Given a,b,c,d satisfy a + b = c + d , a² + b² = c² + d² .

Show that : a²⁰¹⁸ + b²⁰¹⁸ = c²⁰¹⁸ + d²⁰¹⁸

Given 2 straight lines AB and CD

Show that : If  \(AB\perp CD\) then AC² - AD² = BC² - BD²

Find m,n are positive integer satisfy 

Given square ABCD (AB = BC = CD = DA = a) . E is a point on AD ( E \(\ne\) A). Bisector of \(\angle EBA,\angle EBC\) cut DA,DA at M,N

a) Show that : \(BE\perp MN\)

b) Where does point E on AD for value of S_DMN maximum?

Show that :

\(\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+....+\dfrac{1}{2009^3}< \dfrac{1}{4}\)

Solve this disequation :

 \(\dfrac{x-2}{4}-\dfrac{2}{3}>\dfrac{5x-9}{12}\)

Given isosceles trapezoid ABCD, BD is the large bottom, O is the intersection of 2 diagonal line, \(\angle BOC=60^0\) . Call I,M,N,P,Q in order is midpoint of BC,OA,OB,AB,CD. Prove that :

a) \(DM\perp AC\)

b) \(\Delta MNP\) is an equilateral triangle

c) \(\angle MQP=\angle QND=\angle NMC\)

d) Prove that : Intersection H of 3 height in triangle MNQ and 2 point I,O are on a line .

An triangle has height and median divide the angle at the top into 3 equal angles. Calculate all angle of that triangle .

Given quadrilateral ABCD has \(\angle B+\angle C=180^0\) and BC = CD . Prove that AC is the bisector of \(\angle A\).

P/s : Solve by 2 ways

Prove that : 4ⁿ + 15n - 10 \(⋮\) 9 , \(n\in N\)

Prove that with n is natural number that :

a) a² - a \(⋮\) 2    ;    a³ - a \(⋮\) 3     ;     a⁴ - a \(⋮\) 4

b)  a³ - a \(⋮\) 6    ;    a³ - 7a \(⋮\)  6    ;    a³ + 11a \(⋮\)  6

With x + y + z = 0

Show that : x³ + x²z + y²z - xyz + y³ = 0

Given expression B = (x² + 1)(y² + 1) - (x + 4)(x - 4) - (y - 5)(y + 5) 

Prove that : B \(\ge\) 42 \(\forall x,y\)  . Which value of x,y that B = 42 .

Given x,y \(\in\) Z . Show that :

a) If A = 5x + y \(⋮\) 19 then B = 4x - 3y \(⋮\) 19

b) C = 4x + 3y \(⋮\) 13 then D = 7x + 2y \(⋮\) 13 

Given a + b + c = 2009

Show that : \(\dfrac{a^3+b^3+c^3-3abc}{a^2+b^2+c^2-ab-bc-ca}=2009\)

Can or not if three needles of the clock is located on a straight line . 

Calculate A know :

\(A=\dfrac{\left(1+\dfrac{1}{4}\right)\left(3^4+\dfrac{1}{4}\right)\left(5^4+\dfrac{1}{4}\right)....\left(29^4+\dfrac{1}{4}\right)}{\left(2^4+\dfrac{1}{4}\right)\left(4^4+\dfrac{1}{4}\right)\left(6^4+\dfrac{1}{4}\right).....\left(30^4+\dfrac{1}{4}\right)}\)

Given right isosceles triangle ABC at A. On AB,AC take D,E so that AD = AE. Line through D \(\perp BE\) and cut BC at I , Line through A \(\perp BE\) cut BC at K. Call M is the intersection of AK and CD.

a) Show that : \(\Delta ABE=\Delta ACD\)

b) \(\Delta MAC\) isosceles

c) M is the midpoint of CD , K is the midpoint of  IC

d) Call G is the intersection of DK and IM , MK cut GC at F.

Show that : FM = FK

By denote t = x - 3

Solve that equation : (x - 1)³ + (x - 4)³ = 8