Prove that: \(a^4+b^4+c^4+d^4\ge4abcd\)

Prove that: \(a^4+b^4+2\ge4ab\)

Prove that: \(\left(a+b+c\right)^3\ge a^3+b^3+c^3+24abc\) with a,b,c \(\ge0\)

Prove that: \(8\left(a^3+b^3+c^3\right)\ge\left(a+b\right)^3+\left(b+c\right)^3+\left(c+a\right)^3\) with a,b,c > 0

Given a + b + c = 0. Prove that: \(ab+bc+ca\le0\)

Prove that: \(a^4+b^4+c^4\ge abc\left(a+b+c\right)\)

Given a,b,c > 0. Prove that: \(a^2+b^2+c^2+2abc+1\ge2\left(ab+bc+ca\right)\)

A standard chess desk has 8*8 square tiles. How many squares are there total?

A class has 30 students. Prove that we'll find two students in that class started with a same letter

Prove that \(x,y\in\varnothing\): \(4x^2-7y^2=6\)

Prove that \(x,y\in\varnothing\): \(9x^2-8y^2=15\)

Prove that: \(\dfrac{a^2+b^2}{2}\ge\left(\dfrac{a+b}{2}\right)^2\)

Compare: \(\sqrt{8}\) and \(\sqrt{5}+1\) solve in at least 2 ways.

Given \(\dfrac{\overline{ab}}{\overline{bc}}=\dfrac{b}{c}\). Prove that: \(\dfrac{a}{b}=\dfrac{b}{c}\left(c\ne0\right)\)

Given abc = 2. Compact: \(\dfrac{a}{ab+a+2}+\dfrac{b}{bc+b+1}+\dfrac{2c}{ac+2c+2}\)

Given \(a+b+c=a^2+b^2+c^2=1\) and x:y:z = a:b:c

Prove that: \(\left(x+y+z\right)^2=x^2+y^2+z^2\)

Given \(\dfrac{xy+1}{y}=\dfrac{yz+1}{z}=\dfrac{zx+1}{x}\)

Prove that:

1) x = y = z

2) \(x^2y^2z^2=1\)

Find the last 2-digit numbers of: \(D=1978^{1986^8}\)

Find \(n\in N\) such that: \(\left(2^{3n+4}+3^{2n+1}\right)⋮19\)

Find \(n\in N\) such that: \(\left(n2^{2n}+1\right)⋮3\)