Found problems: 1488
2004 Brazil National Olympiad, 4
Consider all the ways of writing exactly ten times each of the numbers $0, 1, 2, \ldots , 9$ in the squares of a $10 \times 10$ board.
Find the greatest integer $n$ with the property that there is always a row or a column with $n$ different numbers.
2012 Vietnam Team Selection Test, 3
There are $42$ students taking part in the Team Selection Test. It is known that every student knows exactly $20$ other students. Show that we can divide the students into $2$ groups or $21$ groups such that the number of students in each group is equal and every two students in the same group know each other.
2013 South East Mathematical Olympiad, 4
There are $12$ acrobats who are assigned a distinct number ($1, 2, \cdots , 12$) respectively. Half of them stand around forming a circle (called circle A); the rest form another circle (called circle B) by standing on the shoulders of every two adjacent acrobats in circle A respectively. Then circle A and circle B make up a formation. We call a formation a “[i]tower[/i]” if the number of any acrobat in circle B is equal to the sum of the numbers of the two acrobats whom he stands on. How many heterogeneous [i]towers[/i] are there?
(Note: two [i]towers[/i] are homogeneous if either they are symmetrical or one may become the other one by rotation. We present an example of $8$ acrobats (see attachment). Numbers inside the circle represent the circle A; numbers outside the circle represent the circle B. All these three formations are “[i]towers[/i]”, however they are homogeneous [i]towers[/i].)
2010 Vietnam Team Selection Test, 3
We call a rectangle of the size $1 \times 2$ a domino. Rectangle of the $2 \times 3$ removing two opposite (under center of rectangle) corners we call tetramino. These figures can be rotated.
It requires to tile rectangle of size $2008 \times 2010$ by using dominoes and tetraminoes. What is the minimal number of dominoes should be used?
2010 Contests, 1
In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each.
It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.
2005 Postal Coaching, 13
Let $a_1 < a_2 < .... < a_n < 2n$ ne $n$ positive integers such that $a_j$ does not divide $a_k$ or $j \not= k$. Prove that $a_1 \geq 2^{k}$ where $k$ is defined by the condition $3^{k} < 2n < 3^{k+1}$ and show that it is the best estimate for $a_1$
2010 South africa National Olympiad, 6
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.
2002 China Girls Math Olympiad, 8
Assume that $ A_1, A_2, \ldots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $ l$ taken arbitrarily on the plane, assume that projections of $ A_1, A_2, \ldots, A_8$ on the line are $ P_1, P_2, \ldots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $ P_{i_1}, P_{i_2}, \ldots, P_{i_8}$ according to the direction of line $ l.$ Thus we get one permutation for $ 1, 2, \ldots, 8,$ namely, $ i_1, i_2, \ldots, i_8.$ In the figure, this permutation is $ 2, 1, 8, 3, 7, 4, 6, 5.$ Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $ N_8 \equal{} N(A_1, A_2, \ldots, A_8).$ Find the maximal value of $ N_8.$
2004 APMO, 3
Let a set $S$ of 2004 points in the plane be given, no three of which are collinear. Let ${\cal L}$ denote the set of all lines (extended indefinitely in both directions) determined by pairs of points from the set. Show that it is possible to colour the points of $S$ with at most two colours, such that for any points $p,q$ of $S$, the number of lines in ${\cal L}$ which separate $p$ from $q$ is odd if and only if $p$ and $q$ have the same colour.
Note: A line $\ell$ separates two points $p$ and $q$ if $p$ and $q$ lie on opposite sides of $\ell$ with neither point on $\ell$.
2005 Hungary-Israel Binational, 3
There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?
2000 Polish MO Finals, 2
In the unit squre For the given natural number $n \geq 2$ find the smallest number $k$ that from each set of $k$ unit squares of the $n$x$n$ chessboard one can achoose a subset such that the number of the unit squares contained in this subset an lying in a row or column of the chessboard is even
2003 ITAMO, 6
Every of $n$ guests invited to a dinner has got an invitation denoted by a number from $1$ to $n$. The guests will be sitting around a round table with $n$ seats. The waiter has decided to derve them according to the following rule. At first, he selects one guest and serves him/her at any place. Thereafter, he selects the guests one by one: having chosen a guest, he goes around the table for the number of seats equal to the preceeding guest's invitation number (starting from the seat of the preceeding guest), and serves the guest there.
