Regonial Spanish Olympiad

These are some of the Regional Spanish Mathematical Olympiad

1 We have a rectangle of 2n2 * n squares, in nn 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 1xy+1yz+1zx=162\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} = \frac{1}{6^{2}}xyz+yxz+zxy=7262\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy} = \frac{7^{2}}{6^{2}}1(xy)2+1(yz)2+1(zx)2=72362\frac{1}{(xy)^{2}}+\frac{1}{(yz)^{2}}+ \frac{1}{(zx)^{2}} = \frac{7^{2}}{36^{2}}

3 For every positive integer n1n \geq 1 we denote an=n4+n2+1a_{n} = n^4 + n^2 + 1. Find the greatest common divisor of an a_{n} and an+1a_{n+1} in function of nn.

Note by Jordi Bosch
4 years, 6 months ago

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I like question 2!

Calvin Lin Staff - 4 years, 6 months ago

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picture pictureFor the square ABCDABCD we have three choices.For the square BCFEBCFE we have two choices.Similarly,for the square GDCHGDCH we have two choices.Now,for the square HCIFHCIF we have two choices or one choice depending on the fact whether the squares,GDCHGDCH and BCFEBCFE have the same colour or different colours.WHAT TO DO NOW?

Adarsh Kumar - 4 years, 6 months ago

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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.

Jordi Bosch - 4 years, 6 months ago

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For 1. I got 23n 2*3^n ,

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

for 3. I got n2+n+1 n^2 + n + 1 if 7∤2n+1 7 \not | 2n+1 and 7(n2+n+1) 7(n^2 + n + 1) if 72n+1 7|2n+1

Siddhartha Srivastava - 4 years, 6 months ago

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Under which condition does 72n+17|2n+1?

Jordi Bosch - 4 years, 6 months ago

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When 2n+10(mod7) 2n + 1 \equiv 0 \pmod7 or 2n6(mod7) 2n \equiv 6 \pmod7 or n3(mod7) n \equiv 3 \pmod7

Siddhartha Srivastava - 4 years, 6 months ago

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@Siddhartha Srivastava Nicely done! that is if n3(mod7)n \equiv 3 \pmod{7} then the greatest common divisor of ana_{n} and an+1a_{n+1} is 7(n2+n+1)7(n^2+n+1), else the gcd between ana_{n} and an+1a_{n+1} is (n2+n+1)(n^2+n+1).

Jordi Bosch - 4 years, 6 months ago

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

A Brilliant Member - 4 years, 5 months ago

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

akash deep - 4 years, 6 months ago

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Correct :)

Krishna Sharma - 4 years, 6 months ago

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Are there any other solutions?

Calvin Lin Staff - 4 years, 6 months ago

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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,-3,6)

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

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

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

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

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

Krishna Sharma - 4 years, 6 months ago

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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

akash deep - 4 years, 5 months ago

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I want learn these sums with enthusiasm.

Anuvind Shrivastava - 4 years, 6 months ago

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