35*) Given \(\Delta ABC\). Prove that: \(S_{ABC}\le\dfrac{\sqrt{3}}{12}\left(AB^2+BC^2+CA^2\right)\)

34) Solve the equations:
\(\left\{{}\begin{matrix}\sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2\\\sqrt{7y}\left(1-\dfrac{1}{x-y}\right)=4\sqrt{2}\end{matrix}\right.\)

Find the area of a right triangle at A with perimeter 72 cm. The difference between the median line and the height derived from A is 7 cm.

Given \(A=\left(x-2y\right)^2+\left(x-3\right)^2+\left(y-1\right)^2+3\). The least possible value of A is:...

Given a polynomial f(x) such that \(x^3+2x^2\left(4y-1\right)-4xy^2-f\left(x\right)=-5^3+4xy\left(2x-y\right)\). The value of \(f\left(\dfrac{f\left(-5\right)}{5^2}\right)\)=...

How many liters of a 92 - octane gasoline should be mixed with 200 liters of a 98 - octane gasoline to produce a mixture that is 96 - octane gasoline.

Let the weight triangle ABC (CA = CB), the height BD. On the edges BA, BC take the two points E and F respectively so that BE = BF = BD. Via E, draw a straight line parallel to AC cut BC at N, cut BD at K. Pass F draw a straight line parallel to AC cut AB at M, cut BD at I.

Calculate the lengths of sides AB and BC if you know EM = 9cm, FN = 12cm and IK = 6cm.

Compact

A = \(\left(a+b+c\right)^2+\left(a+b-c\right)^2-2\left(a-b\right)^2\)

In the figure, two regular pentagons have been attached to a regular hexagon to create a 12-gon. If each regular polygon has side length 3 feet, what is the perimeter of the resulting 12-gon?

Of 1000 people surveyed, one-third of the 630 people who reported owning a cat also own a dog. If each person surveyed owns a cat, a dog or both, how many own a dog?