MathDB

3

Part of 2005 Croatia National Olympiad

Problems(4)

sum of three Egypt fractions less than 1

Source: Croatian NMC 2005, 1st Grade

5/8/2007
If k,l,mk, l, m are positive integers with 1k+1l+1m<1\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1, find the maximum possible value of 1k+1l+1m\frac{1}{k}+\frac{1}{l}+\frac{1}{m}.
inequalities
inequality for logarit

Source: Croatian NMC 2005, 2nd Grade

5/8/2007
If a,b,ca, b, c are real numbers greater than 11, prove that for any real number rr (log⁡abc)r+(log⁡bca)r+(log⁡cab)r≥3⋅2r.(\log_{a}bc)^{r}+(\log_{b}ca)^{r}+(\log_{c}ab)^{r}\geq 3 \cdot 2^{r}.
inequalitieslogarithmsinequalities proposed
points inside a trihedral angle

Source: Croatian NMC 2005, 3rd Grade

5/8/2007
Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a fixed positive value aa.
geometry unsolvedgeometry
multiple of 2^{2005} consists of the digits 2 and 5

Source: Croatian NMC 2005, 4 th Grade

5/9/2007
Show that there is a unique positive integer which consists of the digits 22 and 55, having 20052005 digits and divisible by 220052^{2005}.
pigeonhole principleinductionnumber theory proposednumber theory