Found problems: 9
2019 China Team Selection Test, 2
Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$.
(1) Find the smallest integer $k$, such that if the minimum number in $A,B\in S$ respectively are both $\geq k$, then $A\rightarrow B$ implies $B\rightarrow A$.
(2) For the $k$ obtained in (1), how many tuples can we pick from $S$, such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$.
2016 Tournament Of Towns, 7
a.) There are $2n+1$ ($n>2$) batteries. We don't know which batteries are good and which are bad but we know that the number of good batteries is greater by $1$ than the number of bad batteries. A lamp uses two batteries, and it works only if both of them are good. What is the least number of attempts sufficient to make the lamp work?
b.) The same problem but the total number of batteries is $2n$ ($n>2$) and the numbers of good and bad batteries are equal.
[i]Proposed by Alexander Shapovalov[/i]
2025 Bulgarian Spring Mathematical Competition, 9.3
In a country, there are towns, some of which are connected by roads. There is a route (not necessarily direct) between every two towns. The Minister of Education has ensured that every town without a school is connected via a direct road to a town that has a school. The Minister of State Optimization wants to ensure that there is a unique path between any two towns (without repeating traveled segments), which may require removing some roads.
Is it always possible to achieve this without constructing additional schools while preserving what the Minister of Education has accomplished?
2024 India Iran Friendly Math Competition, 5
Let $n \geq k$ be positive integers and let $a_1, \dots, a_n$ be a non-increasing list of positive real numbers. Prove that there exists $k$ sets $B_1, \dots, B_k$ which partition the set $\{1, 2, \dots, n\}$ such that $$\min_{1 \le j \le k} \left(\sum_{i \in B_j} a_i \right) \geq \min_{1 \le j \le k} \left(\frac{1}{2k+1-2j} \cdot \sum^n_{i=j} a_i\right).$$
[i]Proposed by Navid Safaei[/i]
2024 India Iran Friendly Math Competition, 1
A league consists of $2024$ players. A [i]round[/i] involves splitting the players into two different teams and having every member of one team play with every member of the other team. A round is called [i]balanced[/i] if both teams have an equal number of players. A tournament consists of several rounds at the end of which any two players have played each other. The committee organised a tournament last year which consisted of $N$ rounds. Prove that the committee can organise a tournament this year with $N$ balanced rounds.
[i]Proposed by Anant Mudgal and Navilarekallu Tejaswi[/i]
2001 Saint Petersburg Mathematical Olympiad, 9.7
300 students participate on the international math olympiad. Every student speaks in exactly two of the official languages of the olympiad and every language is spoken by 100 people (it is known that students speak only the official languages). Prove that the students can be sited on a circular table, such that no two neighbors spoke the same language.
2019 Tournament Of Towns, 5
A magician and his assistent are performing the following trick.There is a row of 12 empty closed boxes. The magician leaves the room, and a person from the audience hides a coin in each of two boxes of his choice, so that the assistent knows which boxes contain coins. The magician returns, and the assistant is allowed to open one box that does not contain a coin. Next, the magician selects 4 boxes, which are simultaneously opened. The goal of the magician is to open both boxes that contain coins. Devise a method that will allow the magician and his assistant to always succesfully perform the trick.
2019 China Team Selection Test, 2
Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$.
(1) Find the smallest integer $k$, such that if the minimum number in $A,B\in S$ respectively are both $\geq k$, then $A\rightarrow B$ implies $B\rightarrow A$.
(2) For the $k$ obtained in (1), how many tuples can we pick from $S$, such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$.
2011 QEDMO 9th, 8
There are $256$ lumps of metal that have different weights in pairs. With the help of a beam balance , one may now compare every two lumps. Find the smallest number $m$ such that you can be sure to find the heaviest as well as the lightest lump with the weighing process.