Found problems: 14842
2007 Portugal MO, 5
Rua do Antonio has $100$ houses numbered from $1$ to $100$. Any house numbered with the difference between the numbers of two houses of the same color is a different color. Prove that on Rua do Antonio there are houses of at least five different colors.
2019 Canada National Olympiad, 3
You have a $2m$ by $2n$ grid of squares coloured in the same way as a standard checkerboard. Find the total number of ways to place $mn$ counters on white squares so that each square contains at most one counter and no two counters are in diagonally adjacent white squares.
2023 Ukraine National Mathematical Olympiad, 10.8
Consider a complete graph on $4046$ nodes, whose edges are colored in some colors. Let's call this graph $k$-good if we can split all its nodes into $2023$ pairs so that there are exactly $k$ distinct colors among the colors of $2023$ edges that connect the nodes from the same pairs. Is it possible that the graph is $999$-good and $1001$-good but not $1000$-good?
[i]Proposed by Anton Trygub[/i]
2015 Azerbaijan JBMO TST, 2
$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.
2009 India IMO Training Camp, 6
Prove The Following identity:
$ \sum_{j \equal{} 0}^n \left ({3n \plus{} 2 \minus{} j \choose j}2^j \minus{} {3n \plus{} 1 \minus{} j \choose j \minus{} 1}2^{j \minus{} 1}\right ) \equal{} 2^{3n}$.
The Second term on left hand side is to be regarded zero for j=0.
2003 Argentina National Olympiad, 2
On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.
2016 Singapore Junior Math Olympiad, 4
A group of tourists get on $10$ buses in the outgoing trip. The same group of tourists get on $8$ buses in the return trip. Assuming each bus carries at least $1$ tourist, prove that there are at least $3$ tourists such that each of them has taken a bus in the return trip that has more people than the bus he has taken in the outgoing trip.
1954 Moscow Mathematical Olympiad, 286
Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.
2023 BMT, Tie 1
Mataio has a weighted die numbered $1$ to $6$, where the probability of rolling a side $n$ for $1 \le n \le 6$ is inversely proportional to the value of $n$. If Mataio rolls the die twice, what is the probability that the sum of the two rolls is $7$?
2004 Brazil Team Selection Test, Problem 2
Show that there exist infinitely many pairs of positive integers $(m,n)$ such that $\binom m{n-1}=\binom{m-1}n$.
1992 India National Olympiad, 7
Let $n\geq 3$ be an integer. Find the number of ways in which one can place the numbers $1, 2, 3, \ldots, n^2$ in the $n^2$ squares of a $n \times n$ chesboard, one on each, such that the numbers in each row and in each column are in arithmetic progression.
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?
2018 Argentina National Olympiad, 5
In the plane you have $2018$ points between which there are not three on the same line. These points are colored with $30$ colors so that no two colors have the same number of points. All triangles are formed with their three vertices of different colors. Determine the number of points for each of the $30$ colors so that the total number of triangles with the three vertices of different colors is as large as possible.
2004 BAMO, 3
NASA has proposed populating Mars with $2,004$ settlements. The only way to get from one settlement to another will be by a connecting tunnel. A bored bureaucrat draws on a map of Mars, randomly placing $N$ tunnels connecting the settlements in such a way that no two settlements have more than one tunnel connecting them. What is the smallest value of $N$ that guarantees that, no matter how the tunnels are drawn, it will be possible to travel between any two settlements?
2010 Portugal MO, 1
There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.
2016 BMT Spring, 7
Find the coefficient of $x^2$ in the following polynomial
$$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$
2016 Gulf Math Olympiad, 4
4. Suppose that four people A, B, C and D decide to play games of tennis
doubles. They might first play the team A and B against the team C
and D. Next A and C might play B and D. Finally A and D might play
B and C. The advantage of this arrangement is that two conditions are
satisfied.
(a) Each player is on the same team as each other player exactly once.
(b) Each player is on the opposing team to each other player exactly
twice.
Is it possible to arrange a collection of tennis matches satisfying both
condition (a) and condition (b) in the following circumstances?
(i) There are five players.
(ii) There are seven players.
(iii) There are nine players.
2018 HMIC, 5
Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.
India EGMO 2023 TST, 3
Let $N \geqslant 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
[i]Proposed by Sutanay Bhattacharya and Anant Mudgal[/i]
1969 Yugoslav Team Selection Test, Problem 6
Let $E$ be the set of $n^2+1$ closed intervals on the real axis. Prove that there exists a subset of $n+1$ intervals that are monotonically increasing with respect to inclusion, or a subset of $n+1$ intervals none of which contains any other interval from the subset.
2016 Czech And Slovak Olympiad III A, 6
We put a figure of a king on some $6 \times 6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence $35$ jumps visited each box of the chessboard just once.
2008 Bosnia and Herzegovina Junior BMO TST, 4
On circle are $ 2008$ blue and $ 1$ red point(s) given. Are there more polygons which have a red point or those which dont have it??
LMT Team Rounds 2021+, 10
Aidan and Andrew independently select distinct cells in a $4 $ by $4$ grid, as well as a direction (either up, down, left, or right), both at random. Every second, each of them will travel $1$ cell in their chosen direction. Find the probability that Aidan and Andrew willmeet (be in the same cell at the same time) before either one of them hits an edge of the grid. (If Aidan and Andrew cross paths by switching cells, it doesn’t count as meeting.)
1989 Putnam, A5
Show that we can find $\alpha>0$ such that, given any point $P$ inside a regular $2n+1$-gon which is inscribed in a circle radius $1$, we can find two vertices of the polygon whose distance from $P$ differ by less than $\frac1n-\frac\alpha{n^3}$.
2011 Argentina National Olympiad, 2
Three players $A,B$ and $C$ take turns removing stones from a pile of $N$ stones. They move in the order $A,B,C,A,B,C,…A$. The game begins, and the one who takes out the last stone loses the game. The players $A$ and $C$ team up against $B$ , they agree on a joint strategy. $B$ can take in each play $1,2,3,4$ or $5$ stones, while $A$ and $C$, they can each get $1,2$ or $3$ stones each turn. Determine for what values of $N$ have winning strategy $A$ and $C$, and for what values the winning strategy is from $B$.
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