# Tagged Questions

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### Recurrence relation and combinatorics

I am reading p.4 of the article http://mercercountymathcircle.files.wordpress.com/2014/03/recurrence_relations.pdf which consider the following problem: Find the units digit of ...
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### Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
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### The rows continue to be different to each other

In each position of an $n \times n$ matrix there is a number. We know that all the rows of the matrix are different from each other. Show that we can delete a column so that the rows of the matrix ...
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### Find value of $n$ with given conditions

The 4-digit positive number $n$'s digit sum is $20$. The sum of the first two digits is $11$, the sum of the first and the last digit as well. The first digit is the last digit $+3$. What is the ...
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### Distributing cards among players

Moderator Note: This is a current contest question on codechef.com. N players sit around a round table. There are $n \cdot m$ cards with unique numbers of range $1\ldots n\cdot m$. Each player ...
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### Number of queries required to find the function.

this is a slight variation to question $3$ of the Nordic mathematical olympiad of 2010.(in short that one deals with bijections and this one deals with any kind of function).We have 2010 buttons and ...
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### A function on sets which is constant for all permutations

Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions: ...
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### A bijection defined on the set of configuration of lamps

Consider $n$ lamps clockwise numbered from $1$ to $n$ on a circle. Let $\xi$ to be a configuration where $0 \le \ell \le n$ random lamps are turned on. A cool procedure consists in perform, ...
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### Possible arrangements of marbles in bags?

I've come across a question on a math test asking, "How many different ways can you put a dozen identical marbles into six bags so that each bag has atleat one marble in it?". I would imagine that ...
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### Ratio of sum in black and white squares

For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once). Let $S_1$ be the sum of the numbers put in the ...
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### Cutting a chessboard into domino pieces!

A friend of mine gave me this problem from a european olympiad: Suppose we have a $8\times8$ chessboard. Each edge has a number; the number of ways of dividing this chessboard into $1\times2$ and ...
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### A game on a graph

Alice and Bob play a game on a complete graph ${G}$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses ...
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### Existence of sets with four common elements among 16000 sets.

In the Duma there are $1600$ delegates, who have formed $16000$ committees of $80$ persons each. Prove that one can find two committees having no fewer than four common members. Source: All-Russian ...
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### Number of non-decreasing sequences

How do I find the number of non-decreasing sequences of length $N$, such that all number in the sequences lie in the range $[a, b]$. Also, the frequency of the most frequently occurring element should ...
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### simple games with cute winning strategies?

Im thinking of games of two players ($A$ goes first and $B$ second) like the following: There are 35 chips in a table, during each turn a player can remove 1,2,3 or 4 chips. Prove player $B$ can ...
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### Complex Combinatorics Hexagon/Triangle Contest Problem

The problem is as follows: The six sides of convex hexagon $A_1A_2A_3A_4 A_5A_6$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings ...
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### Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$.

Prove that a sequence of $11$ numbers always contains six numbers summing up to a multiple of $6$. This is a problem from a selection to IMO 2014. I like thinking about this problem, it is ...
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### From any list of $131$ positive integers with prime factor at most $41$, $4$ can always be chosen such that their product is a perfect square

Author's note:I don't want the whole answer,but a guide as to how I should think about this problem. BdMO 2010 In a set of $131$ natural numbers, no number has a prime factor greater than 42. ...
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### 2014 USAMO Problem :with Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points $$\dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots$$ in the plane with the following property: For any three distinct ...
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### How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
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### Lines covering points on napkin

Suppose we place a $100\times 100$ napkin on an infinite lattice plane. What is the minimum number of lines that can always cover all the lattice points lying inside or on the border of the napkin, no ...
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### 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 ...
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### How many zeros does this expression end in?

How many zeroes does $$\frac{50!}{2^95^5}$$ end in?