0
votes
1answer
17 views

Bound the Number of Acute-angled Triangles

I encounter the following problem with solution. But I do not quite understand the argument for 5, 10 points and eventually 100 points. Can someone elucidate the details? Problem In a plane there ...
1
vote
1answer
58 views

How many zeros does this expression end in?

How many zeroes does $$\frac{50!}{2^95^5}$$ end in?
1
vote
1answer
31 views

a spider has 1 sock and 1 shoe for each leg. then find out the the total possibilities.

a spider has one sock and one shoe for each of its 8 legs.in how many different orders can the spider put on its shocks and shoes; assuming that on each leg ;the shock must be put on before the shoe? ...
0
votes
3answers
80 views

$1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$

This is a previous AIME question. $1-x+x^2-x^3+. . .-x^{17}=a_0+a_1y+a_2y^2+. . .+a_{17}y^{17},y=x+1$. Then what is $a_{17}$? Is anything wrong with the following method? $1-x+x^2-x^3+. . ...
3
votes
2answers
141 views

Find the number of series

Find the number of series $(a_1,..., a_{2n})$ that have terms from ${\{0,...9\}}$ so that: $$ 11|\sum_{i=1}^{n}a_i-\sum_{i=n+1}^{2n}a_i $$ (this is not a homework) There is a similar problem ...
10
votes
1answer
103 views

Question concerning finite intersecting sets

Let $\{X_i\}_{i=1}^{\infty}$, $\{Y_j\}_{j=1}^{\infty}$ be finite sets of cardinality at most $n$. If for any finite $F$, there are $i,j \in \mathbb{N}$ such that $F \cap X_i \cap Y_j = \emptyset$, ...
2
votes
4answers
65 views

determining the amount of total questions needed in a game given the probabilty

I'm creating a game and can't seem to quite figure this out - driving me crazy. There are 8 questions in my game You can play the game an unlimited amount of times the test bank doesn't change. so ...
2
votes
1answer
37 views

Relabelling players in a tournament

BdMO 2014 $n$ players take part in a chess tournament where each player plays with all others only once and the only outcomes of the games are win and loss.Prove that it is possible,after the ...
1
vote
2answers
97 views

A counting problem using Burnside's lemma

Suppose we have 12 objects (say, 6 indistinguishable black ones and 6 indistinguishable white ones). How many seatings at a round table can we form from them? The answer is $80$, but how could this ...
4
votes
1answer
115 views

If one eats $100$ chocolates in $58$ days,then he must be eating exactly 15 chocolates in some consecutive days

BdMO 2014 Nationals $X$ eats 100 chocolates in 58 days,eating at least 1 chocolate per day.Prove that,in some consecutive days,she ate exactly 15 chocolates. I tried using the pigeonhole ...
1
vote
0answers
231 views

Finding the number of arrangement of N people of different height such that K of them are visible from front

Moderator Note: This is a current contest question on codechef.com. [Initially, I had asked this question in stackoverflow, but someone suggested to post it here, and hence this question is ...
2
votes
1answer
39 views

AIME 1986:different sequences of coin tosses

AIME 1986 Problem-13 In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We ...
1
vote
1answer
56 views

Choosing $2n-1$ points from $n\times n$ grid such that $3$ points always form a right triangle

NOTE: Looking for a hint,not the whole solution. BdMO 2012 Nationals Secondary Consider a $n×n$ grid of points. Prove that no matter how we choose $2n-1$ points from these, there will always ...
0
votes
2answers
59 views

Choosing people around a circular table

There are 20 people around a circular table.We have to choose $3$ of them such that at least $2$ of them are sitting together.In how many ways can this be done? Number of ways of choosing 3 people ...
5
votes
1answer
139 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
3
votes
1answer
189 views

The library with 999 books.

