Found problems: 295
1974 Putnam, B6
For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate
$$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$
for $i=0,1,2$. Your result should be strong enough to permit direct evaluation of the numbers $s_{i,n}$ and to show clearly the relationship of $s_{0,n}, s_{1,n}$ and $s_{2,n}$ to each other for all positive integers $n$. In particular, show the relationships among these three sums for $n = 1000$.
2019 Thailand TST, 2
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 $S$.
2023 Israel National Olympiad, P6
Determine if there exists a set $S$ of $5783$ different real numbers with the following property:
For every $a,b\in S$ (not necessarily distinct) there are $c\neq d$ in $S$ so that $a\cdot b=c+d$.
2022 Turkey EGMO TST, 2
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more element to every 29-element [i]nice [/i] subset while keeping it nice, find the minimum value of $n$.
2012 Dutch BxMO/EGMO TST, 5
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
2020 Iran MO (3rd Round), 4
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
1996 Yugoslav Team Selection Test, Problem 1
Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions:
(a) Any two distinct sets from $\mathfrak F$ have exactly one element in common;
(b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$.
Can $n$ be equal to $1996$?
2015 Ukraine Team Selection Test, 12
For a given natural $n$, we consider the set $A\subset \{1,2, ..., n\}$, which consists of at least $\left[\frac{n+1}{2}\right]$ items. Prove that for $n \ge 2015$ the set $A$ contains a three-element arithmetic sequence.
1986 Czech And Slovak Olympiad IIIA, 4
Let $C_1,C_2$, and $C_3$ be points inside a bounded convex planar set $M$. Rays $l_1,l_2,l_3$ emanating from $C_1,C_2,C_3$ respectively partition the complement of the set $M \cup l_1 \cup l_2 \cup l_3$ into three regions $D_1,D_2,D_3$. Prove that if the convex sets $A$ and $B$ satisfy $A\cap l_j =\emptyset = B\cap l_j$ and $A\cap D_j \ne \emptyset \ne B\cap D_j$ for $j = 1,2,3$, then $A\cap B \ne \emptyset$
1985 All Soviet Union Mathematical Olympiad, 410
Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$
2017 AMC 10, 12
Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$?
$\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$
2010 Bosnia And Herzegovina - Regional Olympiad, 4
It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that
$i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements
$ii)$ Upper bound can be reached for $n=3$
2015 Cono Sur Olympiad, 6
Let $S = \{1, 2, 3, \ldots , 2046, 2047, 2048\}$. Two subsets $A$ and $B$ of $S$ are said to be [i]friends[/i] if the following conditions are true:
[list]
[*] They do not share any elements.
[*] They both have the same number of elements.
[*] The product of all elements from $A$ equals the product of all elements from $B$.
[/list]
Prove that there are two subsets of $S$ that are [i]friends[/i] such that each one of them contains at least $738$ elements.
2009 Jozsef Wildt International Math Competition, W. 10
Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$
[list=1]
[*] Find $|F|$
[*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$
[*] Find $n$, if $|F|=540$ and $n=2k$
[/list]
1959 AMC 12/AHSME, 14
Given the set $S$ whose elements are zero and the even integers, positive and negative. Of the five operations applied to any pair of elements: (1) addition (2) subtraction (3) multiplication (4) division (5) finding the arithmetic mean (average), those elements that only yield elements of $S$ are:
$ \textbf{(A)}\ \text{all} \qquad\textbf{(B)}\ 1,2,3,4\qquad\textbf{(C)}\ 1,2,3,5\qquad\textbf{(D)}\ 1,2,3\qquad\textbf{(E)}\ 1,3,5 $
1999 Bosnia and Herzegovina Team Selection Test, 5
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$
1996 Tuymaada Olympiad, 2
Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?
2016 IMC, 4
Let $n\ge k$ be positive integers, and let $\mathcal{F}$ be a family of finite sets with the following properties:
(i) $\mathcal{F}$ contains at least $\binom{n}{k}+1$ distinct sets containing exactly $k$ elements;
(ii) for any two sets $A, B\in \mathcal{F}$, their union $A\cup B$ also belongs to $\mathcal{F}$.
Prove that $\mathcal{F}$ contains at least three sets with at least $n$ elements.
(Proposed by Fedor Petrov, St. Petersburg State University)
2012 IFYM, Sozopol, 3
Let $A$ be a set of natural numbers, for which for $\forall n\in \mathbb{N}$ exactly one of the numbers $n$, $2n$, and $3n$ is an element of $A$. If $2\in A$, show whether $13824\in A$.
2017 Costa Rica - Final Round, LR2
There is a set of $17$ consecutive positive integers. Let $m$ be the smallest of these numbers. Determine for which values of $m$ the set can be divided into three subsets disjoint, such that the sum of the elements of each subset is the same.
1989 Romania Team Selection Test, 4
A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that
a) $f(4k +2) = 3$ for each nonnegative integer $k$,
b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$
2009 Junior Balkan Team Selection Tests - Romania, 3
Let $A$ be a finite set of positive real numbers satisfying the property:
[i]For any real numbers a > 0, the sets $\{x \in A | x > a\}$ and $\{x \in A | x < \frac{1}{a}\}$ have the cardinals of the same parity.[/i]
Show that the product of all elements in $A$ is equal to $1$.
2019 India PRMO, 20
Consider the set $E$ of all natural numbers $n$ such that whenn divided by $11, 12, 13$, respectively, the remainders, int that order, are distinct prime numbers in an arithmetic progression. If $N$ is the largest number in $E$, find the sum of digits of $N$.
2022 Korea -Final Round, P6
Set $X$ is called [i]fancy[/i] if it satisfies all of the following conditions:
[list]
[*]The number of elements of $X$ is $2022$.
[*]Each element of $X$ is a closed interval contained in $[0, 1]$.
[*]For any real number $r \in [0, 1]$, the number of elements of $X$ containing $r$ is less than or equal to $1011$.
[/list]
For [i]fancy[/i] sets $A, B$, and intervals $I \in A, J \in B$, denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$. Determine the maximum value of $n(A, B)$.
2006 MOP Homework, 1
Let $S$ be a set of rational numbers with the following properties:
(a) $\frac12$ is an element in $S$,
(b) if $x$ is in $S$, then both $\frac{1}{x+1}$ and $\frac{x}{x+1}$ are in $S$.
Prove that $S$ contains all rational numbers in the interval $(0, 1)$.