Olympiad problems

2024 Mongolian Mathematical Olympiad, 2

We call a triangle consisting of three vertices of a pentagon [i]big[/i] if it's area is larger than half of the pentagon's area. Find the maximum number of [i]big[/i] triangles that can be in a convex pentagon. [i]Proposed by Gonchigdorj Sandag[/i]

2023 Grand Duchy of Lithuania, 2

Tags: extremal , graph theory , combinatorics

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

2024 Bulgaria MO Regional Round, 11.3

Tags: extremal , divisor , number theory

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

2018 VTRMC, 6

Tags: sequence , extremal , maximum , minimum value

For $n \in \mathbb{N}$, define $a_n = \frac{1 + 1/3 + 1/5 + \dots + 1/(2n-1)}{n+1}$ and $b_n = \frac{1/2 + 1/4 + 1/6 + \dots + 1/(2n)}{n}$. Find the maximum and minimum of $a_n - b_n$ for $1 \leq n \leq 999$.

2023 Mongolian Mathematical Olympiad, 2

Tags: graph theory , extremal , combinatorics

There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$.

2007 Mathematics for Its Sake, 1

Tags: real analysis , function , differentiability , extremal

Find the number of extrema of the function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ defined as $$ f(x)=\prod_{j=1}^n (x-j)^j, $$ where $ n $ is a natural number.

1986 Traian Lălescu, 1.3

Tags: algebra , real analysis , polynom , supremum , polynomial function , extremal

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

2023 Mongolian Mathematical Olympiad, 3

Tags: combinatorics , extremal , graph

Five girls and five boys took part in a competition. Suppose that we can number the boys and girls $1, 2, 3, 4, 5$ such that for each $1 \leq i,j \leq 5$, there are exactly $|i-j|$ contestants that the girl numbered $i$ and the boy numbered $j$ both know. Let $a_i$ and $b_i$ be the number of contestants that the girl numbered $i$ knows and the number of contestants that the boy numbered $i$ knows respectively. Find the minimum value of $\max(\sum\limits_{i=1}^5a_i, \sum\limits_{i=1}^5b_i)$. (Note that for a pair of contestants $A$ and $B$, $A$ knowing $B$ doesn't mean that $B$ knows $A$ and a contestant cannot know themself.)