Found problems: 1488
2006 South East Mathematical Olympiad, 4
Given a circle with its perimeter equal to $n$( $n \in N^*$), the least positive integer $P_n$ which satisfies the following condition is called the “[i]number of the partitioned circle[/i]”: there are $P_n$ points ($A_1,A_2, \ldots ,A_{P_n}$) on the circle; For any integer $m$ ($1\le m\le n-1$), there always exist two points $A_i,A_j$ ($1\le i,j\le P_n$), such that the length of arc $A_iA_j$ is equal to $m$. Furthermore, all arcs between every two adjacent points $A_i,A_{i+1}$ ($1\le i\le P_n$, $A_{p_n+1}=A_1$) form a sequence $T_n=(a_1,a_2,,,a_{p_n})$ called the “[i]sequence of the partitioned circle[/i]”. For example when $n=13$, the number of the partitioned circle $P_{13}$=4, the sequence of the partitioned circle $T_{13}=(1,3,2,7)$ or $(1,2,6,4)$. Determine the values of $P_{21}$ and $P_{31}$, and find a possible solution of $T_{21}$ and $T_{31}$ respectively.
1993 All-Russian Olympiad Regional Round, 11.5
The expression $ x^3 \plus{} . . . x^2 \plus{} . . . x \plus{} ... \equal{} 0$ is written on the blackboard. Two pupils alternately replace the dots by real numbers. The first pupil
attempts to obtain an equation having exactly one real root. Can his opponent spoil his efforts?
2005 MOP Homework, 6
A $10 \times 10 \times 10$ cube is made up up from $500$ white unit cubes and $500$ black unit cubes, arranged in such a way that every two unit cubes that shares a face are in different colors. A line is a $1 \times 1 \times 10$ portion of the cube that is parallel to one of cube’s edges. From the initial cube have been removed $100$ unit cubes such that $300$ lines of the cube has exactly one missing cube.
Determine if it is possible that the number of removed black unit cubes is divisible by $4$.
2006 MOP Homework, 2
Mykolka the numismatist possesses $241$ coins, each worth an integer number of turgiks. The total value of the coins is $360$ turgiks. Is it necessarily true that the coins can be divided into three groups of equal total value?
2003 Bulgaria National Olympiad, 1
Let $x_1, x_2 \ldots , x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p+x_q+x_r$ (with $1\le p<q<r\le 5$) are equal to $0$, then $x_1=x_2=\cdots=x_5=0$.
2014 Saudi Arabia IMO TST, 2
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned in games against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of his points against the other nine of the ten). What was the total number of players in the tournament?
2010 China Team Selection Test, 3
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.
1994 Kurschak Competition, 2
Prove that if we erase $n-3$ diagonals of a regular $n$-gon, then we may still choose $n-3$ of the remaining diagonals such that they don't intersect inside the $n$-gon; but it is possible to erase $n-2$ diagonals such that this statement doesn't hold.
1979 IMO Longlists, 3
Is it possible to partition $3$-dimensional Euclidean space into $1979$ mutually isometric subsets?
2005 Singapore MO Open, 4
Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.
2011 Postal Coaching, 4
Consider $2011^2$ points arranged in the form of a $2011 \times 2011$ grid. What is the maximum number of points that can be chosen among them so that no four of them form the vertices of either an isosceles trapezium or a rectangle whose parallel sides are parallel to the grid lines?
2013 Polish MO Finals, 6
For each positive integer $n$ determine the maximum number of points in space creating the set $A$ which has the following properties:
$1)$ the coordinates of every point from the set $A$ are integers from the range $[0, n]$
$2)$ for each pair of different points $(x_1,x_2,x_3), (y_1,y_2,y_3)$ belonging to the set $A$ it is satisfied at least one of the following inequalities $x_1< y_1, x_2<y_2, x_3<y_3$ and at least one of the following inequalities $x_1>y_1, x_2>y_2,x_3>y_3$.
