Found problems: 14842
2013 Iran MO (3rd Round), 5
Consider a graph with $n$ vertices and $\frac{7n}{4}$ edges.
(a) Prove that there are two cycles of equal length.
(25 points)
(b) Can you give a smaller function than $\frac{7n}{4}$ that still fits in part (a)? Prove your claim.
We say function $a(n)$ is smaller than $b(n)$ if there exists an $N$ such that for each $n>N$ ,$a(n)<b(n)$
(At most 5 points)
[i]Proposed by Afrooz Jabal'ameli[/i]
1980 Swedish Mathematical Competition, 3
Let $T(n)$ be the number of dissimilar (non-degenerate) triangles with all side lengths integral and $\leq n$. Find $T(n+1)-T(n)$.
2002 Kurschak Competition, 3
Prove that the edges of a complete graph with $3^n$ vertices can be partitioned into disjoint cycles of length $3$.
2020 Iran Team Selection Test, 1
A weighted complete graph with distinct positive wights is given such that in every triangle is [i]degenerate [/i] that is wight of an edge is equal to sum of two other. Prove that one can assign values to the vertexes of this graph such that the wight of each edge is the difference between two assigned values of the endpoints.
[i]Proposed by Morteza Saghafian [/i]
2014 Contests, 1
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
2012 BMT Spring, 4
There are 1$2$ people labeled $1, ..., 12$ working together on $12$ missions, with people $1, ... , i $working on the $i$th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability $1/2$. Given that the first $11$ missions succeed, and the $12$th mission fails, what is the probability that person $12$ is the spy?
2021 Science ON all problems, 2
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/i]
MathLinks Contest 3rd, 3
An integer point of the usual Euclidean $3$-dimensional space is a point whose three coordinates are all integers. A set $S$ of integer points is called a [i]covered [/i] set if for all points $A, B$ in $S$ each integer point in the segment $[AB]$ is also in $S$.
Determine the maximum number of elements that a covered set can have if it does not contain $2004$ collinear points.
2012 Saint Petersburg Mathematical Olympiad, 6
On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.
KoMaL A Problems 2019/2020, A. 767
In an $n\times n$ array all the fields are colored with a different color. In one move one can choose a row, move all the fields one place to the right, and move the last field (from the right) to the leftmost field of the row; or one can choose a column, move all the fields one place downwards, and move the field at the bottom of the column to the top field of the same column. For what values of $n$ is it possible to reach any arrangement of the $n^2$ fields using these kinds of steps?
[i]Proposed by Ádám Schweitzer[/i]
2000 Estonia National Olympiad, 5
At a given plane with $2,000$ lines, all those with an odd number of different points of intersection with intersecting lines.
a) Can there be an odd number of red lines if in the plane given there are no parallel lines?
b) Can there be an odd number of red lines if none of any 3 given lines intersect at one point?
2017 Portugal MO, 3
In an athletics tournament, five teams participate. Each athlete has a shirt numbered with a positive integer, and all athletes on the same team have different numbers. Each athlete participates in a single event and only one athlete from each team is present in each event. Emídio noticed that the sum of the athletes' jersey numbers in each event is always $20$. What is the maximum number of athletes in the tournament?
2013 Iran MO (3rd Round), 3
$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order.
A set of overtaking actions is called "small" if any car overtakes at most once.
A set of overtaking actions is called "complete" if any car overtakes exactly once.
If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that
\[\mid F\mid=2\mid G\mid\]
(20 points)
[i]Proposed by Morteza Saghafian[/i]
2010 Contests, 2
Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue.
Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.
2010 Contests, 4
How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6)$ are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6$ is from the set $ \{1,2,3,4\}$ and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for $ j \equal{} 1,2,3,4,5,6$ (where $ a_7$ is to be taken as $ a_1$) are all equal to one another?
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
1978 Austrian-Polish Competition, 9
In a convex polygon $P$ some diagonals have been drawn, without intersections inside $P$. Show that there exist at least two vertices of $P$, neither one of them being an endpoint of any one of those diagonals.
2010 IberoAmerican, 3
Around a circular table sit $12$ people, and on the table there are $28$ vases. Two people can see each other, if and only if there is no vase lined with them. Prove that there are at least two people who can be seen.
2014 Rioplatense Mathematical Olympiad, Level 3, 1
Let $n \ge 3$ be a positive integer. Determine, in terms of $n$, how many triples of sets $(A,B,C)$ satisfy the conditions:
$\bullet$ $A, B$ and $C$ are pairwise disjoint , that is, $A \cap B = A \cap C= B \cap C= \emptyset$.
$\bullet$ $A \cup B \cup C= \{ 1 , 2 , ... , n \}$.
$\bullet$ The sum of the elements of $A$, the sum of the elements of $B$ and the sum of the elements of $C$ leave the same remainder when divided by $3$.
Note: One or more of the sets may be empty.
2022 Germany Team Selection Test, 3
A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or
[*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter.
[i]Proposed by Aron Thomas[/i]
2023 BMT, 4
A grasshopper is traveling on the coordinate plane, starting at the origin $(0, 0)$. Each hop, the grasshopper chooses to move $1$ unit up, down, left, or right with equal probability. The grasshopper hops $4$ times and stops at point $P$. Compute the probability that it is possible to return to the origin from $P$ in at most $3$ hops.
1977 All Soviet Union Mathematical Olympiad, 250
Given scales and a set of $n$ different weights. We take weights in turn and add them on one of the scales sides. Let us denote "$L$" the scales state with the left side down, and "$R$" -- with the right side down.
a) Prove that you can arrange the weights in such an order, that we shall obtain the sequence $LRLRLRLR...$ of the scales states. (That means that the state of the scales will be changed after putting every new weight.)
b) Prove that for every $n$-letter word containing $R$'s and $L$'s only you can arrange the weights in such an order, that the sequence of the scales states will be described by that word.
2022 Purple Comet Problems, 28
Six gamers play a round-robin tournament where each gamer plays one game against each of the other five gamers. In each game there is one winner and one loser where each player is equally likely to win that game, and the result of each game is independent of the results of the other games. The probability that the tournament will end with exactly one gamer scoring more wins than any other player is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2025 Bangladesh Mathematical Olympiad, P5
In an $N \times N$ table consisting of small unit squares, some squares are coloured black and the other squares are coloured white. For each pair of columns and each pair of rows, the four squares on the intersections of these rows and columns must not all be of the same colour. What is the largest possible value of $N$?
2019 IMO Shortlist, C8
Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.
Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.