Found problems: 1488
2005 APMO, 4
In a small town, there are $n \times n$ houses indexed by $(i, j)$ for $1 \leq i, j \leq n$ with $(1, 1)$ being the house at the top left corner, where $i$ and $j$ are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by $(1, c)$, where $c \leq \frac{n}{2}$. During each subsequent time interval $[t, t+1]$, the fire fighters defend a house which is not yet on fire while the fire spreads to all undefended [i]neighbors[/i] of each house which was on fire at time t. Once a house is defended, it remains so all the time. The process ends when the fire can no longer spread. At most how many houses can be saved by the fire fighters?
A house indexed by $(i, j)$ is a [i]neighbor[/i] of a house indexed by $(k, l)$ if $|i - k| + |j - l|=1$.
2004 China Team Selection Test, 2
There are $ n \geq 5$ pairwise different points in the plane. For every point, there are just four points whose distance from which is $ 1$. Find the maximum value of $ n$.
2003 All-Russian Olympiad, 4
Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.
2003 All-Russian Olympiad Regional Round, 11.4
Points $ A_1,A_2,...,A_n$ and $ B_1,B_2,...,B_n$ are given on a plane. Show that the points $ B_i$ can be renumbered in such a way that the angle between vectors $ A_iA_j^{\longrightarrow}$ and $ B_iB_j^{\longrightarrow}$ is acute or right whenever $ i\neq j$.
2014 Saudi Arabia IMO TST, 3
There are $2015$ coins on a table. For $i = 1, 2, \dots , 2015$ in succession, one must turn over exactly $i$ coins. Prove that it is always possible either to make all of the coins face up or to make all of the coins face down, but not both.
1997 Iran MO (3rd Round), 3
Let $d$ be a real number such that $d^2=r^2+s^2$, where $r$ and $s$ are rational numbers. Prove that we can color all points of the plane with rational coordinates with two different colors such that the points with distance $d$ have different colors.
1993 Irish Math Olympiad, 3
If $ 1 \le r \le n$ are integers, prove the identity:
$ \displaystyle\sum_{d\equal{}1}^{\infty}\binom {n\minus{}r\plus{}1}{d} \binom {r\minus{}1} {d\minus{}1}\equal{}\binom {n}{r}.$
2013 ELMO Shortlist, 5
There is a $2012\times 2012$ grid with rows numbered $1,2,\dots 2012$ and columns numbered $1,2,\dots, 2012$, and we place some rectangular napkins on it such that the sides of the napkins all lie on grid lines. Each napkin has a positive integer thickness. (in micrometers!)
(a) Show that there exist $2012^2$ unique integers $a_{i,j}$ where $i,j \in [1,2012]$ such that for all $x,y\in [1,2012]$, the sum \[ \sum _{i=1}^{x} \sum_{j=1}^{y} a_{i,j} \] is equal to the sum of the thicknesses of all the napkins that cover the grid square in row $x$ and column $y$.
(b) Show that if we use at most $500,000$ napkins, at least half of the $a_{i,j}$ will be $0$.
[i]Proposed by Ray Li[/i]
2004 Hong kong National Olympiad, 4
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
2015 Bangladesh Mathematical Olympiad, 1
[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b]
A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:
Witness [b]One[/b] said exactly one of the four witnesses is a liar.
Witness [b]Two[/b] said exactly two of the four witnesses is a liar.
Witness [b]Three[/b] said exactly three of the four witnesses is a liar.
Witness [b]Four[/b] said exactly four of the four witnesses is a liar.
Assume that each of the statements is either true or false.How many of the winesses are liars?
2014 Postal Coaching, 4
Show that the number of ordered pairs $(S,T)$ of subsets of $[n]$ satisfying $s>|T|$ for all $s\in S$ and $t>|S|$ for all $t\in T$ is equal to the Fibonacci number $F_{2n+2}$.
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Moderator says:
http://www.artofproblemsolving.com/Forum/viewtopic.php?p=296007#p296007
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=515970&hilit=Putnam+1990[/color]
2007 IberoAmerican, 6
Let $ \mathcal{F}$ be a family of hexagons $ H$ satisfying the following properties:
i) $ H$ has parallel opposite sides.
ii) Any 3 vertices of $ H$ can be covered with a strip of width 1.
Determine the least $ \ell\in\mathbb{R}$ such that every hexagon belonging to $ \mathcal{F}$ can be covered with a strip of width $ \ell$.
Note: A strip is the area bounded by two parallel lines separated by a distance $ \ell$. The lines belong to the strip, too.