2005 MOP Homework, 2
Set $S=\{1,2,...,2004\}$. We denote by $d_1$ the number of subset of $S$ such that the sum of elements of the subset has remainder $7$ when divided by $32$. We denote by $d_2$ the number of subset of $S$ such that the sum of elements of the subset has remainder $14$ when divided by $16$. Compute $\frac{d_1}{d_2}$.
2004 India IMO Training Camp, 4
Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.
2006 India IMO Training Camp, 3
Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds
[list][b](a)[/b] $|B| > 25$,
[b](b)[/b] $F(B)={\O}$,
[b](c)[/b] $\bigcap_{j\in F(B)} A_j \neq {\O}$.[/list]
Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.
2004 Korea - Final Round, 3
2004 computers make up a network using several cables. If for a subset $S$ in the set of all computers, there isn't a cable that connects two computers in $S$, $S$ is called independant. One lets the arbitrary independant set consists at most 50 computers, and uses the least number of cables.
(1) Let $c(L)$ be the number of cables which connects the computer $L$. Prove that for two computers $A,B$, $c(A)=c(B)$ if there is a cable which connects $A$ and $B$, $|c(A)-c(B)|\leq 1$ otherwise.
(2) Determine the number of used cables.
2014 South East Mathematical Olympiad, 8
Define a figure which is constructed by unit squares "cross star" if it satisfies the following conditions:
$(1)$Square bar $AB$ is bisected by square bar $CD$
$(2)$At least one square of $AB$ lay on both sides of $CD$
$(3)$At least one square of $CD$ lay on both sides of $AB$
There is a rectangular grid sheet composed of $38\times 53=2014$ squares,find the number of such cross star in this rectangle sheet
2005 Tuymaada Olympiad, 1
The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers?
[i]Proposed by S. Berlov[/i]
2014 China Western Mathematical Olympiad, 3
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
2006 Italy TST, 1
Let $S$ be a string of $99$ characters, $66$ of which are $A$ and $33$ are $B$. We call $S$ [i]good[/i] if, for each $n$ such that $1\le n \le 99$, the sub-string made from the first $n$ characters of $S$ has an odd number of distinct permutations. How many good strings are there? Which strings are good?
2001 Korea - Final Round, 3
For a positive integer $n \ge 5$, let $a_i,b_i (i = 1,2, \cdots ,n)$ be integers satisfying the
following two conditions:
[list]
(a) The pairs $(a_i,b_i)$ are distinct for $i = 1,2,\cdots,n$;
(b) $|a_1b_2-a_2b_1| = |a_2b_3-a_3b_2| = \cdots = |a_nb_1-a_1b_n| = 1$.
[/list]
Prove that there exist indices $i,j$ such that $1<|i-j|<n-1$ and $|a_ib_j-a_jb_i|=1$.
2006 India IMO Training Camp, 3
Let $A_1,A_2,\ldots,A_n$ be subsets of a finite set $S$ such that $|A_j|=8$ for each $j$. For a subset $B$ of $S$ let $F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}$. Suppose for each subset $B$ of $S$ at least one of the following conditions holds
[list][b](a)[/b] $|B| > 25$,
[b](b)[/b] $F(B)={\O}$,
[b](c)[/b] $\bigcap_{j\in F(B)} A_j \neq {\O}$.[/list]
Prove that $A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}$.
2010 China Team Selection Test, 3
Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying
(1) for each $n_i$, its digits belong to the set $\{1,2\}$;
(2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right.
Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.
1978 IMO Longlists, 36
The integers $1$ through $1000$ are located on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e.,$1, 16, 31, \cdots$ ) is marked. The marking is continued until an already marked number is reached. How many of the numbers will be left unmarked?
2005 MOP Homework, 1
Two rooks on a chessboard are said to be attacking each other if they are placed in the same row or column of the board.
(a) There are eight rooks on a chessboard, none of them attacks any other. Prove that there is an even number of rooks on black fields.
(b) How many ways can eight mutually non-attacking rooks be placed on the 9 £ 9 chessboard so that all eight rooks are on squares of the same color.