Find all $n$ for which he can select the guests in such an order to serve all the guests.
2001 Tournament Of Towns, 2
At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group. Prove that at least 3/4 of all “F” marks were given to the same student.
2002 India IMO Training Camp, 12
Let $a,b$ be integers with $0<a<b$. A set $\{x,y,z\}$ of non-negative integers is [i]olympic[/i] if $x<y<z$ and if $\{z-y,y-x\}=\{a,b\}$. Show that the set of all non-negative integers is the union of pairwise disjoint olympic sets.
1981 Canada National Olympiad, 5
$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the festival last?
1972 IMO Longlists, 33
A rectangle $ABCD$ is given whose sides have lengths $3$ and $2n$, where $n$ is a natural number. Denote by $U(n)$ the number of ways in which one can cut the rectangle into rectangles of side lengths $1$ and $2$.
$(a)$ Prove that
\[U(n + 1)+U(n -1) = 4U(n);\]
$(b)$ Prove that
\[U(n) =\frac{1}{2\sqrt{3}}[(\sqrt{3} + 1)(2 +\sqrt{3})^n + (\sqrt{3} - 1)(2 -\sqrt{3})^n].\]
2009 China Girls Math Olympiad, 7
On a $ 10 \times 10$ chessboard, some $ 4n$ unit squares are chosen to form a region $ \mathcal{R}.$ This region $ \mathcal{R}$ can be tiled by $ n$ $ 2 \times 2$ squares. This region $ \mathcal{R}$ can also be tiled by a combination of $ n$ pieces of the following types of shapes ([i]see below[/i], with rotations allowed).
Determine the value of $ n.$
2007 India Regional Mathematical Olympiad, 4
How many 6-digit numbers are there such that-:
a)The digits of each number are all from the set $ \{1,2,3,4,5\}$
b)any digit that appears in the number appears at least twice ?
(Example: $ 225252$ is valid while $ 222133$ is not)
[b][weightage 17/100][/b]
2007 Bundeswettbewerb Mathematik, 4
A regular hexagon, as shown in the attachment, is dissected into 54 congruent equilateral triangles by parallels to its sides. Within the figure we yield exactly 37 points which are vertices of at least one of those triangles. Those points are enumerated in an arbitrary way. A triangle is called [i]clocky[/i] if running in a clockwise direction from the vertex with the smallest assigned number, we pass a medium number and finally reach the vertex with the highest number. Prove that at least 19 out of 54 triangles are clocky.
2003 India IMO Training Camp, 5
On the real number line, paint red all points that correspond to integers of the form $81x+100y$, where $x$ and $y$ are positive integers. Paint the remaining integer point blue. Find a point $P$ on the line such that, for every integer point $T$, the reflection of $T$ with respect to $P$ is an integer point of a different colour than $T$.
2011 Poland - Second Round, 2
$\forall n\in \mathbb{Z_{+}}-\{1,2\}$ find the maximal length of a sequence with elements from a set $\{1,2,\ldots,n\}$, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form $x,y,x,y$ ($x\neq y$).
2008 Spain Mathematical Olympiad, 3
Every point in the plane is coloured one of seven distinct colours. Is there an inscribed trapezoid whose vertices are all of the same colour?
1999 Tuymaada Olympiad, 1
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.
[i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]
1977 Canada National Olympiad, 7
A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram). A woman lives in the southwest corner of the city and works in the northeast corner. She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice. Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$.
[asy]
unitsize(0.4 cm);
for(int i = 0; i <= 11; ++i) {
draw((i,0)--(i,7));
}
for(int j = 0; j <= 7; ++j) {
draw((0,j)--(11,j));
}
label("$\underbrace{\hspace{4.4 cm}}$", (11/2,-0.5));
label("$\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}$", (11,7/2));
label("$m$ blocks", (11/2,-1.5));
label("$n$ blocks", (14,7/2));
[/asy]
2010 Argentina Team Selection Test, 1
In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each.
It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.