Olympiad problems

2018 Mediterranean Mathematics OIympiad, 4

Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties: $*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering. $*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$. $*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$. (Proposed by Gerhard Woeginger, Austria)

1986 Tournament Of Towns, (114) 1

Tags: algebra , maximum , inequalities

For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?

1985 All Soviet Union Mathematical Olympiad, 400

Tags: algebra , polynomial , trinomial , integer , maximum

The senior coefficient $a$ in the square polynomial $$P(x) = ax^2 + bx + c$$ is more than $100$. What is the maximal number of integer values of $x$, such that $|P(x)|<50$.

2013 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: inequalities , minimum , maximum

If $x$ and $y$ are nonnegative real numbers such that $x+y=1$, determine minimal and maximal value of $$A=x\sqrt{1+y}+y\sqrt{1+x}$$

1998 Moldova Team Selection Test, 11

Tags: geometry , ratio , lattice points , lattice , maximum

Let $A,B,C$ be nodes of the lattice $Z\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \cap BC$. Determine max $\frac{AP}{PE}$ , over all such configurations.