In the town of Capibara there is a library with books in 999 themes. Since Capibara is an international town they have books in various languages. We know that for every language we can find exactly ...
1
vote
1answer
31 views

Compute number of points having same property

I have been given a cuboid which has either green or red color for each point (integer coordinates) in it. I am also given another cuboid whose lower left corner is (x1, y1, z1) and upper right corner ...
1
vote
2answers
85 views

$7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
1
vote
0answers
40 views

Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
1
vote
1answer
55 views

Arranging red and blue tiles in a line with at least 1 blue tile between any 2 red tiles

BdMO 2010 Nationals: Tom and Jerry have $8$ blue tiles and $6$ red tiles.They want to arrange them in a straight line so that between any $2$ red tiles there is always at least $1$ blue tile.In ...
1
vote
1answer
59 views

math contest ranking problem?

A math contest is held among 4 middle schools. Each school enters a team of 3 students. The 12 contestants are ranked from 1 (best performance) to 12 (worst performance). The team that has the overall ...
4
votes
2answers
103 views

Cute coloring problem on a board

Suppose we color an $n\times n$ square board using $n$ colors exactly $n$ times each. Prove that there is either a column or a row containing at least $\lceil \sqrt n \rceil$ different colors. A ...
7
votes
1answer
108 views

South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)

Let $A_n$ be the number of ways to tile a $4×n$ rectangle using $2×1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3. My attempt: Define basic shapes A, B and C, ...
2
votes
1answer
67 views

How to prove $\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2i-1}{i+1} $?

How to prove this closed form involving Catalan numbers? $$\binom{2n}{n}\frac{1}{n+1} = \prod \limits_{i = 2}^n \frac{2 \times (2i-1)}{i+1} $$ I have seen this being used here. Not sure how to derive ...
7
votes
3answers
276 views

A problem with 26 distinct positive integers

I am trying to solve the following problem. Assume that we are given 26 distinct positive integers. Show that either there exist 6 of them $x_1<x_2<x_3<x_4<x_5<x_6$, with $x_1$ ...
2
votes
1answer
53 views

Travelling to the point of origin without using the same road twice

BdMO 2013 Secondary: There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n$. Prove that there is a city such that ...
7
votes
1answer
174 views

Korean Math Olympiad (Construct rectangle)

Prove that an $m$ × $n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of 8 where $m$ > 1 and $n$ > 1. My solution: starting from 2 × 4 and 3 × 8 ...
12
votes
2answers
441 views

If 1 boy knows r girls and 1 girl knows r boys ,then number of boys=girls

Yet another question from BdMO 2013 Nationals: In a class,every boy knows $r$ number of girls and every girl knows $r$ number of boys.Show that there are equal number of boys and girls[Assume that ...
2
votes
0answers
91 views

Rearranging numbered cards to reverse their order

I have been thinking about this question for a long time, but I can't solve it. Here is the question: We have $9$ cards, with numbers one to nine written on them (in the order $1, 2, \ldots , 9$). ...
3
votes
1answer
47 views

Neighbors with very different labels

The cells of a square $2011$ by $2011$ array are labelled with the integers $1,2,\dots, 2011^2$ in such a way that every label is used exactly once. We identity the top and bottom edges and the left ...
4
votes
1answer
99 views

What is the minimum number of locks on the cabinet that would satisfy these conditions?

Eleven scientists want to have a cabinet built where they will keep some top secret work. They want multiple locks installed, with keys distributed in such a way that if any six scientists are present ...
0
votes
0answers
110 views

Iran Math Olympiad 2013 (Perfect Set)

Let $n$ be a natural number and suppose that $w_1,w_2,…,w_n$ are $n$ weights. We call the set of {$w_1,w_2,…,w_n$} to be a Perfect Set if we can achieve all of the 1, 2, …, W weights with sums of ...
1
vote
1answer
47 views

Young Tableaux Generalizing

The entries in a array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? (2010 AMC12 B) The ...
1
vote
2answers
167 views

France Olympiad Team Selection Test 2005

In an international meeting of n ≥ 3 participants, 14 languages are spoken. We know that: - Any 3 participants speak a common language. - No language is spoken by more than half of the participants. ...
3
votes
2answers
112 views

