Found problems: 14842
2006 Singapore Team Selection Test, 2
Let S be a set of sequences of length 15 formed by using the letters a and b
such that every pair of sequences in S differ in at least 3 places. What is the
maximum number of sequences in S?
2007 All-Russian Olympiad, 8
Given an undirected graph with $N$ vertices. For any set of $k$ vertices, where $1\le k\le N$, there are at most $2k-2$ edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges).
[i]I. Bogdanov, G. Chelnokov[/i]
2013 Bosnia Herzegovina Team Selection Test, 3
Prove that in the set consisting of $\binom{2n}{n}$ people we can find a group of $n+1$ people in which everyone knows everyone or noone knows noone.
1998 All-Russian Olympiad, 7
A jeweller makes a chain consisting of $N>3$ numbered links. A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized. What is the maximum number?
2014 CHMMC (Fall), 3
Suppose that in a group of $6$ people, if $A$ is friends with $B$, then $B$ is friends with $A$. If each of the $6$ people draws a graph of the friendships between the other $5$ people, we get these $6$ graphs, where edges represent
friendships and points represent people.
[img]https://cdn.artofproblemsolving.com/attachments/5/5/7265067f585e3dfe77ba94ac6261b4462cd015.png[/img]
If Sue drew the first graph, how many friends does she have?
1994 Tournament Of Towns, (409) 7
In a $10$ by $10$ square grid (which we call “the bay”) you are requested to place ten “ships”: one $1$ by $4$ ship, two $1$ by $3$ ships, three $1$ by $2$ ships and four $1$ by $1$ ships. The ships may not have common points (even corners) but may touch the “shore” of the bay. Prove that
(a) by placing the ships one after the other arbitrarily but in the order indicated above, it is always possible to complete the process;
(b) by placing the ships in reverse order (beginning with the smaller ones), it is possible to reach a situation where the next ship cannot be placed (give an example).
(KN Ignatjev)
2011 Korea National Olympiad, 3
There are $n$ students each having $r$ positive integers. Their $nr$ positive integers are all different. Prove that we can divide the students into $k$ classes satisfying the following conditions.
(a) $ k \le 4r $
(b) If a student $A$ has the number $m$, then the student $B$ in the same class can't have a number $l$ such that
\[ (m-1)! < l < (m+1)!+1 \]
2005 Georgia Team Selection Test, 1
1. The transformation $ n \to 2n \minus{} 1$ or $ n \to 3n \minus{} 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.
2024/2025 TOURNAMENT OF TOWNS, P4
A mother and her son are playing. At first, the son divides a ${300}\mathrm{\;g}$ wheel of cheese into 4 slices. Then the mother divides ${280}\mathrm{\;g}$ of butter between two plates. At last, the son puts the cheese slices on those plates. The son wins if on each plate the amount of cheese is not less than the amount of butter (otherwise the mother wins). Who of them can win irrespective of the opponent's actions?
Alexandr Shapovalov
2006 MOP Homework, 5
For a triple $(m,n,r)$ of integers with $0 \le r \le n \le m-2$, define $p(m,n,r)=\sum^r_{k=0} (-1)^k \dbinom{m+n-2(k+1)}{n} \dbinom{r}{k}$. Prove that $p(m,n,r)$ is positive and that $\sum^n_{r=0} p(m,n,r)=\dbinom{m+n}{n}$.
2021 Harvard-MIT Mathematics Tournament., 10
Let $n>1$ be a positive integer. Each unit square in an $n\times n$ grid of squares is colored either black or white,
such that the following conditions hold:
$\bullet$ Any two black squares can be connected by a sequence of black squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any two white squares can be connected by a sequence of white squares where every two consecutive squares in the sequence share an edge;
$\bullet$ Any $2\times 2$ subgrid contains at least one square of each color.
Determine, with proof, the maximum possible difference between the number of black squares and white squares in this grid (in terms of $n$).
2023/2024 Tournament of Towns, 5
5. Tom has 13 weight pieces that look equal, however 12 of them weigh the same and the 13th piece is fake and weighs more than the others. He also has two balances: one shows correctly which pan is heavier or that their weights are equal, the other one gives the correct result when the weights on the pans differ, and gives a random result when the weights are equal. (Tom does not know which balance is which). Tom can choose the balance before each weighting. Prove that he can surely determine the fake weight piece in three weighings.
Andrey Arzhantsev
Russian TST 2019, P3
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]
2007 China Western Mathematical Olympiad, 1
Let set $ T \equal{} \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$.
2024 Korea Junior Math Olympiad (First Round), 10.
Find the number of cases in which one of the numbers 1, 2, 3, 4, and 5 is written at each vertex of an equilateral triangle so that the following conditions are satisfied. (However, the same number is counted as one when rotated, and the same number can be written multiple times.)
$ \bigstar $ The product of the two numbers written at each end of the sides of an equilateral triangle is an even number.
1991 IMO, 3
Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime.
1957 Putnam, B5
Let $f$ be an increasing mapping from the family of subsets of a given finite set $H$ into itself, i.e. such that for every $X \subseteq Y\subseteq H$ we have $f (X )\subseteq f (Y )\subseteq H .$ Prove that there exists a subset $H_{0}$ of $H$ such that $f (H_{0}) = H_{0}.$
2006 India IMO Training Camp, 3
Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds
[list][b](a)[/b] $|B| > 25$,
[b](b)[/b] $F(B)={\O}$,
[b](c)[/b] $\bigcap_{j\in F(B)} A_j \neq {\O}$.[/list]
Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.
1956 Moscow Mathematical Olympiad, 340
a) * In a rectangle of area $5$ sq. units, $9$ rectangles of area $1$ are arranged. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
b) In a rectangle of area $5$ sq. units, lie $9$ arbitrary polygons each of area $1$. Prove that the area of the overlap of some two of these rectangles is $\ge 1/9$
2023 Assara - South Russian Girl's MO, 4
In a $50 \times 50$ checkered square, each cell is painted in one of $100$ given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a $1 \times 2$ rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given $100$ colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?
2008 VJIMC, Problem 4
The numbers of the set $\{1,2,\ldots,n\}$ are colored with $6$ colors. Let
$$S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}$$and
$$D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.$$Prove that
$$|D|\le2|S|+\frac{n^2}2.$$
2013 AMC 12/AHSME, 3
A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?
$ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 60\qquad\textbf{(E)}\ 70 $
1979 IMO Longlists, 2
For a finite set $E$ of cardinality $n \geq 3$, let $f(n)$ denote the maximum number of $3$-element subsets of $E$, any two of them having exactly one common element. Calculate $f(n)$.
2018 PUMaC Combinatorics A, 4
If $a$ and $b$ are selected uniformly from $\{0,1,\ldots,511\}$ without replacement, the expected number of $1$'s in the binary representation of $a+b$ can be written in simplest from as $\tfrac{m}{n}$. Compute $m+n$.
1984 IMO Longlists, 1
The fraction $\frac{3}{10}$ can be written as the sum of two positive fractions with numerator $1$ as follows: $\frac{3}{10} =\frac{1}{5}+\frac{1}{10}$ and also $\frac{3}{10}=\frac{1}{4}+\frac{1}{20}$. There are the only two ways in which this can be done. In how many ways can $\frac{3}{1984}$ be written as the sum of two positive fractions with numerator $1$?
Is there a positive integer $n,$ not divisible by $3$, such that $\frac{3}{n}$ can be written as the sum of two positive fractions with numerator $1$ in exactly $1984$ ways?