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)

Hey, stop copying the solutions 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

3x + y = 10y + x

=> 2x = 9y 

=> y = 2; x = 9 => x + y = 11

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:

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

Sorry D = 0

6996 is after 6969

1111 is before 1961, maybe?

Four times the number is four 

So the number 1 is the correct answer

37 × 9 = 333

Sorry, choose c

Q = P × 5/4

\(\Rightarrow\) 4Q × 5P = 5P × 5P = (5P)²

Choose d

A + B = Z

Z + P = T

\(\Rightarrow\)A + Z + P + B = Z + T

\(\Rightarrow\)A + 8 = T or A = T - 8

c. X cannot fit inside Y

Can the lines be arcs? Because I can't see the King says it is impossible!

I drawed just  you!

Kar Nola Roz = move four apples (1)

Kir Roz Pala = sell four plates (2)

=> Roz = four

=> plates = Kir/Pala

But Insa Kar Kir = carefully move plates (3)

=> Kir = plates

=> Kar = move (from 1)

=> Nola = apples 

d. B will fit into D

82888

17071

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?

March

Now it is geography!!!

742 (miles/hour)
 

THINK

\(\rightarrow\) THANK

\(\rightarrow\) THARK

\(\rightarrow\) SHARK

\(\rightarrow\) STARK

\(\rightarrow\) START

Are you going to change Math to English?

HOPE IS THE MOST EXCITING THING THERE IS IN LIFE.

-MANDY MOOSE

I KNTHI HATT YAEBM FI MOWEN DNA LDICHNER REWE NI AGERCH, EW UODLW ETG MESOEREWH.
 —SEMAJ URBRETH

\(\Rightarrow\) I THINK THAT MAYBE IF WOMEN AND CHILDREN WERE IN CHANGE, WE WOULD GET SOMEWHERE

- JAMES THURBER

OMG! This is word puzzle, not Math!

1 quarter + 16 dimes + 3 nickels = $200

Female students: \(\dfrac{4}{5}\)

Male students: \(\dfrac{1}{5}\)

The ratio is: \(\dfrac{1}{5}\div\dfrac{4}{5}=\dfrac{1}{4}=1:4\)

First 3 fractions:

2 x 3 = 6 (the 5^th)

3 x 3 = 9 (the 6^th)

4 x 3 = 12 (the 7^th)

Continue with the next 3 fractions:

5 x 6 = 30 (the 8^th)

6 x 6 = 36 (the 9^th)

9 x 6 = 54 (the 10^th)

So \(\dfrac{1}{19}\) is incorrect and should be replaced with \(\dfrac{1}{30}\)

We can also put dot (.): 6.7