The nonnegative real numbers $a$ and $b$ satisfy $a + b = 1$. Prove that: $$\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1$$ When do we have equality in the right inequality and when in the left inequality? Proposed by Walther Janous

. In the isosceles triangle $ABC$ with $AC = BC$ we denote by $D$ the foot of the altitude through $C$. The midpoint of $CD$ is denoted by $M$. The line $BM$ intersects $AC$ in $E$. Prove that the length of $AC$ is three times that of $CE$.

. Anthony denotes in sequence all positive integers which are divisible by $2$. Bertha denotes in sequence all positive integers which are divisible by $3$. Claire denotes in sequence all positive integers which are divisible by $4$. Orderly Dora denotes all numbers written by the other three. Thereby she puts them in order by size and does not repeat a number. What is the $2017th$ number in her list? ¨Proposed by Richard Henner

How many solutions does the equation: $$[\frac{x}{20}]=[\frac{x}{17}]$$have over the set of positve integers? $[a]$ denotes the largest integer that is less than or equal to $a$. Proposed by Karl Czakler