Found problems: 1488
2010 Postal Coaching, 6
Students have taken a test paper in each of $n \ge 3$ subjects. It is known that in any subject exactly three students got the best score, and for any two subjects exactly one student got the best scores in both subjects. Find the smallest $n$ for which the above conditions imply that exactly one student got the best score in each of the $n$ subjects.
2014 European Mathematical Cup, 2
Jeck and Lisa are playing a game on table dimensions $m \times n$ , where $m , n >2$. Lisa starts so that she puts knight figurine on arbitrary square of table.After that Jeck and Lisa put new figurine on table by the following rules:
[b]1.[/b] Jeck puts queen figurine on any empty square of a table which is two squares vertically and one square horizontally distant, or one square vertically and two squares horizontally distant from last knight figurine which Lisa put on the table
[b]2.[/b] Lisa puts knight figurine on any empty square of a table which is in the same row, column or diagonal as last queen figurine Jeck put on the table.
Player which cannot put his figurine loses. For which pairs of $(m,n)$ Lisa has winning strategy?
[i]
Proposed by Stijn Cambie[/i]
2003 Tournament Of Towns, 1
There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?
2014 Contests, 3
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal Rolínek)
1988 IMO Longlists, 78
It is proposed to partition a set of positive integers into two disjoint subsets $ A$ and $ B$ subject to the conditions
[b]i.)[/b] 1 is in $ A$
[b]ii.)[/b] no two distinct members of $ A$ have a sum of the form $ 2^k \plus{} 2, k \equal{} 0,1,2, \ldots;$ and
[b]iii.)[/b] no two distinct members of B have a sum of that form.
Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.
2013 Romania Team Selection Test, 4
Let $n$ be an integer greater than 1. The set $S$ of all diagonals of a $ \left( 4n-1\right) $-gon is partitioned into $k$ sets, $S_{1},S_{2},\ldots ,S_{k},$ so that, for every pair of distinct indices $i$ and $j,$ some diagonal in $S_{i}$ crosses some diagonal in $S_{j};$ that is, the two diagonals share an interior point. Determine the largest possible value of $k $ in terms of $n.$
2007 Balkan MO, 4
For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that
$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.
1985 IMO Longlists, 6
On a one-way street, an unending sequence of cars of width $a$, length $b$ passes with velocity $v$. The cars are separated by the distance $c$. A pedestrian crosses the street perpendicularly with velocity $w$, without paying attention to the cars.
[b](a)[/b] What is the probability that the pedestrian crosses the street uninjured?
[b](b)[/b] Can he improve this probability by crossing the road in a direction other than perpendicular?
2010 Tournament Of Towns, 5
In a tournament with $55$ participants, one match is played at a time, with the loser dropping out. In each match, the numbers of wins so far of the two participants differ by not more than $1$. What is the maximal number of matches for the winner of the tournament?
2006 APMO, 5
In a circus, there are $n$ clowns who dress and paint themselves up using a selection of 12 distinct colours. Each clown is required to use at least five different colours. One day, the ringmaster of the circus orders that no two clowns have exactly the same set of colours and no more than 20 clowns may use any one particular colour. Find the largest number $n$ of clowns so as to make the ringmaster's order possible.
2013 Gulf Math Olympiad, 3
There are $n$ people standing on a circular track. We want to perform a number of [i]moves[/i] so that we end up with a situation where the distance between every two neighbours is the same. The [i]move[/i] that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move.
Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.
1996 Vietnam National Olympiad, 3
Let be given integers k and n such that 1<=k<=n. Find the number of ordered k-tuples (a_1,a_2,...,a_n), where a_1, a_2, ..., a_k are different and in the set {1,2,...,n}, satisfying
1) There exist s, t such that 1<=s<t<=k and a_s>a_t.
2) There exists s such that 1<=s<=k and a_s is not congruent to s mod 2.
P.S. This is the only problem from VMO 1996 I cannot find a solution or I cannot solve. But I'm not good at all in combinatoric...
