Found problems: 1488
2006 China Girls Math Olympiad, 5
The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral points on segment $PQ$. Find the least value of the number of elements of $T$.
1979 IMO Longlists, 32
Let $n, k \ge 1$ be natural numbers. Find the number $A(n, k)$ of solutions in integers of the equation
\[|x_1| + |x_2| +\cdots + |x_k| = n\]
2010 Tuymaada Olympiad, 1
Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):
At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.
Is there a winning strategy available for Sasha?
1990 Kurschak Competition, 3
We would like to give a present to one of $100$ children. We do this by throwing a biased coin $k$ times, after predetermining who wins in each possible outcome of this lottery.
Prove that we can choose the probability $p$ of throwing heads, and the value of $k$ such that, by distributing the $2^k$ different outcomes between the children in the right way, we can guarantee that each child has the same probability of winning.
2010 Tournament Of Towns, 7
Merlin summons the $n$ knights of Camelot for a conference. Each day, he assigns them to the $n$ seats at the Round Table. From the second day on, any two neighbours may interchange their seats if they were not neighbours on the first day. The knights try to sit in some cyclic order which has already occurred before on an earlier day. If they succeed, then the conference comes to an end when the day is over. What is the maximum number of days for which Merlin can guarantee that the conference will last?
1999 IberoAmerican, 3
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.
For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.
2003 Irish Math Olympiad, 5
(a) In how many ways can $1003$ distinct integers be chosen from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers di ffer by $10?$
(b) Show that there are $(3(5151) + 7(1700)) 101^7$ ways to choose $1002$ distinct integers from the set $\{1, 2, ... , 2003\}$ so that no two of the chosen integers diff er by $10.$
2001 Tournament Of Towns, 5
In a chess tournament, every participant played with each other exactly once, receiving $1$ point for a win, $1/2$ for a draw and $0$ for a loss.
[list][b](a)[/b] Is it possible that for every player $P$, the sum of points of the players who were beaten by P is greater than the sum of points of the players who beat $P$?
[b](b)[/b] Is it possible that for every player $P$, the first sum is less than the second one?[/list]
2001 Kurschak Competition, 2
Let $k\ge 3$ be an integer. Prove that if $n>\binom k3$, then for any $3n$ pairwise different real numbers $a_i,b_i,c_i$ ($1\le i\le n$), among the numbers $a_i+b_i$, $a_i+c_i$, $b_i+c_i$, one can find at least $k+1$ pairwise different numbers. Show that this is not always the case when $n=\binom k3$.
1998 China Team Selection Test, 1
Find $k \in \mathbb{N}$ such that
[b]a.)[/b] For any $n \in \mathbb{N}$, there does not exist $j \in \mathbb{Z}$ which satisfies the conditions $0 \leq j \leq n - k + 1$ and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots, \left( \begin{array}{c}
n\\
j + k - 1\end{array} \right)$ forms an arithmetic progression.
[b]b.)[/b] There exists $n \in \mathbb{N}$ such that there exists $j$ which satisfies $0 \leq j \leq n - k + 2$, and $\left(
\begin{array}{c}
n\\
j\end{array} \right), \left( \begin{array}{c}
n\\
j + 1\end{array} \right), \ldots , \left( \begin{array}{c}
n\\
j + k - 2\end{array} \right)$ forms an arithmetic progression.
Find all $n$ which satisfies part [b]b.)[/b]
2012 Portugal MO, 3
Isabel wants to partition the set $\mathbb{N}$ of the positive integers into $n$ disjoint sets $A_{1}, A_{2}, \ldots, A_{n}$. Suppose that for each $i$ with $1\leq i\leq n$, given any positive integers $r, s\in A_{i}$ with $r\neq s$, we have $r+s\in A_{i}$. If $|A_{j}|=1$ for some $j$, find the greatest positive integer that may belong to $A_{j}$.
1987 Kurschak Competition, 3
Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess.
Prove that we can choose three members of the club who play three different games amongst each other.
2001 Tournament Of Towns, 5
The only pieces on an $8\times8$ chessboard are three rooks. Each moves along a row or a column without running to or jumping over another rook. The white rook starts at the bottom left corner, the black rook starts in the square directly above the white rook, and the red rook starts in the square directly to the right of the white rook. The white rook is to finish at the top right corner, the black rook in the square directly to the left of the white rook, and the red rook in the square directly below the white rook. At all times, each rook must be either in the same row or the same column as another rook. Is it possible to get the rooks to their destinations?
