Found problems: 1488
2011 Vietnam Team Selection Test, 1
A grasshopper rests on the point $(1,1)$ on the plane. Denote by $O,$ the origin of coordinates. From that point, it jumps to a certain lattice point under the condition that, if it jumps from a point $A$ to $B,$ then the area of $\triangle AOB$ is equal to $\frac 12.$
$(a)$ Find all the positive integral poijnts $(m,n)$ which can be covered by the grasshopper after a finite number of steps, starting from $(1,1).$
$(b)$ If a point $(m,n)$ satisfies the above condition, then show that there exists a certain path for the grasshopper to reach $(m,n)$ from $(1,1)$ such that the number of jumps does not exceed $|m-n|.$
1988 IMO Longlists, 57
$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as
\[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|.
\]
$ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.
2011 China Team Selection Test, 2
Let $\ell$ be a positive integer, and let $m,n$ be positive integers with $m\geq n$, such that $A_1,A_2,\cdots,A_m,B_1,\cdots,B_m$ are $m+n$ pairwise distinct subsets of the set $\{1,2,\cdots,\ell\}$. It is known that $A_i\Delta B_j$ are pairwise distinct, $1\leq i\leq m, 1\leq j\leq n$, and runs over all nonempty subsets of $\{1,2,\cdots,\ell\}$. Find all possible values of $m,n$.
2010 Contests, 3
[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity.
[b](b)[/b]What is the smallest area possible of pentagons with integral coordinates.
Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.
2007 Kurschak Competition, 3
Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties:
[list]
[*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points,
[*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]
2008 India Regional Mathematical Olympiad, 6
Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$.
[16 points out of 100 for the 6 problems]
2011 Finnish National High School Mathematics Competition, 5
Two players, the builder and the destroyer, plays the following game. Builder starts and players chooses alternatively different elements from the set $\{0,1,\ldots,10\}.$ Builder wins if some four integer of those six integer he chose forms an arithmetic sequence. Destroyer wins if he can prevent to form such an arithmetic four-tuple. Which one has a winning strategy?
2007 Estonia National Olympiad, 5
The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.
1989 Kurschak Competition, 3
We play the following game in a Cartesian coordinate system in the plane. Given the input $(x,y)$, in one step, we may move to the point $(x,y\pm 2x)$ or to the point $(x\pm 2y,y)$. There is also an additional rule: it is not allowed to make two steps that lead back to the same point (i.e, to step backwards).
Prove that starting from the point $\left(1;\sqrt 2\right)$, we cannot return to it in finitely many steps.
2000 France Team Selection Test, 1
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.
1990 IMO Longlists, 45
The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules:
[i](i)[/i] The tourist has n keys, each key with a different color.
[i](ii)[/i] Once a key is used, it is not permitted to change until it is destroyed.
[i](iii)[/i] Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it.
Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.
1993 Korea - Final Round, 1
Consider a $9 \times 9$ array of white squares. Find the largest $n \in\mathbb N$ with the property: No matter how one chooses $n$ out of 81 white squares and color in black, there always remains a $1 \times 4$ array of white squares (either vertical or horizontal).
2006 China Team Selection Test, 1
Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.
2022 Israel TST, 3
A class has 30 students. To celebrate 'Tu BiShvat' each student chose some dried fruits out of $n$ different kinds. Say two students are friends if they both chose from the same type of fruit. Find the minimal $n$ so that it is possible that each student has exactly \(6\) friends.
2004 China Girls Math Olympiad, 1
We say a positive integer $ n$ is [i]good[/i] if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k \plus{} a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
1979 IMO Longlists, 62
$T$ is a given triangle with vertices $P_1,P_2,P_3$. Consider an arbitrary subdivision of $T$ into finitely many subtriangles such that no vertex of a subtriangle lies strictly between two vertices of another subtriangle. To each vertex $V$ of the subtriangles there is assigned a number $n(V)$ according to the following rules:
$(\text{i})$ If $V$ = $P_i$, then $n(V) = i$.
$(\text{ii})$ If $V$ lies on the side $P_i P_j$ of $T$, then $n(V) = i$ or $j$.
$(\text{iii})$ If $V$ lies inside the triangle $T$, then $n(V)$ is any of the numbers $1,2,3$.
Prove that there exists at least one subtriangle whose vertices are numbered $1, 2, 3$.
2005 IberoAmerican, 2
A flea jumps in a straight numbered line. It jumps first from point $0$ to point $1$. Afterwards, if its last jump was from $A$ to $B$, then the next jump is from $B$ to one of the points $B + (B - A) - 1$, $B + (B - A)$, $B + (B-A) + 1$.
Prove that if the flea arrived twice at the point $n$, $n$ positive integer, then it performed at least $\lceil 2\sqrt n\rceil$ jumps.
2006 China National Olympiad, 6
Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.
1999 China Team Selection Test, 3
Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:
[b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$;
[b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$
[b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such
that $M \subset A_k$.
Find the smallest possible value of $n$.
2003 Finnish National High School Mathematics Competition, 5
Players Aino and Eino take turns choosing numbers from the set $\{0,..., n\}$ with $n\in \Bbb{N}$ being fixed in advance.
The game ends when the numbers picked by one of the players include an arithmetic progression of length $4.$
The one who obtains the progression wins.
Prove that for some $n,$ the starter of the game wins. Find the smallest such $n.$
1993 All-Russian Olympiad Regional Round, 9.8
Number $ 0$ is written on the board. Two players alternate writing signs and numbers to the right, where the first player always writes either $ \plus{}$ or $ \minus{}$ sign, while the second player writes one of the numbers $ 1, 2, ... , 1993$,writing each of these numbers exactly once. The game ends after $ 1993$ moves. Then the second player wins the score equal to the absolute value of the expression obtained thereby on the board. What largest score can he always win?
2005 China National Olympiad, 3
As the graph, a pond is divided into 2n (n $\geq$ 5) parts. Two parts are called neighborhood if they have a common side or arc. Thus every part has three neighborhoods. Now there are 4n+1 frogs at the pond. If there are three or more frogs at one part, then three of the frogs of the part will jump to the three neighborhoods repsectively. Prove that for some time later, the frogs at the pond will uniformily distribute. That is, for any part either there are frogs at the part or there are frogs at the each of its neighborhoods.
[img]http://www.mathlinks.ro/Forum/files/china2005_2_214.gif[/img]
1997 Finnish National High School Mathematics Competition, 5
For an integer $n\geq 3$, place $n$ points on the plane in such a way that all the distances between the points are at most one and exactly $n$ of the pairs of points have the distance one.
2006 Hungary-Israel Binational, 3
A group of $ 100$ students numbered $ 1$ through $ 100$ are playing the following game. The judge writes the numbers $ 1$, $ 2$, $ \ldots$, $ 100$ on $ 100$ cards, places them on the table in an arbitrary order and turns them over. The students $ 1$ to $ 100$ enter the room one by one, and each of them flips $ 50$ of the cards. If among the cards flipped by student $ j$ there is card $ j$, he gains one point. The flipped cards are then turned over again. The students cannot communicate during the game nor can they see the cards flipped by other students. The group wins the game if each student gains a point. Is there a strategy giving the group more than $ 1$ percent of chance to win?
2013 Finnish National High School Mathematics Competition, 2
In a particular European city, there are only $7$ day tickets and $30$ day tickets to the public transport. The former costs $7.03$ euro and the latter costs $30$ euro. Aina the Algebraist decides to buy at once those tickets that she can travel by the public transport the whole three year (2014-2016, 1096 days) visiting in the city. What is the cheapest solution?