2004 IberoAmerican, 1
It is given a 1001*1001 board divided in 1*1 squares. We want to amrk m squares in such a way that:
1: if 2 squares are adjacent then one of them is marked.
2: if 6 squares lie consecutively in a row or column then two adjacent squares from them are marked.
Find the minimun number of squares we most mark.
2006 Team Selection Test For CSMO, 4
All the squares of a board of $(n+1)\times(n-1)$ squares
are painted with [b]three colors[/b] such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of $n$.
2012 Philippine MO, 1
A computer generates even integers half of the time and another computer generates even integers a third of the time. If $a_i$ and $b_i$ are the integers generated by the computers, respectively, at time $i$, what is the probability that $a_1b_1 +a_2b_2 +\cdots + a_kb_k$ is an even integer.
2005 ISI B.Stat Entrance Exam, 7
Q. For integers $m,n\geq 1$, Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets:
$A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$
$B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}$ given that $\alpha _i \geq 0$ and $\alpha_ i\in \mathbb{Z}$ for all $i$
$C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$
$(a)$ Define a one-one onto map from $A_{m,n}$ onto $B_{m+1,n-1}$.
$(b)$ Define a one-one onto map from $A_{m,n}$ onto $C_{m,n+m-1}$.
$(c)$ Find the number of elements of the sets $A_{m,n}$ and $B_{m,n}$.
2011 Puerto Rico Team Selection Test, 6
Two children take turns breaking chocolate bar that is 5*10 squares. They can only break the bar using the divisions between squares and can only do 1 break at a time.. The first player that when breaking the chocolate bar breaks off only a single square wins. Is there a winning strategy for any player?
1999 All-Russian Olympiad, 2
Each rational point on a real line is assigned an integer. Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.
1993 Turkey MO (2nd round), 3
$n\in{Z^{+}}$ and $A={1,\ldots ,n}$. $f: N\rightarrow N$ and $\sigma: N\rightarrow N$ are two permutations, if there is one $k\in A$ such that $(f\circ\sigma)(1),\ldots ,(f\circ\sigma)(k)$ is increasing and $(f\circ\sigma)(k),\ldots ,(f\circ\sigma)(n)$ is decreasing sequences we say that $f$ is good for $\sigma$. $S_\sigma$ shows the set of good functions for $\sigma$.
a) Prove that, $S_\sigma$ has got $2^{n-1}$ elements for every $\sigma$ permutation.
b)$n\geq 4$, prove that there are permutations $\sigma$ and $\tau$ such that, $S_{\sigma}\cap S_{\tau}=\phi$
.
2005 South africa National Olympiad, 3
A warehouse contains $175$ boots of size $8$, $175$ boots of size $9$ and $200$ boots of size $10$. Of these $550$ boots, $250$ are for the left foot and $300$ for the right foot. Let $n$ denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.)
(a) Is $n=50$ possible?
(b) Is $n=51$ possible?
2011 Postal Coaching, 6
In a party among any four persons there are three people who are mutual acquaintances or mutual strangers. Prove that all the people can be separated into two groups $A$ and $B$ such that in $A$ everybody knows everybody else and in $B$ nobody knows anybody else.
2005 MOP Homework, 3
In a television series about incidents in a conspicuous town there are $n$ citizens staging in it, where $n$ is an integer greater than $3$. Each two citizens plan together a conspiracy against one of the other citizens. Prove that there exists a citizen, against whom at least $\sqrt{n}$ other citizens are involved in the conspiracy.
2008 Spain Mathematical Olympiad, 3
Let $p\ge 3$ be a prime number. Each side of a triangle is divided into $p$ equal parts, and we draw a line from each division point to the opposite vertex. Find the maximum number of regions, every two of them disjoint, that are formed inside the triangle.
2006 Estonia National Olympiad, 5
The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules:
1) A is a word.
2) If s is a word, then $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and $ x \oplus y$ denotes the concatenation of x and y.
The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules:
1) A is a word.
2) If s is a word, $ s \oplus s$ and $ s \oplus \bar{s}$ are words, where $ \bar{s}$ is defined as above and $ x \oplus y$ is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x.
Prove that the two languages consist of the same words.
2004 All-Russian Olympiad, 1
Let $ M \equal{} \{ x_1..., x_{30}\}$ a set which consists of 30 distinct positive numbers, let $ A_n,$ $ 1 \leq n \leq 30,$ the sum of all possible products with $ n$ elements each of the set $ M.$ Prove if $ A_{15} > A_{10},$ then $ A_1 > 1.$