Prove that if a , b , c are positive numbers and a + b + c = 1 then :
\(\left(a+\dfrac{1}{a}\right)^2+\left(b+\dfrac{1}{b}\right)^2+\left(c+\dfrac{1}{c}\right)^2>33\)
Compare two expressions A and B :
\(A=124.\left(\dfrac{1}{1.1985}+\dfrac{1}{2.1986}+\dfrac{1}{3.1987}+...+\dfrac{1}{16.2000}\right)\)
\(B=\dfrac{1}{1.17}+\dfrac{1}{2.18}+\dfrac{1}{3.19}+...+\dfrac{1}{1984.2000}\)
Consider the expression :
\(S=\dfrac{1}{2^0}+\dfrac{2}{2^1}+\dfrac{3}{2^2}+\dfrac{4}{2^3}+...+\dfrac{1992}{2^{1991}}\)
Prove that : S < 4
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Simplified :
\(x^2-1-\dfrac{x^4-3x^2-4}{x^2+1}\)
find x :
\(\left(x+1\right)\left(x-3\right)-\left(x+5\right)\left(x-5\right)\left(x-2\right)=0\)
Polynomial Analysis into Factors :
\(x^3-x^2-4x^2+8x+4\)
a , Given two real numbers x , y satisfy :
\(x^2-2y^2=xy\) ( x + y other than 0 and y other than 0 )
Calculate the value of the expression :
\(P=\dfrac{x-y}{x+y}\)
b , Find the integer (x , y) pair satisfying :
\(x^2+xy-2016x-2017y-2018=0\)
Prove that \(n^2+11n+39\) not divisible by 49 for every natural number n .
Find the maximum value of the expression :
\(P=\dfrac{3x^2+6x+11}{x^2+2x+3}\)
\(p=\dfrac{3x^2+6x+10}{x^2+2x+3}\)
Solve the following simultaneous equations .
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given the positive integers a , b , c such that \(2a^a+b^b=3c^c\) . Evaluate the expression \(P=2015^{a-b}+2016^{b-c}+2017^{c-a}\)
Given a fixed triangle ABC of height h . Find the set of points M with the sum of the distances to 3 of the triangle by the constant m ( m > h ) .
Let the triangle ABC have the area S . The points D , E , F are in the order of AB , BC , CA such that AD = DB , BE = 1 part 2 EC , CF = 1 part 2 FA . Straight segments AE , BF , CD intersect to form a triangle . Calculate the area of the triangl e.
Given a fixed ABCD rectangle. Find the set of points M so that :
a , \(MA^2+MC^2=MB^2+MD^2\)
b , \(MA+MC=MB+MD\)
Let a , b and c be positive real numbers such that \(a^2+b^2+c^2=12\) . Prove that \(\dfrac{a+b}{4+bc}+\dfrac{b+c}{4+ca}+\dfrac{c+a}{4+ab}\ge\dfrac{3}{2}\) .