Chapter 7

Prove that \((S-a)(S-b)(S-c)(S-d)>81 a b c d\) where \(S=a+b+c+d\).

If \(a^{2}+b^{2}+c^{2}=1\) and \(x^{2}+y^{2}+z^{2}=1\), show that \(a x+b y+c z \leq 1\).

If \(x\) and \(y\) are positive real numbers and \(m\) and \(n\) are any positive integers, then \(\frac{x^{n} y^{m}}{\left(1+x^{2 n}\right)\left(1+y^{2 m}\right)}>\frac{1}{4}\). (True/False)

For \(a \geq 0, b \geq 0\) and \(x>y>0\), prove that \(\left(a^{x}+b^{x}\right)^{\frac{1}{x}} \leq\left(a^{y}+b^{y}\right)^{\frac{1}{y}}\).

$$ 5 x-y=3, y^{2}-6 x^{2}=25 $$

$$ x^{4}+y^{4}=706, x+y=8 $$

$$ 3 x+4 y=18, \frac{1}{x}+\frac{1}{y}=\frac{5}{6} $$

$$ \frac{x^{2}}{y}+\frac{y^{2}}{x}=18, x+y=12 $$

$$ \left(3-\frac{6 y}{x+y}\right)^{2}+\left(3+\frac{6 y}{x-y}\right)^{2}=82,3 x+7 y=26 $$

$$ \sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{5}{2}, x+y=10 $$

$$ (x+y)^{\frac{2}{3}}+2(x-y)^{\frac{2}{3}}=3\left(x^{2}-y^{2}\right)^{\frac{1}{3}}, 3 x-2 y=13 $$

$$ x^{2}+y(x+1)=17, y^{2}+x(y+1)=13 \text { . } $$

$$ x y+3 y^{2}-x+4 y-7=0,2 x y+y^{2}-2 x-2 y+1=0 $$

$$ 3 x^{2}+x y+y^{2}=15,31 x y-3 x^{2}-5 y^{2}=45 \text { . } $$

$$ 3 x^{2}+2 y^{2}=50, x y-3 y^{2}=1 $$

$$ x+y+x y=11, x^{2} y+x y^{2}=30 $$

$$ x^{3}-y^{3}=127, x^{2} y-x y^{2}=42 \text { . } $$

$$ x^{4}+x^{2} y^{2}+y^{4}=2613, x^{2}+x y+y^{2}=67 $$

$$ \frac{1}{x^{3}}-\frac{1}{y^{3}}=91, \frac{1}{x}-\frac{1}{y}=1 $$

$$ (3 x)^{\log 3}=(4 y)^{\log 4}, 4^{\log x}=3^{\log y} $$

$$ 2^{x}+2^{y}=20, x+y=6 $$

$$ 6^{x}\left(\frac{2}{3}\right)^{y}-3 \cdot 2^{x+y}-8 \cdot 3^{x-y}+24=0, x y=2 $$

$$ \text { Find the positive solutions of the system of equations } x^{x+y}=y^{n}, y^{x+y}=x^{2 n} y^{n}, \text { where } n>0 \text { . } $$

$$ 2 x+y-2 z=0,7 x+6 y-9 z=0, x^{3}+y^{3}+z^{3}=1728 $$

$$ 9 x+y-8 z=0,4 x-8 y+7 z=0, y z+z x+x y=47 $$

$$ x(x+y+z)=4, y(x+y+z)=9, z(x+y+z)=12 $$

$$ x(y+z)=3, y(z+x)=4, z(x+y)=5 $$

$$ x y=x+y, y z=2(y+z), z x=3(z+x) $$

$$ (x+y)^{2}-z^{2}=-9,(y+z)^{2}-x^{2}=15,(z+x)^{2}-y^{2}=3 $$

$$ x y+x+y=23, x z+z+x=41, y z+y+z=27 \text { . } $$

$$ x^{2}+y^{2}+z^{2}=84, x+y+z=14, x z=y^{2} $$

$$ x+y+z=7, x y+x z=y z-2, x^{2}+y^{2}+z^{2}=21 $$

$$ (x-2)^{2}+(y-3)^{2}+(z-1)^{2}=24, x y+y z+z x=63,2 x+3 y+z=30 . $$

$$ x+y-4 x y=0, y+z-6 y z=0, z+x-8 z x=0 $$

$$ x z+y=7 z, y z+x=8 z, x+y+z=12 $$

Equation \(\ln x=x\) has how many solutions?

Equation \(x \ln x=1\) has how many solutions?

$$ e^{|x|}=|x| $$

$$ \text { Solve } 3^{x}+4^{x}=5^{x} $$

Solve \(2^{x}

Solve \(3^{x}+4^{x}>7\)

Solve \(\left[x^{2}+x+1\right]=2\) ([ ] denotes Greatest Integer Function).

Let \(\\{x\\}\) and \([x]\) denote the fractional and integral parts of a real number \(x\) respectively. Solve \(4\\{x\\}=x+[x]\).

Solve \(y=2[x]+3, y=3[x-2]+5\) ([ ] denotes Greatest Integer Function).

Solve \((x)^{2}=[x]^{2}+2 x\), where \([x]\) is the greatest integer \(\leq x\) and \((x)\) is the least integer \(\geq x\).

Solve \(\left[x^{2}\right]+2[x]-3 x=0, \quad 0 \leq x \leq 2([]\) denotes Greatest Integer Function).

Solve \(y=4-[x]^{2},[y]+y=6([]]\) denotes Greatest Integer Function).

Solve the equation \(\left[\frac{3 x-1}{4}\right]+\left[\frac{3 x+1}{4}\right]+\left[\frac{3 x-1}{2}\right]=\frac{6 x+3}{5}\) ([ ] denotes Greatest integer function).

Solve the system of equations:- \(x+[y]+\\{z\\}=1.1\) \([x]+\\{y\\}+z=2.2\) \(\\{x\\}+y+[z]=3.3\) ([ ] denotes Greatest integer function, \(\\{\) \\} denotes Fractional part function).

For every +ve integer \(n\), prove that \(\sqrt{4 n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4 n+2}\). Hence, prove that \([\sqrt{n}+\sqrt{n+1}]=[\sqrt{4 n+1}]\), where [] denotes greatest integer function.