Found problems: 800
Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$.
[i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
Let each of the vertices of a regular $9$-gon (polygon of 9 equal sides and equal angles) be coloured black or white .
$(a).$ Show that there are two adjacent verices of same colour.
$(b).$ Show there are three vertices of the same colour forming an isosceles triangle.
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.
A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\in S$.
For each integer $n\ge 2$, find the largest integer $k$ (in terms of $n$) such that there exists a set of $n$ distinct points that is [i]not[/i] overdetermined, but has $k$ overdetermined subsets.
[i]Proposed by Carl Schildkraut[/i]
An [i]anti-Pascal[/i] triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from $1$ to $10$.
\[\begin{array}{
c@{\hspace{4pt}}c@{\hspace{4pt}}
c@{\hspace{4pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{4pt}}c
} \vspace{4pt}
& & & 4 & & & \\\vspace{4pt}
& & 2 & & 6 & & \\\vspace{4pt}
& 5 & & 7 & & 1 & \\\vspace{4pt}
8 & & 3 & & 10 & & 9 \\\vspace{4pt}
\end{array}\]
Does there exist an anti-Pascal triangle with $2018$ rows which contains every integer from $1$ to $1 + 2 + 3 + \dots + 2018$?
[i]Proposed by Morteza Saghafian, Iran[/i]
We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black.
Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black?
[It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]
For every natural number $a$, consider the set $S(a)=\{a^n+a+1|n=2,3,\ldots\}$. Does there exist an infinite set $A\subset\mathbb N$ with the property that for any two distinct
elements $x,y\in A$, $x$ and $y$ are coprime and $S(x)\cap S(y)=\emptyset$?
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
A grid consists of all points of the form $(m, n)$ where $m$ and $n$ are integers with $|m|\le 2019,|n| \le 2019$ and $|m| +|n| < 4038$. We call the points $(m,n)$ of the grid with either $|m| = 2019$ or $|n| = 2019$ the [i]boundary points[/i]. The four lines $x = \pm 2019$ and $y= \pm 2019$ are called [i]boundary lines[/i]. Two points in the grid are called [i]neighbours [/i] if the distance between them is equal to $1$.
Anna and Bob play a game on this grid.
Anna starts with a token at the point $(0,0)$. They take turns, with Bob playing first.
1) On each of his turns. Bob [i]deletes [/i] at most two boundary points on each boundary line.
2) On each of her turns. Anna makes exactly three [i]steps[/i] , where a [i]step [/i] consists of moving her token from its current point to any neighbouring point, which has not been deleted.
As soon as Anna places her token on some boundary point which has not been deleted, the game is over and Anna wins.
Does Anna have a winning strategy?
[i]Proposed by Demetres Christofides, Cyprus[/i]
Let $G$ be a tournoment such that it's edges are colored either red or blue.
Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.
Does there exists a positive irrational number ${x},$ such that there are at most finite positive integers ${n},$ satisfy that for any integer $1\leq k\leq n,$ $\{kx\}\geq\frac 1{n+1}?$
A finite list of rational numbers is written on a blackboard. In an [i]operation[/i], we choose any two numbers $a$, $b$, erase them, and write down one of the numbers \[
a + b, \; a - b, \; b - a, \; a \times b, \; a/b \text{ (if $b \neq 0$)}, \; b/a \text{ (if $a \neq 0$)}.
\] Prove that, for every integer $n > 100$, there are only finitely many integers $k \ge 0$, such that, starting from the list \[ k + 1, \; k + 2, \; \dots, \; k + n, \] it is possible to obtain, after $n - 1$ operations, the value $n!$.
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.
Let $q$ be a positive integer which is not a perfect cube. Prove that there exists a positive constant $C$ such that for all natural numbers $n$, one has
$$\{ nq^{\frac{1}{3}} \} + \{ nq^{\frac{2}{3}} \} \geq Cn^{-\frac{1}{2}}$$
where $\{ x \}$ denotes the fractional part of $x$.
Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.