Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Let $n \in N$ such that $1 + 2 + ... + n$ is divisible by $3$. Integers $a_1\ge a_2\ge a_3\ge 2$ have sum $n$ and they satisfy $1 + 2 + ... + a_1\le \frac{1}{3}( 1 + 2 + ... + n ) $ and $1 + 2 + ... + (a_1+ a_2) \le \frac{2}{3}( 1 + 2 + ... + n )$. Prove that there is a partition of $\{ 1 , 2 , ... , n\}$ in three subsets $A_1, A_2, A_3$ with cardinals $| A_i| = a_i, i = 1 , 2 , 3$, and with equal sums of their elements .

Let $n$ be a positive integer. What is the smallest value of $m$ with $m > n$ such that the set $M = \{n, n + 1, ..., m\}$ can be partitioned into subsets so that in each subset, there is a number which equals to the sum of all other numbers of this subset?

let $ V$ be a finitive set and $ g$ and $ f$ be two injective surjective functions from $ V$to$ V$.let $ T$ and $ S$ be two sets such that they are defined as following" $ S \equal{} \{w \in V: f(f(w)) \equal{} g(g(w))\}$ $ T \equal{} \{w \in V: f(g(w)) \equal{} g(f(w))\}$ we know that $ S \cup T \equal{} V$, prove: for each $ w \in V : f(w) \in S$ if and only if $ g(w) \in S$

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 $3^{2022}$.

$(HUN 3)$ 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.

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 $3^{2022}$.

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)$.

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.

Let be a function $ s:\mathbb{N}^2\longrightarrow \mathbb{N} $ that sends $ (m,n) $ to the number of solutions in $ \mathbb{N}^n $ of the equation: $$ x_1+x_2+\cdots +x_n=m $$ [b]1)[/b] Prove that: $$ s(m+1,n+1)=s(m,n)+s(m,n+1) =\prod_{r=1}^n\frac{m-r+1}{r} ,\quad\forall m,n\in\mathbb{N} $$ [b]2)[/b] Find $ \max\left\{ a_1a_2\cdots a_{20}\bigg| a_1+a_2+\cdots +a_{20}=2007, a_1,a_2,\ldots a_{20}\in\mathbb{N} \right\} . $

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)$.

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.

For positive integers $n, k, l$, we define the number of $l$-tuples of positive integers $(a_1,a_2,\cdots a_l)$ satisfying the following as $Q(n,k,l)$. (i): $n=a_1+a_2+\cdots +a_l$ (ii): $a_1>a_2>\cdots > a_l > 0$. (iii): $a_l$ is an odd number. (iv): There are $k$ odd numbers out of $a_i$. For example, from $9=8+1=6+3=6+2+1$, we have $Q(9,1,1)=1$, $Q(9,1,2)=2$, $Q(9,1,3)=1$. Prove that if $n>k^2$, $\sum_{l=1}^n Q(n,k,l)$ is $0$ or an even number.

For any integer $r \geq 1$, determine the smallest integer $h(r) \geq 1$ such that for any partition of the set $\{1, 2, \cdots, h(r)\}$ into $r$ classes, there are integers $a \geq 0 \ ; 1 \leq x \leq y$, such that $a + x, a + y, a + x + y$ belong to the same class. [i]Proposed by Romania[/i]

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.

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]

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]

Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

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.

Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$. (a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$. (b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

For every positive integer $n$, let $p(n)$ denote the number of sets $\{x_1, x_2, \dots, x_k\}$ of integers with $x_1 > x_2 > \dots > x_k > 0$ and $n = x_1 + x_3 + x_5 + \dots$ (the right hand side here means the sum of all odd-indexed elements). As an example, $p(6) = 11$ because all satisfying sets are as follows: $$\{6\}, \{6, 5\}, \{6, 4\}, \{6, 3\}, \{6, 2\}, \{6, 1\}, \{5, 4, 1\}, \{5, 3, 1\}, \{5, 2, 1\}, \{4, 3, 2\}, \{4, 3, 2, 1\}.$$ Show that $p(n)$ equals to the number of partitions of $n$ for every positive integer $n$.