Found problems: 800
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the $k$-th move, Jumpy swaps the positions of the two walnuts adjacent to walnut $k$.
Prove that there exists a value of $k$ such that, on the $k$-th move, Jumpy swaps some walnuts $a$ and $b$ such that $a<k<b$.
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
A [i]triangulation[/i] of a convex polygon $\Pi$ is a partitioning of $\Pi$ into triangles by diagonals having no common points other than the vertices of the polygon. We say that a triangulation is a [i]Thaiangulation[/i] if all triangles in it have the same area.
Prove that any two different Thaiangulations of a convex polygon $\Pi$ differ by exactly two triangles. (In other words, prove that it is possible to replace one pair of triangles in the first Thaiangulation with a different pair of triangles so as to obtain the second Thaiangulation.)
[i]Proposed by Bulgaria[/i]
Find all positive integers $(a,b,c)$ such that
$$ab-c,\quad bc-a,\quad ca-b$$ are all powers of $2$.
[i]Proposed by Serbia[/i]
Find, with proof, all pairs of positive integers $(n,d)$ with the following property: for every integer $S$, there exists a unique non-decreasing sequence of $n$ integers $a_1,a_2,...,a_n$ such that $a_1 + a_2 + ... + a_n = S$ and $a_n-a_1=d.$
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?
Let $n$ be an integer greater than $1$ and let $X$ be an $n$-element set. A non-empty collection of subsets $A_1, ..., A_k$ of $X$ is tight if the union $A_1 \cup \cdots \cup A_k$ is a proper subset of $X$ and no element of $X$ lies in exactly one of the $A_i$s. Find the largest cardinality of a collection of proper non-empty subsets of $X$, no non-empty subcollection of which is tight.
[i]Note[/i]. A subset $A$ of $X$ is proper if $A\neq X$. The sets in a collection are assumed to be distinct. The whole collection is assumed to be a subcollection.
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
Find all functions $f: \mathbb R^+ \rightarrow \mathbb R^+$ satisfying the following condition: for any three distinct real numbers $a,b,c$, a triangle can be formed with side lengths $a,b,c$, if and only if a triangle can be formed with side lengths $f(a),f(b),f(c)$.
A sequence of integers $x_1, x_2, \dots, x_k$ is called fibtastic if the difference between any two consecutive elements in the sequence is a Fibonacci number.
The integers from $1$ to $2024$ are split into two groups, each written in increasing order.
Group A is $a_1, a_2, \dots, a_m$ and Group B is $b_1, b_2, \dots, b_n.$
Find the largest integer $M$ such that we can guarantee that we can pick $M$ consecutive elements from either Group A or Group B which form a fibtastic sequence.
As an illustrative example, if a group of numbers is $2, 4, 11, 12, 13, 16, 18, 27, 29, 30,$ the
longest fibtastic sequence is $11, 12, 13, 16, 18,$ which has length $5.$
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]
[i]Proposed by Denmark[/i]
Let $a_1$, $a_2$, $a_3$, $\ldots$ be a sequence of positive integers such that $a_1=2021$ and
$$\sqrt{a_{n+1}-a_n}=\lfloor \sqrt{a_n} \rfloor. $$
Show that there are infinitely many odd numbers and infinitely many even numbers in this sequence.
[i] Proposed by Li4, Tsung-Chen Chen, and Ming Hsiao.[/i]
Let $H = \{ \lfloor i\sqrt{2}\rfloor : i \in \mathbb Z_{>0}\} = \{1,2,4,5,7,\dots \}$ and let $n$ be a positive integer. Prove that there exists a constant $C$ such that, if $A\subseteq \{1,2,\dots, n\}$ satisfies $|A| \ge C\sqrt{n}$, then there exist $a,b\in A$ such that $a-b\in H$. (Here $\mathbb Z_{>0}$ is the set of positive integers, and $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.)
The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection.
Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$.
[i]Proposed by Warut Suksompong, Thailand[/i]
[b]2/1/32.[/b] Is it possible to fill in a $2020$ x $2020$ grid with the integers from $1$ to $4,080,400$ so that the sum of each row is $1$ greater than the previous row?
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.
Let $n \ge 2$ be an integer. A cube of size $n \times n \times n$ is dissected into $n^3$ unit cubes. A nonzero real number is written at the center of each unit cube so that the sum of the $n^2$ numbers in each slab of size $1 \times n \times n$, $n \times 1 \times n$, or $n \times n \times 1$ equals zero. (There are a total of $3n$ such slabs, forming three groups of $n$ slabs each such that slabs in the same group are parallel and slabs in different groups are perpendicular.)
Could it happen that some plane in three-dimensional space separates the positive and the negative written numbers? (The plane should not pass through any of the numbers.)
[i]Nikolai Beluhov[/i]
Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied:
[list=1]
[*] Each number in the table is congruent to $1$ modulo $n$.
[*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$.
[/list]
Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.
Consider $2018$ pairwise crossing circles no three of which are concurrent. These circles subdivide the plane into regions bounded by circular $edges$ that meet at $vertices$. Notice that there are an even number of vertices on each circle. Given the circle, alternately colour the vertices on that circle red and blue. In doing so for each circle, every vertex is coloured twice- once for each of the two circle that cross at that point. If the two colours agree at a vertex, then it is assigned that colour; otherwise, it becomes yellow. Show that, if some circle contains at least $2061$ yellow points, then the vertices of some region are all yellow.
Proposed by [i]India[/i]
Is it possible to write the numbers $17$,$18$,$19$,...$32$ in a $4*4$ grid of unit squares with one number in each square such that if the grid is divided into four $2*2$ subgrids of unit squares ,then the product of numbers in each of the subgrids divisible by $16$?
Consider arrangements of the $9$ numbers $1, 2, 3, \dots, 9$ in a $3 \times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\{a_1, a_2, a_3\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.