**A. 623.** Let \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) be three distinct positive reals. The *logarithmic mean *of \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) is defined by

\(\displaystyle L(a,b,c) = 2\left(
\frac{a}{(\ln a-\ln b)(\ln a-\ln c)} +
\frac{b}{(\ln b-\ln c)(\ln b-\ln a)} +
\frac{c}{(\ln c-\ln a)(\ln c-\ln b)} \right).\)

Prove that \(\displaystyle \sqrt[3]{abc} < L(a,b,c) < \frac{a+b+c}{3}\).

(5 points)

**A. 624.** \(\displaystyle a)\) Prove that for every infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) there exists some \(\displaystyle C>0\) such that for every positive integer \(\displaystyle r\) there are positive integers \(\displaystyle n\), \(\displaystyle m\) satisfying \(\displaystyle |n-m|\ge r\) and \(\displaystyle |x_n-x_m|<\frac{C}{|n-m|}\).

\(\displaystyle b)\) Show that for every \(\displaystyle C>0\) there exists an infinite sequence \(\displaystyle x_1,x_2,\ldots\in[0,1]\) and a positive integer \(\displaystyle r\) such that \(\displaystyle |x_n-x_m|>\frac{C}{|n-m|}\) holds true for every pair \(\displaystyle n\), \(\displaystyle m\) of positive integers with \(\displaystyle |n-m|\ge r\).

(CIIM6, Costa Rica)

(5 points)

**A. 625.** Let \(\displaystyle n\ge2\), and let \(\displaystyle \mathcal{S}\) be a family of some subsets of \(\displaystyle \{1,2,\ldots,n\}\) with the property that \(\displaystyle |A\cup B\cup C\cup D|\le n-2\) for all \(\displaystyle A,B,C,D\in\mathcal{S}\). Show that \(\displaystyle |\mathcal{S}|\le 2^{n-2}\).

(CIIM6, Costa Rica)

(5 points)

**B. 4653.** How many ordered triples of positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) are there for which \(\displaystyle [a,b,c]=10!\) and \(\displaystyle (a,b,c)=1\)? (\(\displaystyle (a,b,c)\) denotes the greatest common divisor, and \(\displaystyle [a,b,c]\) denotes the least common multiple.)

(4 points)

**C. 1250.** The sides of a triangle are \(\displaystyle a=2t-1\), \(\displaystyle b=t^2-1\), \(\displaystyle c=t^2-t+1\), where \(\displaystyle t>1\) is a real number. Prove that the radius of the inscribed circle of the triangle is \(\displaystyle (t-1)\frac{\sqrt 3}2\).

(5 points)

This problem is for grade 11 - 12 students only.