Olympiad problems

2006 MOP Homework, 5

Tags: inequalities , max , sum , maximal , algebra

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

2014 Danube Mathematical Competition, 4

Tags: combinatorics , cardinality , lattice points , maximal , parallelogram , geometry , combinatorial geometry

Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.

2015 IMAR Test, 2

Tags: maximal , vertices , graph theory , graph , combinatorics

Let $n$ be a positive integer and let $G_n$ be the set of all simple graphs on $n$ vertices. For each vertex $v$ of a graph in $G_n$, let $k(v)$ be the maximal cardinality of an independent set of neighbours of $v$. Determine $max_{G \in G_n} \Sigma_{v\in V (G)}k(v)$ and the graphs in $G_n$ that achieve this value.

2016 Balkan MO Shortlist, C2

Tags: maximal , combinatorics

There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal $k$ such that the following holds: There are $k$ customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.

2017 Federal Competition For Advanced Students, P2, 6

Tags: subset , maximal , combinatorics

Let $S = \{1,2,..., 2017\}$. Find the maximal $n$ with the property that there exist $n$ distinct subsets of $S$ such that for no two subsets their union equals $S$. Proposed by Gerhard Woeginger

2024 ITAMO, 6

Tags: inequalities , integer , extermal , fraction , maximal

For each integer $n$, determine the smallest real number $M_n$ such that \[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\] for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.

2020 German National Olympiad, 1

Tags: geometry , angle , circles , maximal

Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.

2011 German National Olympiad, 4

Tags: geometry , point , set , angle , maximal , sum

There are two points $A$ and $B$ in the plane. a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$ b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.

2024 ISI Entrance UGB, P7

Tags: parabola , conic , applications of calculus , maximal

Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.

2013 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: number theory , maximal

Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers