Found problems: 9
2019 Teodor Topan, 4
Ana choses two real numbers $ y>0,x $ and Bogdan repeatedly tries to guess these in the following manner: at step $ j $ he choses a real number $ b_j, $ asks her if $ b_j=x+jy, $ and she tells him the truth.
[b]a)[/b] If $ x=0, $ can Bogdan find Ana's numbers in a finite number of steps?
[b]b)[/b] If $ x\neq 0, $ can Bogdan find Ana's numbers in a finite number of steps?
2022 BMT, 8
Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$
2015 Romania National Olympiad, 4
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
2003 Alexandru Myller, 4
A professor organized five exams for a class consisting of at least two students. Before starting the first test, he deduced that there will be at least two students from that class that will have the same amount of passed exams.
What is the minimum numer of students that class could have had such that the conclusion of the professor's reasoning was correct.
2009 Romania National Olympiad, 4
We say that a natural number $ n\ge 4 $ is [i]unusual[/i] if, for any $ n\times n $ array of real numbers, the sum of the numbers from any $ 3\times 3 $ compact subarray is negative, and the sum of the numbers from any $ 4\times 4 $ compact subarray is positive.
Find all unusual numbers.
2022 BMT, 9
Lysithea and Felix each have a take-out box, and they want to select among $42$ different types of sweets to put in their boxes. They each select an even number of sweets (possibly $0$) to put in their box. In each box, there is at most one sweet of any type, although the boxes may have sweets of the same type in common. The total number of sweets they take out is $42.$ Let $N$ be the number of ways can they select sweets to take out. Compute the remainder when $N$ is divided by $42^2 - 1.$
2022 BMT, 6
Bayus has eight slips of paper, which are labeled 1$, 2, 4, 8, 16, 32, 64,$ and $128.$ Uniformly at random, he draws three slips with replacement; suppose the three slips he draws are labeled $a, b,$ and $c.$ What is the probability that Bayus can form a quadratic polynomial with coefficients $a, b,$ and $c,$ in some order, with $2$ distinct real roots?
2022 BMT, 5
Given a positive integer $n,$ let $s(n)$ denote the sum of the digits of $n.$ Compute the largest positive integer $n$ such that $n = s(n)^2 + 2s(n) - 2.$
2005 Alexandru Myller, 4
Prove that there exists an undirected graph having $ 2004 $ vertices such that for any $ \in\{ 1,2,\ldots ,1002 \} , $ there exists at least two vertices whose orders are $ n. $