# 2014-12 Rational ratios in a triangle

Determine all triangles ABC such that all of $$\frac{AB}{BC}, \frac{BC}{CA}, \frac{CA}{AB}, \frac{\angle A}{\angle B}, \frac{\angle B}{\angle C}, \frac{\angle C}{\angle A}$$ are rational.

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# 2012-18 Diagonal

Let $$r_1,r_2,r_3,\ldots$$ be a sequence of all rational numbers in $$(0,1)$$ except finitely many numbers. Let $$r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots$$ be a decimal representation of $$r_j$$. (For instance, if $$r_1=\frac{1}{3}=0.333333\cdots$$, then $$a_{1,k}=3$$ for any $$k$$.)

Prove that the number $$0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots$$ given by the main diagonal cannot be a rational number.

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# 2010-5 Dependence over Q

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

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# 2009-9 min or max

Suppose that * is an associative and commutative binary operation on the set of rational numbers such that

1. 0*0=0
2. (a+c)*(b+c)=(a*b)+c for all rational numbers a,b,c.

Prove that either

1. a*b=max(a,b) for all rational numbers a,b, or
2. a*b=min(a,b) for all rational number a,b.
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# 2009-7 A rational problem

Let n>1 be an integer and let x>1 be a real number. Prove that if
$$\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}$$
is a rational number, then x is rational.

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Let a, b, c, d be positive rational numbers. Prove that if $$\sqrt a+\sqrt b+\sqrt c+\sqrt d$$ is rational, then each of $$\sqrt a$$, $$\sqrt b$$, $$\sqrt c$$, and $$\sqrt d$$ is rational.