Given quadrilateral ABCD , \(\widehat{A}=\widehat{B}=90^0\) , \(\widehat{C}=2\widehat{D}\)

a) Calculate : \(\widehat{C}\) and \(\widehat{D}\)

b) If AC = 2BC. Prove that \(\Delta ACD\) is the equilateral triangle .

This is an ancient Mathematic problem , perhaps some people already knew this :

Prove that : For every natural number n> 4, fraction \(\dfrac{4}{n}\) equals 3 fractions have a numerator of 1 .

Solve the equation :

-a²b - ab² - a²x - b²x + ax² + bx² = 0 

Given \(x=\dfrac{b^2+c^2-a^2}{2bc}\) ; \(y=\dfrac{a^2-\left(b-c\right)^2}{\left(b+c\right)^2-a^2}\)

Calculator : \(P=x+y+xy\)

Given acute triangle ABC and AA' ; BB' ; CC' perpendicular at BC ; AC ; AB 

Which triangle ABC could be to \(\dfrac{\left(AB+BC+CA\right)^2}{AA'+BB'+CC'}\) reach the smallest value.

Given \(x\ne y\ne z\) and \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=0\)

Calculator A know :

A = \(\dfrac{yz}{x^2+2yz}+\dfrac{xz}{y^2+2xz}+\dfrac{xy}{z^2+2xy}\)

Prove that with n is a positive number : \(\sqrt[3]{\left(n+1\right)^2}-\sqrt[3]{n^2}< \dfrac{2}{3\sqrt[3]{n}}< \sqrt[3]{n^2}-\sqrt[3]{\left(n-1\right)^2}\)

Given a,b,c are positive real numbers so that abc = 1

Prove that : \(\dfrac{b+c}{\sqrt{a}}+\dfrac{c+a}{\sqrt{b}}+\dfrac{a+b}{\sqrt{c}}\ge\sqrt{a}+\sqrt{b}+\sqrt{c}+3\)

Good luck :) 

Given two positive real numbers x,y so that \(x+y\ge10\) 

Find the smallest value of P , know P = \(2x+y+\dfrac{30}{x}+\dfrac{5}{y}\)

Prove that with all values of x,y 

So that : Value of A = 4x(x + y)(x + y + z)(x + z) + y²z² is non-negative

Prove that :) with n \(\in\) Z , so that :

n².(n⁴ - 1) \(⋮\) 60 

Perfect :v 

***/ Give a,b,c > 0

Prove that : \(\dfrac{abc}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\le\dfrac{\left(a+b\right)\left(a+b+2c\right)}{\left(3a+3b+2c\right)^2}\le\dfrac{1}{8}\)

Prove that : With a,b,c,d are positive real number so that :

\(\sqrt{\left(a^2+c^2\right)\left(b^2+d^2\right)}+\sqrt{\left(a^2+d^2\right)\left(b^2+d^2\right)}\ge\left(a+b\right)\left(c+d\right)\)

Give a + b + c + d = 0.
Be write : a² + b² + c² + d² by total of three cube.

Given triangle ABC isosceles square at C, M is a point anywhere on AB. Draw MR \(\perp\) AC. MS \(\perp\) BC

a) Prove that: CM and RS intersect at the midpoint each way

b) Call O is the midpoint of AB , which triangle ORS is ? Why ?

Given triangle ABC isosceles square at B. From D of AB , draw DE \(\perp\) AC at E, ED ray cut CB at F. Call M,N,P,Q are midpoints of AD , DF , FC , CA.

Prove that : MNPQ is the square 

( IMO . Shorlist 1990 )

Supports a,b,c,d are non-negetive numbers so that ab + bc + cd + da = 1

Prove that : \(\dfrac{a^3}{b+c+d}+\dfrac{b^3}{a+c+d}+\dfrac{c^3}{a+b+d}+\dfrac{d^3}{a+b+c}\ge\dfrac{1}{3}\)

Give x + y = 3 and x² + y² = 5 .

Calculate : x³ + y³

2. Prove that , with erery real numbers x,y so that expressions are square numbers :

a) A = (x + y)(x + 2y)(x + 3y)(x + 4y) + y⁴

b) B = x(x + 1)(x + 2)(x + 3) + 1 

c) C = 4x(x + y)(x + y + z)(x + z) + y²z²

1. Find the smallest value of these expressions :

A = x² + 3x + 7

B = 2x² + 9y² - 6xy - 6x - 12y + 2004