Algebra (Thailand Math POSN 1st elimination round 2014)

Way too easier than last year.

Write the full solution.

1.) Let $a,b,c,d$ be real numbers such that $\displaystyle \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{a}$. Find the maximum value of $\displaystyle \frac{ab-3bc+ca}{a^{2}-b^{2}+c^{2}}$.

2.) If $(x+y+z)\left(\displaystyle \frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) = 1$ , prove that $(x+y)(y+z)(z+x) = 0$ .

3.) Find all roots of $\displaystyle \left\lfloor \frac{x}{2} \right\rfloor - \left\lfloor \frac{x}{3} \right\rfloor = \frac{x}{7}$ .

4.) (Someone posted this problem before in Brilliant.) Find all polynomials $P(x)$ with real coefficients such that $P(1) = 210$ and

$(x+10)P(2x) = (8x-32)P(x+6)$ .

5.) If $x,y,z$ are positive real numbers such that $\displaystyle x+\frac{y}{z} = y+\frac{z}{x} = z+\frac{x}{y} = 2$ . Find the value of $x+y+z$ .

Check out all my notes and stuffs for more math problems!

Thailand Math POSN 2013

Thailand Math POSN 2014

#Algebra

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Is the first answer -1??

A Former Brilliant Member - 6 years, 7 months ago

Nope there's another case of a,b,c,d you didn't consider.

Samuraiwarm Tsunayoshi - 6 years, 7 months ago

For1st I got 3

5th - answer 3

Krishna Sharma - 6 years, 7 months ago

@Krishna Sharma – Yep, but how will you prove that no.5 has exactly 1 answer?

Samuraiwarm Tsunayoshi - 6 years, 7 months ago

@Samuraiwarm Tsunayoshi – We can apply AM-GM first on $x,y,z$ and then on $\frac{x}{y},\frac{y}{z},\frac{z}{x}$ and add both and say that since it is at its minimum so there is equality between the terms.

Aneesh Kundu - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$