Found problems: 1782
We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.
Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same.
Let $ n\ge 2$ be an integer. What is the minimal and maximal possible rank of an $ n\times n$ matrix whose $ n^{2}$ entries are precisely the numbers $ 1, 2, \ldots, n^{2}$?
Prove that for each positive integer n,the equation
$x^{2}+15y^{2}=4^{n}$
has at least $n$ integer solution $(x,y)$
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:
\[f(x-y+f(y))=f(x)+f(y).\]
Let $G$ be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let $O$ be a fixed vertex of $G$. For an arbitrary positive number $n$, let $V_{n}$ be the number of vertices which can be reached from $O$ passing through at most $n$ edges ( $O$ counts). Find the smallest possible value of $V_{n}$.
Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n\minus{}1}\plus{}1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j \equal{} A_k$.
For which numbers $ n$ is there a positive integer $ k$ with the following property: The sum of digits for $ k$ is $ n$ and the number $ k^2$ has sum of digits $ n^2.$
Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that
\[\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.\]
The positive sequence $\{a_n\}$ satisfies:$a_1=1+\sqrt 2$ and $(a_n-a_{n-1})(a_n+a_{n-1}-2\sqrt n)=2(n\geq 2).$
(1)Find the general formula of $\{a_n\}$;
(2)Find the set of all the positive integers $n$ so that $\lfloor a_n\rfloor=2022$.
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).
Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.
Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying:
i) $f(1)=f(2)=1$;
ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$.
For each integer $m\ge 2$, find the value of $f(2^m)$.
Is it possible to place the numbers $0,1,2,\dots,9$ on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?
A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.
1. The transformation $ n \to 2n \minus{} 1$ or $ n \to 3n \minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.
The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$
\[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]
In a simple graph $G$, we call $t$ pairwise adjacent vertices a $t$[i]-clique[/i]. If a vertex is connected with all other vertices in the graph, we call it a [i]central[/i] vertex. Given are two integers $n,k$ such that $\dfrac {3}{2} \leq \dfrac{1}{2} n < k < n$. Let $G$ be a graph on $n$ vertices such that
[b](1)[/b] $G$ does not contain a $(k+1)$-[i]clique[/i];
[b](2)[/b] if we add an arbitrary edge to $G$, that creates a $(k+1)$-[i]clique[/i].
Find the least possible number of [i]central[/i] vertices in $G$.
The sequence $ a_n$ satisfies $ a_{m\plus{}n}\plus{} a_{m\minus{}n}\equal{}\frac12(a_{2m}\plus{}a_{2n})$ for all $ m\geq n\geq 0$. If $ a_1\equal{}1$, find $ a_{1995}$.
Let $a,b,c$ be 3 integers, $b$ odd, and define the sequence $\{x_n\}_{n\geq 0}$ by $x_0=4$, $x_1=0$, $x_2=2c$, $x_3=3b$ and for all positive integers $n$ we have
\[ x_{n+3} = ax_{n-1}+bx_n + cx_{n+1} . \]
Prove that for all positive integers $m$, and for all primes $p$ the number $x_{p^m}$ is divisible by $p$.
Let $ p$ be a positive integer. Define the function $ f: \mathbb{N}\to\mathbb{N}$ by $ f(n)\equal{}a_1^p\plus{}a_2^p\plus{}\cdots\plus{}a_m^p$, where $ a_1, a_2,\ldots, a_m$ are the decimal digits of $ n$ ($ n\equal{}\overline{a_1a_2\ldots a_m}$). Prove that every sequence $ (b_k)^\infty_{k\equal{}0}$ of positive integer that satisfy $ b_{k\plus{}1}\equal{}f(b_k)$ for all $ k\in\mathbb{N}$, has a finite number of distinct terms. $ \mathbb{N}\equal{}\{1,2,3\ldots\}$
Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.
Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n\minus{}1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.
Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$
With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.
There are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays. The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together. Wendy notices that there are even numbers of groups she could select whether she chooses $4$ or $5$ students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\\\dbinom{16}5&=4368.\end{align*}
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even. This gives her an idea for a problem for the $2008$ Jupiter Falls High School Math Meet team test:
\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]
Wendy works the solution out correctly. What is her answer?