Found problems: 133
In a regular 100-gon, 41 vertices are colored black and the remaining 59 vertices are colored white. Prove that there exist 24 convex quadrilaterals $Q_{1}, \ldots, Q_{24}$ whose corners are vertices of the 100-gon, so that
[list]
[*] the quadrilaterals $Q_{1}, \ldots, Q_{24}$ are pairwise disjoint, and
[*] every quadrilateral $Q_{i}$ has three corners of one color and one corner of the other color.
[/list]
Show that the set $\{1,2,....,2^n\}$ can be partitioned in two classes, none of which contains an arithmetic progression of length $2n$.
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
[i]Proposed by Poland.[/i]
Prove that among any $ 16 $ numbers smaller than $ 101 $ there are four of them that have the property that the sum of two of them is equal to the sum of the other two.
Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$.
(Alberto Alfarano)
Consider positive integers $2,3,\ldots,n-1,n$ where $n\ge96.$ Consider any partition in two (sub)sets. Show that at least one of these two sets always contains two numbers and their product. Show that the statement does not hold for $n=95,$ e.g. there is a partition without the mentioned property.
We call an even positive integer $n$ [i]nice[/i] if the set $\{1, 2, \dots, n\}$ can be partitioned into $\frac{n}{2}$ two-element subsets, such that the sum of the elements in each subset is a power of $3$. For example, $6$ is nice, because the set $\{1, 2, 3, 4, 5, 6\}$ can be partitioned into subsets $\{1, 2\}$, $\{3, 6\}$, $\{4, 5\}$. Find the number of nice positive integers which are smaller than $3^{2022}$.
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called [i]historic[/i] if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
A well-known puzzle asks for a partition of a cross into four parts which are to be reassembled into a square. One solution is exhibited on the picture.
[img]https://cdn.artofproblemsolving.com/attachments/9/1/3b8990baf5e37270c640e280c479b788d989ba.png[/img]
Show that there are infinitely many solutions. (Some solutions split the cross into four equal parts!)
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
$(HUN 3)$ In the plane $4000$ points are given such that each line passes through at most $2$ of these points. Prove that there exist $1000$ disjoint quadrilaterals in the plane with vertices at these points.
Is it possible to partition the set of positive integer numbers into two classes, none of which contains an infinite arithmetic sequence (with a positive ratio)? What is we impose the extra condition that in each class $\mathcal{C}$ of the partition, the set of difference
\begin{align*} \left\{ \min \{ n \in \mathcal{C} \mid n >m \} -m \mid m \in \mathcal{C} \right \} \end{align*}
be bounded?
The set of all $10$-digit numbers may be represented as a union of two subsets: the subset $M$ consisting of all $10$-digit numbers, each of which may be represented as a product of two $5$-digit numbers, and the subset $N$ , containing the remaining $10$-digit numbers . Which of the sets $M$ and $N$ contains more elements?
(S. Fomin , Leningrad)
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
[i]Proposed by Poland.[/i]
The sum of several (not necessarily different) positive integers not exceeding $10$ is equal to $S$.
Find all possible values of $S$ such that these numbers can always be partitioned into two groups with the sum of the numbers in each group not exceeding $70$.
[i](I. Voronovich)[/i]
The Euclidian three-dimensional space has been partitioned into three nonempty sets $A_1,A_2,A_3$. Show that one of these sets contains, for each $d > 0$, a pair of points at mutual distance $d$.
Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence.
Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths.
[asy]
size(4cm);
draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin);
draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin);
draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin);
draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin);
draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin);
draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin);
draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin);
[/asy]
[i]Proposed by Zixiang Zhou, Canada[/i]
We are given the natural numbers $1=a_1,\, \, a_2,...,a_n$, for which
$a_i\leq a_{i+1}\leq 2a_i$
for $i=1,2,...,n-1$ and the sum $\sum_{i=1}^n a_i$ is even. Prove that these numbers can be partitioned into two groups with equal sum.
Could the three-dimensional space be expressed as the union of disjoint circumferences?
Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.
There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$, and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group.
Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $.
We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.
Prove that the following statement holds for all odd integers $n \ge 3$:
If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.