2014 Iran MO (2nd Round), 3
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.
2013 South africa National Olympiad, 5
Some coins are placed on a $20 \times 13$ board. Two coins are called [i]neighbours[/i] if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?
2007 Federal Competition For Advanced Students, Part 2, 2
38th Austrian Mathematical Olympiad 2007, round 3 problem 5
Given is a convex $ n$-gon with a triangulation, that is a partition into triangles through diagonals that don’t cut each other. Show that it’s always possible to mark the $ n$ corners with the digits of the number $ 2007$ such that every quadrilateral consisting of $ 2$ neighbored (along an edge) triangles has got $ 9$ as the sum of the numbers on its $ 4$ corners.
1995 Belarus National Olympiad, Problem 8
Five numbers 1,2,3,4,5 are written on a blackboard. A student may
erase any two of the numbers a and b on the board and write the
numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?
1983 USAMO, 1
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.
1970 Regional Competition For Advanced Students, 2
In the plane seven different points $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3$ are given. The points $P_1, P_2, P_3, P_4$ are on the straight line $p$, the points $Q_1, Q_2, Q_3$ are not on $p$. By each of the three points $Q_1, Q_2, Q_3$ the perpendiculars are drawn on the straight lines connecting points different of them. Prove that the maximum's number of the possibles intersections of all perpendiculars is to 286, if the points $Q_1, Q_2, Q_3$ are taken in account as intersections.
2001 IberoAmerican, 3
Let $S$ be a set of $n$ elements and $S_1,\ S_2,\dots,\ S_k$ are subsets of $S$ ($k\geq2$), such that every one of them has at least $r$ elements.
Show that there exists $i$ and $j$, with $1\leq{i}<j\leq{k}$, such that the number of common elements of $S_i$ and $S_j$ is greater or equal to: $r-\frac{nk}{4(k-1)}$
1997 Federal Competition For Advanced Students, Part 2, 2
We define the following operation which will be applied to a row of bars being situated side-by-side on positions $1, 2, \ldots ,N$. 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 $1, \ldots,M$. From an initial number $a_0 > 0$ of bars there originates a sequence $(a_n)_{n \geq 0}$, where an is the number of bars after having applied the operation $n$ times.
[b](a)[/b] Prove that for no $n > 0$ can we have $a_n = 1997$.
[b](b)[/b] Determine all natural numbers that can only occur as $a_0$ or $a_1$.
2006 China Team Selection Test, 3
$d$ and $n$ are positive integers such that $d \mid n$. The n-number sets $(x_1, x_2, \cdots x_n)$ satisfy the following condition:
(1) $0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq n$
(2) $d \mid (x_1+x_2+ \cdots x_n)$
Prove that in all the n-number sets that meet the conditions, there are exactly half satisfy $x_n=n$.
2014 Iran MO (3rd Round), 3
We have a $10 \times 10$ table. $T$ is a set of rectangles with vertices from the table and sides parallel to the sides of the table such that no rectangle from the set is a subrectangle of another rectangle from the set. $t$ is the maximum number of elements of $T$.
(a) Prove that $t>300$.
(b) Prove that $t<600$.
[i]Proposed by Mir Omid Haji Mirsadeghi and Kasra Alishahi[/i]
1996 Vietnam Team Selection Test, 2
There are some people in a meeting; each doesn't know at least 56 others, and for any pair, there exist a third one who knows both of them. Can the number of people be 65?
2014 Portugal MO, 6
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?
2009 Croatia Team Selection Test, 2
In each field of 2009*2009 table you can write either 1 or -1.
Denote Ak multiple of all numbers in k-th row and Bj the multiple of all numbers in j-th column.
Is it possible to write the numbers in such a way that
$ \sum_{i\equal{}1}^{2009}{Ai}\plus{} \sum_{i\equal{}1}^{2009}{Bi}\equal{}0$?