2013 Iran MO (3rd Round), 5
A subsum of $n$ real numbers $a_1,\dots,a_n$ is a sum of elements of a subset of the set $\{a_1,\dots,a_n\}$. In other words a subsum is $\epsilon_1a_1+\dots+\epsilon_na_n$ in which for each $1\leq i \leq n$ ,$\epsilon_i$ is either $0$ or $1$.
Years ago, there was a valuable list containing $n$ real not necessarily distinct numbers and their $2^n-1$ subsums. Some mysterious creatures from planet Tarator has stolen the list, but we still have the subsums.
(a) Prove that we can recover the numbers uniquely if all of the subsums are positive.
(b) Prove that we can recover the numbers uniquely if all of the subsums are non-zero.
(c) Prove that there's an example of the subsums for $n=1392$ such that we can not recover the numbers uniquely.
Note: If a subsum is sum of element of two different subsets, it appears twice.
Time allowed for this question was 75 minutes.
2018 Cono Sur Olympiad, 3
Define the product $P_n=1! \cdot 2!\cdot 3!\cdots (n-1)!\cdot n!$
a) Find all positive integers $m$, such that $\frac {P_{2020}}{m!}$ is a perfect square.
b) Prove that there are infinite many value(s) of $n$, such that $\frac {P_{n}}{m!}$ is a perfect square, for at least two positive integers $m$.
2004 China Team Selection Test, 2
Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.
2012 International Zhautykov Olympiad, 2
A set of (unit) squares of a $n\times n$ table is called [i]convenient[/i] if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a [i]convenient [/i] set made of $m$ squares, which becomes in[i]convenient [/i] when any of its squares is removed.
2016 Germany National Olympiad (4th Round), 2
A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days.
Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day.
Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.
1997 Federal Competition For Advanced Students, P2, 5
We define the following operation which will be applied to a row of bars being situated side-by-side on positions $ 1,2,...,N$. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by-side in such a way that all bars form a new row and are situated on positions $ 1,...,M.$
From an initial number $ a_0>0$ of bars there originates a sequence $ (a_n)_{n \ge 0},$ where $ a_n$ is the number of bars after having applied the operation $ n$ times.
$ (a)$ Prove that for no $ n>0$ can we have $ a_n\equal{}1997.$
$ (b)$ Determine all natural numbers that can only occur as $ a_0$ or $ a_1$.
1993 India National Olympiad, 5
Show that there is a natural number $n$ such that $n!$ when written in decimal notation ends exactly in 1993 zeros.
2002 Turkey MO (2nd round), 1
Let $(a_1, a_2,\ldots , a_n)$ be a permutation of $1, 2, \ldots , n,$ where $n \geq 2.$ For each $k = 1, \ldots , n$, we know that $a_k$ apples are placed at the point $k$ on the real axis. Children named $A,B,C$ are assigned respective points $x_A, x_B, x_C \in \{1, \ldots , n\}.$ For each $k,$ the children whose points are closest to $ k$ divide $a_k$ apples equally among themselves. We call $(x_A, x_B, x_C)$ a [i]stable configuration[/i] if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of $n$ for which a stable configuration exists for some distribution $(a_1, \ldots, a_n)$ of the apples.
2013 European Mathematical Cup, 3
We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code $(1101)$ contains $10$ palindromes, of which $6$ are nice.)
a) What is the least number of palindromes in a code?
b) What is the least number of nice palindromes in a code?
2013 Romania Team Selection Test, 1
Fix a point $O$ in the plane and an integer $n\geq 3$. Consider a finite family $\mathcal{D}$ of closed unit discs in the plane such that:
(a) No disc in $\mathcal{D}$ contains the point $O$; and
(b) For each positive integer $k < n$, the closed disc of radius $k + 1$ centred at $O$ contains the centres of at least $k$ discs in $\mathcal{D}$.
Show that some line through $O$ stabs at least $\frac{2}{\pi} \log \frac{n+1}{2}$ discs in $\mathcal{D}$.
2011 Philippine MO, 5
The chromatic number $\chi$ of an (infinite) plane is the smallest number of colors with which we can color the points on the plane in such a way that no two points of the same color are one unit apart.
Prove that $4 \leq \chi \leq 7$.
2009 Croatia Team Selection Test, 2
On sport games there was 1991 participant from which every participant knows at least n other participants(friendship is mutual). Determine the lowest possible n for which we can be sure that there are 6 participants between which any two participants know each other.