\(\dfrac{x}{y+z+t}=\dfrac{y}{z+t+x}=\dfrac{z}{t+x+y}=\dfrac{t}{x+y+z}\)

\(\Rightarrow\dfrac{y+z+t}{x}=\dfrac{z+t+x}{y}=\dfrac{t+x+y}{z}=\dfrac{x+y+z}{t}\)

\(\Rightarrow\dfrac{y+z+t}{x}+1=\dfrac{z+t+x}{y}+1=\dfrac{t+x+y}{z}+1=\dfrac{x+y+z}{t}+1\)

\(\Rightarrow\dfrac{x+y+z+t}{x}=\dfrac{z+t+x+y}{y}=\dfrac{t+x+y+z}{z}=\dfrac{x+y+z+t}{t}\)

Nếu \(x+y+z+t=0\Rightarrow P=-4\)

Nếu \(x+y+z+t\ne0\Rightarrow x=y=z=t\Rightarrow P=4\)

Vậy P nguyên

Ta có đẳng thức: \(a^{102}+b^{102}=\left(a^{101}+b^{101}\right)\left(a+b\right)-ab\left(a^{100}+b^{100}\right)\) với mọi số a,b

Kết hợp với: \(a^{100}+b^{100}=a^{101}+b^{101}=a^{102}+b^{102}\)

\(\Rightarrow1=\left(a+b\right)-ab\Leftrightarrow\left(a-1\right)\left(b-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a=1\Rightarrow1+b^{100}=1+b^{101}=1+b^{102}\Rightarrow b=1\\b=1\Rightarrow1+a^{100}=1+a^{101}=1+a^{102}\Rightarrow a=1\end{matrix}\right.\)

Do đó: \(P=a^{2014}+b^{2014}=1^{2004}+1^{2005}=2\)

vâng

c, Từ công thức: \(R=\rho\dfrac{1}{S}\Leftrightarrow S=\dfrac{\rho\iota}{R}=\dfrac{1,1.10^{-6}.50}{50}=1,1.10^{-6}m^2=1,1mm^2\)

d, Đèn có: \(U_{đm}=3V;P_{đm}=3W\Rightarrow I_{đm}=\dfrac{P_{dm}}{U_{dm}}=1A\)

Để đèn sáng bình thường, ta có: \(I=I_{đm}=1A;U_đ=U_{đm}=3V\)

Giá trị của biến trở: \(R_b=\dfrac{U_b}{I}=\dfrac{9}{1}=9\Omega\)

Vậy phải điều chỉnh biến trở có giá trị \(9\Omega\) thì đèn sáng bình thường

Ta có: \(2x^3-1=15\Leftrightarrow x^3=8\Rightarrow x=2\)

\(\Rightarrow\dfrac{18}{9}=\dfrac{y-25}{16}=\dfrac{z+9}{25}\Rightarrow\left\{{}\begin{matrix}\dfrac{y-25}{16}=2\Rightarrow y=57\\\dfrac{z+9}{25}=2\Rightarrow z=41\end{matrix}\right.\)

Vậy \(B=x+y+z=2+57+41=100\)

Có \(\left(x+\dfrac{1}{2}\right)^2\ge0\forall\Rightarrow\left(x+\dfrac{1}{2}\right)^2+\dfrac{5}{4}\ge\dfrac{5}{4}\forall x\)

\(A=\left[\left(x+\dfrac{1}{2}\right)^2+\dfrac{5}{4}\right]^2\ge\left(\dfrac{5}{4}\right)^2=\dfrac{25}{16}\forall x\)

Dấu "=" xảy ra \(\Leftrightarrow x=\dfrac{1}{2}\)

Vậy min \(A=\dfrac{25}{16}\Leftrightarrow x=\dfrac{-1}{2}\)

\(a,P\left(x\right)+Q\left(x\right)=x^4-x^3+3x^2+3x-1\)

\(b,H\left(x\right)=Q\left(x\right)+2x^4+2=-2x^4-x^2+x-2+2x^4+2=-x^2+x\)

\(c,H\left(x\right)=-x^2+x=x\left(1-x\right)=0\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)