# Understand the problem

##### Source of the problem

##### Topic

##### Difficulty Level

##### Suggested Book

# Start with hints

**Notice that you can represent higher powers by smaller powers.**Suppose \( 20x + \frac{19}{x} = r \) where r is rational. Multiply both sides by x to get \( 20 x^2 = r \cdot x – 19 \) or \( x^2 = r_1 x + r_2 \) (that is divide by 20 to have r/20 as the coefficient of x and -19/20 as the x free number. both of these are rationals. therefore we name them \( r_1 \) and \(r_2 \) )

Now use the fact that \( x^5 \) is rational.

Suppose \( x^5 = r_3 \) (some rational number). Then \( x^2 \cdot x^2 \cdot x = r_3 \) But we know \( x^2 = r_1 x + r_2 \) Replacing we have \( (r_1 \cdot x + r_2 )^2 \cdot x = r_3\) Expanding we have \( r_1^2 x^3 + 2r_1r_2 \cdot x^2 + r_2^2 x = r_3 \) Again we will replace \( x^2 \) to have \( r_1^2 \cdot (r_1 x + r_2) x + 2r_1r_2 \cdot (r_1 x + r_2) + r_2^2 x = r_3 \)

Expand \( r_1^2 \cdot (r_1 x + r_2) x + 2r_1r_2 \cdot (r_1 x + r_2) + r_2^2 x = r_3 \) \( r_1^3 x^2 + r_1^2 r_2 x + 2r_1^2 r_2 x + r_2^2 x + 2 r_1 r_2^2 = r_3 \) Making one final replacement of x^2 and noting that squaring, adding, multiplying rationals gives rationals we have \( r_1^3 (r_1 x + r_2) + r_1^2 r_2 x + 2r_1^2 r_2 x + r_2^2 x + 2 r_1 r_2^2 = r_3 \) \( r_1^4 x + r_1^3r_2 + r_1^2 r_2 x + 2r_1^2 r_2 x + r_2^2 x + 2 r_1 r_2^2 = r_3 \) or we have x as a ratio of rationals.

Hence x is rational.

# Watch video

# Connected Program at Cheenta

#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.