In triangle $\, ABC, \,$ angle $\,A\,$ is twice angle $\,B,\,$ angle $\,C\,$ is obtuse, and the three side lengths $\,a,b,c\,$ are integers. Determine, with proof, the minimum possible perimeter.

Click for solution i think that is 77. I am certain? let (a,b,c) the sides of the triangle with minimum perimeter. suppose that gcd(a,b,c) = 1, (if exist p>1 such that a = px, b = py, c = pz, we have that (x,y,z) is other solution with x+y+z < a+b+c). we have that a² = b²(b+c), because m(A) = 2.m(B). if gcd(b,c)>1, we have gcd(a,b,c)>1. this implies that gcd(b,c) = 1. so, gcd(b,b+c) = 1 and b= m², b+c = n² and a = mn. by the sine law, we have n/m = a/b = 2cos(B). C = 180 - 3B > 90, that implies B < 30°. so, sqrt(3) < n/m < 2. is easy to see that for m = 1,2,3 this inequality no have integer solutions. so, m>=4, n>=7. the pair (m,n) = (4,7) get the solution (a,b,c) = (28,16,33).

For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that \[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \] where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \]is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

Click for solution Here's how I would do it (you can fill in the details): Proceed by (strong) induction on m. If m=1 then it's trivial (everything is 0 mod 1!) If m is composite with at least 2 prime factors, use Chinese Remainder Theorem with two relatively prime factors of m. If m is a power of 2, it's trivial (sequence eventually becomes 0 mod m) If m is a power of an odd prime p, use fact that $2^{m(1-\frac{1}{p})} \equiv 1\,(\text{mod}\, m)$, and that if a = b (mod r) and 2^r = 1 (mod m), then 2^a = 2^b (mod m).

Let $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$, where $m$ and $n$ are positive integers. Prove that $a^m + a^n \geq m^m + n^n$.

Let $\, D \,$ be an arbitrary point on side $\, AB \,$ of a given triangle $\, ABC, \,$ and let $\, E \,$ be the interior point where $\, CD \,$ intersects the external common tangent to the incircles of triangles $\, ACD \,$ and $\, BCD$. As $\, D \,$ assumes all positions between $\, A \,$ and $\, B \,$, prove that the point $\, E \,$ traces the arc of a circle.

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