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

33) Solve the equations:
\(\left\{{}\begin{matrix}x^2+2xy-y^2=2\\x^2+xy+y^2=3\end{matrix}\right.\)

32) Solve the equation (And you will know the surprising age of Tran Anh):

\(\sqrt{x-89}+\sqrt{91-x}=x^2-180x+8102\)

(P/s: Please do this math problem by yourself, step-to-step, and don't try to copy from the Internet, please)

32) Let x,y > 0 such that x + y \(\ge\) 10.

Find the minimum value of \(P=2x+y+\dfrac{30}{x}+\dfrac{5}{y}\)

31) Let x,y are 2 positive real integers such that \(\sqrt{xy}\cdot\left(x-y\right)=x+y\)

Find the minimum value of \(P\) = x + y

30) Let a, b, c are the length of a triangle. Prove that:

\(\dfrac{ab}{a+b-c}+\dfrac{bc}{b+c-a}+\dfrac{ca}{c+a-b}\ge a+b+c\)

29) Find all positive prime number x such that x² + 2^x is a prime number

28) Given x² + 5y² - 4xy - x + 2y - 6 = 0

Find the max and min values of A = x - 2y

27) Given x, y, z \(\in Z\) and

a = x² - yz                             c = z² - xy 

b = y² - xz

Prove that: a³ + b³ + c³ - 3abc is a square number.

26) Find x, y, z such that:

\(\left\{{}\begin{matrix}x+y+xy=8\\y+z+yz=15\\z+x+zx=35\end{matrix}\right.\)

25) Two old men A and B met on the bus. A asked B if he could guess exactly the ages of A's sons and gave B clues:

     A: I have three sons. The product of their ages is 36 and the sum of their ages is the number of the bus we are sitting in.

     B: Of course I know the number of the bus, but I can't still guess their ages. Can you give one more detail?

     A: The biggest child has blue eyes

     B: Now I get it! Their ages are...

Knowing that A and B are very intelligent. Can you explain how B found their ages and tell me what is the number of the bus?

24) Each letters M, A, T, H presents a number. Knowing that: MATH × 4 = HTAM, what is MATH?

23) There are two numbers: one has 6 divisors and one has 4 divisors. What is the minimum divisors of their product?

22) Tran Anh knows he is too old, weak and gonna die soon. So, in his will, he left all his money for his sons and daughters:

- $1000 for the first one and 10% of the remaining

- $2000 for the second one and 10% of the remaining

- $3000 for the third one and 10% of the remaining, and so on

Knowing that after dividing the money, they realise that they get the same amount of money. Find out how much money and how many children do old Tran have?

(After you find out the answer, you will unfold his surprising, unbelievable untolds of Tran Anh)

21) Find the smallest prime number \(\overline{abc}\) such that a, \(\overline{ab}\), \(\overline{abc}\), c, \(\overline{cb}\), \(\overline{cba}\) are also prime number

20) In a class, 14 students collect stamps, 16 students collect postcards, 5 students collect both and 4 students collect none of them. How many students are there in this class? 

19) When asked "is 1 dollar equal to 1 cent?", Tran Anh confidently answered "YES" and proved the equality like this:

As we all know: 1$ = 100c

\(\Leftrightarrow\) \(\dfrac{1}{10}\)$ = 10c

\(\Leftrightarrow\) \(\left(\dfrac{1}{10}\right)^2\)$ = 10²c

\(\Leftrightarrow\) \(\dfrac{1}{100}\)$ = 100c = 1$

\(\Leftrightarrow\) 1c = 1$ (?!??!)

SO please, can you point out the mistake(s) for Tran Anh?

p/s: Tran Anh is too old (542 years old) so he is a doting person 

18) On the island of Nevermind, some people are liars who always lie. The remaining habitants of the island are truthlovers who tell only the truth. Three habitants of the island, A, B, and C met this morning.

A said: “All of us are liars”.

B said: “Only one of us is a truthlover”.

Who is the truthlover?

17) The digits 1, 2, 3, 4, 5, and 6 are each placed in one of the letters so that the multiplication below is correct.

\(\overline{ab}\times c=\overline{def}\)

What is the digit represented by “c” ?