These are some of the Regional Spanish Mathematical Olympiad

1 We have a rectangle of \(2 * n\) squares, in \(n\) columns of 2 squares each. We have 3 colors to paint the squares, and we want that every two squares that share a side don't have the same color. How many diferent ways are there?

2 Find the real solutions of \[\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} = \frac{1}{6^{2}}\]\[\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy} = \frac{7^{2}}{6^{2}}\]\[\frac{1}{(xy)^{2}}+\frac{1}{(yz)^{2}}+ \frac{1}{(zx)^{2}} = \frac{7^{2}}{36^{2}}\]

3 For every positive integer \(n \geq 1\) we denote \(a_{n} = n^4 + n^2 + 1\). Find the greatest common divisor of \( a_{n}\) and \(a_{n+1}\) in function of \(n\).

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## Comments

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TopNewestI like question 2!

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picture

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The problem quickly follows if you consider that a row is already painted and you study the possibilities for next row.

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For 1. I got \( 2*3^n \),

for 2. I got \( (x,y,z) = (-2,-3,6) \) and permutations and

for 3. I got \( n^2 + n + 1 \) if \( 7 \not | 2n+1 \) and \( 7(n^2 + n + 1) \) if \( 7|2n+1 \)

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Under which condition does \(7|2n+1\)?

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When \( 2n + 1 \equiv 0 \pmod7 \) or \( 2n \equiv 6 \pmod7 \) or \( n \equiv 3 \pmod7 \)

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can u elaborate the answer for the first question , please?

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for 2 i got the solutions: (x.y.z)=( -2,-3,6)

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Correct :)

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Are there any other solutions?

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We can make a cubic polynomial with x,y,z as root, we get can get 6 pairs of x,y,z(3 roots all real) by interchanging roots

\((-2,-3,6)\)

\((-2, 6, -3)\)

\((-3,-2,6)\)

\((-3,6,-2)\)

\((6,-2,-3)\)

\((6,-3,-2)\)

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no as @krishna sharma says we can obtain the values for x+y+z, xy+yz+xz and xyz and then declare a polynomial with roots x,y,z and place the coefficient values which will be x+y+z, xy+yz+xz and xyz and then use factor theorem to get the result

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I want learn these sums with enthusiasm.

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