Friday, December 23, 2022

Prove that the polynomial $x^{9999} + x^{8888} + x^{7777} + ... + x^{1111} + 1$ $x^9999+x^8888+x^7777+...+x^1111+1$ is divisible by $x^{9} + x^{8} + x^{7} + ... . + x + 1$ $x^9+x^8+x^7+....+x+1$

Problem: 01

Prove that the polynomial

x^{9999} + x^{8888} + x^{7777} + ... + x^{1111} + 1

$x^9999+x^8888+x^7777+...+x^1111+1$ is divisible by

x^{9} + x^{8} + x^{7} + ... . + x + 1

$x^9+x^8+x^7+....+x+1$

Solution

Thursday, December 22, 2022

Find all function $f (x)$ $f(x)$ if $(x - y) f (x + y) - (x + y) f (x - y) = 4 x y (x^{2} - y^{2})$ $(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)$

Solution

If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$

As we had: $x + \frac{1}{x} = 2$ $x+1/x=2$ $\to {(x + \frac{1}{x})}^{2} = 4$ $rightarrow(x+1/x)^2=4$ $\leftrightarrow x^{2} + \frac{1}{x^{2}} = 2$ $leftrightarrowx^2+1/x^2=2$

We continue with $(x^{2} + \frac{1}{x^{2}}) (x + \frac{1}{x}) = 4$ $(x^2+1/x^2)(x+1/x)=4$ $\leftrightarrow x^{3} + x + \frac{1}{x} + \frac{1}{x^{3}} = 4$ $leftrightarrowx^3+x+1/x+1/x^3=4$

$\leftrightarrow x^{3} + \frac{1}{x^{3}} = 2$ $leftrightarrowx^3+1/x^3=2$

Vietnamese Olympiad 2022: Find the value of $\frac{1}{a^{2023}} + \frac{1}{b^{2023}} + \frac{1}{c^{2023}}$ $1/(a^2023)+1/(b^2023)+1/(c^2023)$

If $a + b + c = 2022$ $a+b+c=2022$ and $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{2022}$ $1/a+1/b+1/c=1/2022$ Find the value of $\frac{1}{a^{2023}} + \frac{1}{b^{2023}} + \frac{1}{c^{2023}}$ $1/(a^2023)+1/(b^2023)+1/(c^2023)$

Solution

Math Book Cambodia

Pages

Friday, December 23, 2022

Prove that the polynomial $x^{9999} + x^{8888} + x^{7777} + ... + x^{1111} + 1$ $x^9999+x^8888+x^7777+...+x^1111+1$ is divisible by $x^{9} + x^{8} + x^{7} + ... . + x + 1$ $x^9+x^8+x^7+....+x+1$

Problem: 01

Solution

Thursday, December 22, 2022

Find all function $f (x)$ $f(x)$ if $(x - y) f (x + y) - (x + y) f (x - y) = 4 x y (x^{2} - y^{2})$ $(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)$

Sunday, September 11, 2022

If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$

Friday, September 9, 2022

Vietnamese Olympiad 2022: Find the value of $\frac{1}{a^{2023}} + \frac{1}{b^{2023}} + \frac{1}{c^{2023}}$ $1/(a^2023)+1/(b^2023)+1/(c^2023)$

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Friday, December 23, 2022

Prove that the polynomial x9999+x8888+x7777+...+x1111+1x^9999+x^8888+x^7777+...+x^1111+1 is divisible by x9+x8+x7+....+x+1x^9+x^8+x^7+....+x+1

Problem: 01

Solution

Thursday, December 22, 2022

Find all function f(x)f(x) if (x−y)f(x+y)−(x+y)f(x−y)=4xy(x2−y2)(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)

Sunday, September 11, 2022

If x+1x=2x+1/x=2 Find the value of x5+1x5x^5+1/x^5

Friday, September 9, 2022

Vietnamese Olympiad 2022: Find the value of 1a2023+1b2023+1c20231/(a^2023)+1/(b^2023)+1/(c^2023)

Prove that the polynomial $x^{9999} + x^{8888} + x^{7777} + ... + x^{1111} + 1$ $x^9999+x^8888+x^7777+...+x^1111+1$ is divisible by $x^{9} + x^{8} + x^{7} + ... . + x + 1$ $x^9+x^8+x^7+....+x+1$

Find all function $f (x)$ $f(x)$ if $(x - y) f (x + y) - (x + y) f (x - y) = 4 x y (x^{2} - y^{2})$ $(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2)$

If $x + \frac{1}{x} = 2$ $x+1/x=2$ Find the value of $x^{5} + \frac{1}{x^{5}}$ $x^5+1/x^5$

Vietnamese Olympiad 2022: Find the value of $\frac{1}{a^{2023}} + \frac{1}{b^{2023}} + \frac{1}{c^{2023}}$ $1/(a^2023)+1/(b^2023)+1/(c^2023)$