Demonstrate inequality :

\(C=\dfrac{1}{2!}+\dfrac{5}{3!}+\dfrac{11}{4!}+...+\dfrac{n^2+n-1}{\left(n+1\right)!}< 2\) ( n positive integer )

Demonstrate inequalities :

\(\dfrac{1}{3}.\dfrac{4}{6}.\dfrac{7}{9}.\dfrac{10}{12}...\dfrac{208}{210}< \dfrac{1}{25}\)

Solve the following equation : 

( x + 3 ) ^ 4 + ( x + 5 ) ^ 4 = 16

For three numbers a , b , c other 0 satisfy : a + b + c different 0 and \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}\)  .

Prove that : \(\dfrac{1}{a^{2005}}+\dfrac{1}{b^{2005}}+\dfrac{1}{c^{2005}}=\dfrac{1}{a^{2005}+b^{2005}+c^{2005}}\)

For xyzt = 24 . Calculate :

\(P=\dfrac{6x}{xyz+3xy+6x+6}+\dfrac{4y}{yzt+4yz+12y+24}+\dfrac{4x}{ztx+zt+4z+12}+\dfrac{2t}{txy+2tx+2t+8}\)

Solve the following equation :

a , \(\dfrac{5x-150}{50}+\dfrac{5x-102}{49}+\dfrac{5x-56}{48}+\dfrac{5x-12}{47}+\dfrac{5x-660}{46}=0\)

b , \(\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2011}+\dfrac{1}{2012}\right)503x=1+\dfrac{2014}{2}+\dfrac{2015}{3}+...+\dfrac{4023}{2011}+\dfrac{4024}{2012}\)

c , \(\left(\dfrac{8}{1.9}+\dfrac{8}{9.17}+\dfrac{8}{17.25}+...+\dfrac{8}{49.57}\right)+\dfrac{58}{57}+2\left(x-1\right)=\dfrac{2x+7}{3}+\dfrac{5x-8}{4}\)

For the ABCD parallelogram the area is equal 120 \(cm^2\) . Let E be the midpoint of AB , I is the intersection of DE and AC . Calculate \(S_{AIE}\) .

Polynomial Analysis into Factors :

\(A=x^4+6x^3+7x^2-6x+1\)

Prove that for every natural number m, n then :

\(x^{6m+4}+x^{6m+2}+1\) divide by \(x^4+x^2+1\)

For polynomials \(F\left(x\right)\) have integer coefficients . Know that \(F\left(0\right)\) , \(F\left(1\right)\) are odd numbers . We all know that polynomials \(F\left(x\right)\) there is no solution .

Find the largest value and the smallest value of :

\(K=\dfrac{x+1}{x^2+3}\)

For quadrilateral ABCD , E is the intersection of AB and CD , F is intersection of AD and BC , I and K are respectively the midpoint of BD and AC .
a , The points M belong to the inner domain of the quadrilateral and have the property

\(S_{MAB}+S_{MCD}=\dfrac{1}{2}S_{ABCD}\) on what road ?

b , Call N is the midpoint of EF . Prove that the points I , K , N are linear .

Prove that n is the natural number we always have :

\(\dfrac{1}{5}+\dfrac{1}{13}+\dfrac{1}{25}+...+\dfrac{1}{n^2+\left(n+1\right)^2}< \dfrac{1}{2}\)

Prove that in every triangle we always have :

\(60< \dfrac{aA+bB+cC}{a+b+c}< 90\)  where A, B, C are the angles of triangle and a, b, c are edges opposite to that angle .

Solve the equation : \(n+S\left(n\right)=1982\) with \(S\left(n\right)\) is the set of digits of n ( n is a non-negative integer ) .

Question 1: Give triangle ABC in A . Call H , K is the midpoint of BC and AC respectively
a) Prove that the ABHK quadrilateral is trapezoid .
b) On the opposite beam of HA , take the point E such that H is the midpoint of AE . Proving the ABEC quadrilateral is a diamond .
c) Draw HN is the high line of the triangle AHB . Let I be the midpoint of AN , on the opposite beam of BH , taking the point M such that B is the midpoint of MH. Prove MN perpendicular to HI.

The sum of the six non - negative integers is equal to their product . Find those numbers .

perform calculation :

\(\left(1-\dfrac{1}{1+2}\right)\left(1-\dfrac{1}{1+2+3}\right)...\left(1-\dfrac{1}{1+2+3+...+1986}\right)\)

Prove that if x + y = 1 and xy are different from 0 then :

\(\dfrac{y}{x^3-1}-\dfrac{x}{y^3-1}=\dfrac{2\left(x-y\right)}{x^2y^2+3}\)