Step 1: Change 534 to base 10

534₆ = 5 x 6² + 3 x 6¹ + 4 x 6⁰ = 202

Step 2: Change 202 to base 4

202 = 3 x 4³ + 0 x 4² + 2 x 4¹ + 2 x 4⁰ = 3022₄

Answer: Because a prisoner is always telling the truth and the other person always says so, when the prisoner asks through the prison guards in turn, the answer is always wrong. The question is: Please ask him where is the exit and tell me what his answer is? Prisoners will always get the wrong answer and go in the opposite direction to the answer.

\(\dfrac{3-x}{5}+\dfrac{x+y}{10}=\dfrac{x-y}{4}\)

\(\Leftrightarrow\dfrac{12-4x}{20}+\dfrac{2x+2y}{20}=\dfrac{5x-5y}{20}\)

\(\Leftrightarrow12-4x+2x+2y=5x-5y\)

\(\Leftrightarrow2y+5y=5x+4x-2x-12\)

\(\Leftrightarrow7y=7x-12\Leftrightarrow y=x-\dfrac{12}{7}\)

3−x5+x+y10=x−y4

⇔12−4x20+2x+2y20=5x−5y20

⇔12−4x+2x+2y=5x−5y

⇔2y+5y=5x+4x−2x−12

⇔7y=7x−12⇔y=x−127

There are 6 ways she can arrange them:

Way 1: red, pink, yellow

Way 2: red, yellow, pink

Way 3: pink, red, yellow

Way 4: pink, yellow, red

Way 5: yellow, red, pink

Way 6: yellow, pink, red

Cách 1: đỏ, hồng, vàng

Cách 2: đỏ, vàng, hồng

Cách 3: màu hồng, đỏ, vàng

Cách 4: màu hồng, vàng, đỏ

Cách 5: vàng, đỏ, hồng

Cách 6: vàng, hồng, đỏ

There are 6 ways she can arrange them:

Way 1: red, pink, yellow

Way 2: red, yellow, pink

Way 3: pink, red, yellow

Way 4: pink, yellow, red

Way 5: yellow, red, pink

Way 6: yellow, pink, red

Example:  they're \(\angle B=\angle M;\angle C=\angle Q;\angle A=\angle P\) but they aren't same. This is Similar triangles

Let Oa, Ob, Oc, Od, Oe be the rays in the picture. We have 8 angles with difference degree measures :

\(\widehat{aOb}=10^0;\widehat{bOc}=20^0;\widehat{cOd}=30^0;\widehat{dOe}=50^0\)

\(\widehat{cOe}=80^0;\widehat{aOd}=60^0;\widehat{bOe}=100^0;\widehat{aOe}=110^0\)

The number of pages in her notes is: 

40 . 240 = 9600 [word]

The total speed of writing of Chanel and Marcus is:

36 + 54 = 90 [word/minute]

Time for Chanel and Marcus to finish writing the book is:

9600 : 90 = 106m40s 

Answer: 106minutes 40seconds

Dao Trong Luan must round the answer to 107 minutes

We have the formula to find the number of 3-digit integers: \(x=a+1\)

So the number of 3-digit integers can be formed is: 

\(\left(1+2+3+4+5+6+7+8+9\right)+\left(1\cdot9\right)=54\left(integers\right)\)

ANSWER: 54 integers.

The price of a box of pens now costs: 100% - 10%  90%

The percentage of the pens in the box now is: 100% - 25% = 75%

The percent change in the cost per pen is: \(\left(90\%:75\%\right)-100\%=20\%\)

ANSWER: 20%