Found problems: 5802
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Let $n$ be an positive integer. We call $n$ $\textit{good}$ when there exists positive integer $k$ s.t. $n=k(k+1)$. Does there exist 2022 pairwise distinct $\textit{good}$ numbers s.t. their sum is also $\textit{good}$ number?
A [i]coloring[/i] of the set of integers greater than or equal to $1$, must be done according to the following rule: Each number is colored blue or red, so that the sum of any two numbers (not necessarily different) of the same color is blue. Determine all the possible [i]colorings[/i] of the set of integers greater than or equal to $1$ that follow this rule.
Let $n$ be a positive integer. Consider the sum $x_1y_1 + x_2y_2 +\cdots + x_ny_n$, where that values of the variables $x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n$ are either 0 or 1.
Let $I(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum of the number is odd, and let $P(n)$ be the number of 2$n$-tuples $(x_1, x_2,\ldots, x_n, y_1, y_2,\ldots, y_n)$ such that the sum is an even number. Show that: \[ \frac{P(n)}{I(n)}=\frac{2^n+1}{2^n-1} \]
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is a continuous function. Moreover, assume that for all $x\in\mathbb{R}$, $$0\leqslant \vert f'(x)\vert\leqslant \frac{1}{2}$$ Define a sequence of real numbers $\{a_n\}_{n\in\mathbb{N}}$ by :$$a_1=1~~\text{and}~~a_{n+1}=f(a_n)~\text{for all}~n\in\mathbb{N}$$ Prove that there exists a positive real number $M$ such that for all $n\in\mathbb{N}$, $$\vert a_n\vert \leqslant M$$
Is $\cos 1^\circ$ rational? Prove.
Let $m$ and $n$ be positive integers. If $x_1$, $x_2$, $\cdots$, $x_m$ are positive integers whose arithmetic mean is less than $n+1$ and if $y_1$, $y_2$, $\cdots$, $y_n$ are positive integers whose arithmetic mean is less than $m+1$, prove that some sum of one or more $x$'s equals some sum of one or more $y$'s.
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
In a badminton tournament, each of $n$ players play all the other $n-1$ players. Each game results in either a win, or a loss. The players then write down the names of those whom they defeated, and also of those who they defeated. For example, if $A$ beats $B$ and $B$ beats $C,$ then $A$ writes the names of both $B$ and $C$. Show that there will be one person, who has written down the names of all the other $n-1$ players.
[hide="Clarification"]
Consider a game between $A,B,C,D,E,F,G$ where $A$ defeats $B$ and $C$ and $B$ defeats $E,F$, $C$ defeats $E.$ Then $A$'s list will have $(B,C,E,F)$, and will not include $G.$
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Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.
Proposed by Navid Safaei
Find all permutations $(a_1, a_2, \cdots, a_{2024})$ of $(1, 2, \cdots, 2024)$ such that there exists a polynomial $P$ with integer coefficients satisfying $P(i) = a_i$ for each $i = 1, 2, \cdots, 2024$.
Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.)
[i]Proposed by Ankan Bhattacharya, USA[/i]
Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals.
[i]Vasile Pop[/i]
Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning
\[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \]
Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$.
[i]Yang Liu[/i]
Let $u,v$ be positive integers. Define sequence $\{a_n\}$ as follows: $a_1=u+v$, and for integers $m\ge 1$,
\[\begin{array}{lll}
\begin{cases}
a_{2m}=a_m+u, \\
a_{2m+1}=a_m+v,
\end{cases}
\end{array}\]
Let $S_m=a_1+a_2+\ldots +a_m(m=1,2,\ldots )$. Prove that there are infinitely many perfect squares in the sequence $\{S_n\}$.
Ann and Beto play with a two pan balance scale. They have $2023$ dumbbells labeled with their weights, which are the numbers $1, 2, \dots, 2023$, with none of them repeating themselves. Each player, in turn, chooses a dumbbell that was not yet placed on the balance scale and places it on the pan with the least weight at the moment. If the scale is balanced, the player places it on any pan. Ana starts the game, and they continue in this way alternately until all the dumbbells are placed. Ana wins if at the end the scale is balanced, otherwise Beto win. Determine which of the players has a winning strategy and describe the strategy.
Find all positive integers $ n$ having the following properties:in two-dimensional Cartesian coordinates, there exists a convex $ n$ lattice polygon whose lengths of all sides are odd numbers, and unequal to each other. (where lattice polygon is defined as polygon whose coordinates of all vertices are integers in Cartesian coordinates.)
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
$\textbf{Problem C.1}$
There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ($A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
Prove that every positive integer can be represented in the form $$3^{m_1}\cdot 2^{n_1}+3^{m_2}\cdot 2^{n_2} + \dots + 3^{m_k}\cdot 2^{n_k}$$
where $m_1 > m_2 > \dots > m_k \geq 0$ and $0 \leq n_1 < n_2 < \dots < n_k$ are integers.
Consider a function $f: \mathbb{Z} \rightarrow \mathbb{Z}$. We call an integer $a$ [i]spanning[/i] if for all integers $b \neq a$, there exists a positive integer $k$ with $f^k(a)=b$. Find, with proof, the maximum possible number of [i]spanning[/i] numbers of $f$.
Note: $\mathbb{Z}$ represents the set of all integers, so $f$ is a function from the set of integers to itself. $f^k(a)$ is defined as $f$ applied $k$ times to $a$.
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]
is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
Prove that for any odd $ n $ there exists a unique polynomial $ P (x) $ $ n $ -th degree satisfying the equation $ P \left (x- \frac {1} {x} \right) = x ^ n- \frac {1} {x ^ n}. $ Is this true for any natural number $ n $?
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
For a positive integer $n$ consider any partition of the set $\{ 1,2,\ldots ,2n \}$ into $n$ two-element subsets $P_1,P_2\ldots,P_n$. In each subset $P_i$, let $p_i$ be the product of the two numbers in $P_i$. Prove that
\[\frac{1}{p_1}+\frac{1}{p_2}+\ldots + \frac{1}{p_n}<1 \]