Combinatorial Proof Of A Number Theory Theorem--Confusion

I came across a combinatorial proof of the Fermat's Little Theorem which states that If $p$ is a prime number then the number ($a$$p$-$a$) is a multiple of $p$ for any natural number $a$. Let me ...
0
votes
0answers
21 views

Number Of Triangles of All Sizes in an Equilateral Triangle [duplicate]

https://mail.google.com/mail/u/0/?ui=2&ik=4622e6803e&view=att&th=1422d3806080ed0d&attid=0.1&disp=emb&realattid=ii_1422415e014f71c5&zw&atsh=1 Consider an ...
1
vote
1answer
40 views

Prove that number of $(A, B, C)$ with $A ∩ B ∩ C = \emptyset$, $A ∩ B \ne \emptyset$, $B ∩ C \ne \emptyset$ is $7^n − 2\cdot6^n + 5^n$

Prove that the number of triples $(A, B, C)$ where $A, B, C$ are subsets of $\{1,2,\cdots,n\}$ such that $A ∩ B ∩ C = \emptyset$, $A ∩ B \ne \emptyset$, $B ∩ C \ne \emptyset$ is $7^n − 2\cdot6^n ...
0
votes
1answer
60 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
0
votes
1answer
61 views

Choosing a Set of r elements from a set having n elements.

Define a set $X$={$1$,$2$,$...$,$n$} . Determine the number of ways of selecting a subset of $X$ such that it contains no consecutive integers .
1
vote
1answer
96 views

How find this maximum of $P_{1}+P_{n}$

Question $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth $x$ points ...
0
votes
0answers
51 views

placing integers on circle with no repeated differences.

Is it possible to place 2008 numbers from 1 to 2009 on a circle such that the absolute values of the differences between numbers and their inmediate neighbors are all different? I think this is from ...
0
votes
1answer
49 views

The intersections of three polygons in a square with area $=6$

Let three convex polygons with areas equal to $3$, in a square with area equals to $6$. We need to prove that there are two of them which has their intersection with area is at least $1$. I have no ...
1
vote
3answers
80 views

Finding the possible location of points

The numbers 1,2,....6 are to be placed in some order at the points A,B,.....F in the figure below. How many ways can the numbers be placed so that each sum of consecutive pairs of points is odd?
1
vote
1answer
88 views

Bernie's Breakfast

Moderator Note: This is a current contest question on Brilliant.org. Bernie's Breakfast Buffet offers omelettes as part of their buffet on Saturday and Sunday. They offer 6 different toppings ...
2
votes
3answers
74 views

Related Theorem of Binomial Theorem

Proving that for any whole number $n$, the following identity holds: $$\sum^{n}_{i=1}{n\choose{i}}i=n\times2^{n-1}$$ So, I memorized this formula for preparing for math contests, but I think it's ...
9
votes
2answers
276 views

1965 Putnam, B2

The problem statement: Suppose $n$ players engage in a tournament in which each player plays every other player in exactly one game, to a win or a loss. Let $w_i$ and $l_i$ denote the wins and ...
-4
votes
3answers
178 views

In how many ways can $16 be divided among 4 people? [closed]

In how many ways can $16 be divided among 4 people, assuming that each person has to get something and there are 5 cent coins and up
1
vote
2answers
193 views

Mathematical olympiad combinatorics question

In a problem set about various topics on combinatorics, geometry and algebra, I found this one There is a $6\times6$ grid, each square filled with a grasshopper. After the bell rings, each ...
6
votes
2answers
123 views

Arguing about a homework problem correctness

I've recently completely a homework in a problem solving class, I think my reasoning is correct but my teacher insisted that my answer is incorrect. I'm not sure if I'm correct or not. Question: ...
3
votes
1answer
191 views

Ramsey Type problem (variant of people at a party)

There is $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor-1$ of them, each of whom either knows both or else ...