 1)² + 3(a + 2)² + 4(a + 3)² = (a + 5)² + (3a + 5)²

b) (a + b)(a - b) + (b + c)(b - c) = (a + c)(a - c)

c) (ax + by)² + (ay - bx)² = (a² + b²)(x² + y²)

Given a,b,c,d are positive real numbers , find minimum value of :

\(M=\dfrac{a-d}{d+b}+\dfrac{d-b}{b+c}+\dfrac{b-c}{c+a}+\dfrac{c-a}{a+d}\)

Given a,b,c,d > 0 , Find the smallest value of A know :

A = \(\dfrac{a}{b+c+d}+\dfrac{b}{c+d+a}+\dfrac{c}{a+b+d}+\dfrac{d}{a+b+c}+\dfrac{b+c+d}{a}+\dfrac{c+d+a}{b}+\dfrac{a+b+d}{c}+\dfrac{a+b+c}{d}\)

2/ Find the value of these equations :

A = 3xy(4x² - y²) + 4xy(y² - 3x²)

B = x¹⁰ - 10x⁹ + 10x⁸ - 10x⁷ + ......... - 10x + 10 with x = 9 

1/ Compact : 

A = -2(3x - 2y) - 5(2y - 3x)

b) (4x² - 3y)2y - (3x² - 4y)3y