Found problems: 1704
2022 Novosibirsk Oral Olympiad in Geometry, 6
Anton has an isosceles right triangle, which he wants to cut into $9$ triangular parts in the way shown in the picture. What is the largest number of the resulting $9$ parts that can be equilateral triangles?
A more formal description of partitioning. Let triangle $ABC$ be given. We choose two points on its sides so that they go in the order $AC_1C_2BA_1A_2CB_1B_2$, and no two coincide. In addition, the segments $C_1A_2$, $A_1B_2$ and $B_1C_2$ must intersect at one point. Then the partition is given by segments $C_1A_2$, $A_1B_2$, $B_1C_2$, $A_1C_2$, $B_1A_2$ and $C_1B_2$.
[img]https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png[/img]
1995 Tournament Of Towns, (453) 2
Four grasshoppers sit at the vertices of a square. Every second, one of them jumps over one of the others to the symmetrical point on the other side (if $X$ jumps over $Y$ to the point $X'$, then $X$, $Y$ and $X'$ lie on a straight line and $XY = YX'$). Prove that after several jumps no three grasshoppers can be:
(a) on a line parallel to a side of the square,
(b) on a straight line.
(AK Kovaldzhy)
2013 China Northern MO, 1
Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.
2007 Switzerland - Final Round, 10
The plane is divided into equilateral triangles of side length $1$. Consider a equilateral triangle of side length $n$ whose sides lie on the grid lines. On every grid point on the edge and inside of this triangle lies a stone. In a move, a unit triangle is selected, which has exactly $2$ corners with is covered with a stone. The two stones are removed, and the third corner is turned a new stone was laid. For which $n$ is it possible that after finitely many moves only one stone left?
1963 All Russian Mathematical Olympiad, 039
On the ends of the diameter two "$1$"s are written. Each of the semicircles is divided onto two parts and the sum of the numbers of its ends (i.e. "$2$") is written at the midpoint. Then every of the four arcs is halved and in its midpoint the sum of the numbers on its ends is written. Find the total sum of the numbers on the circumference after $n$ steps.
2017-IMOC, C2
On a large chessboard, there are $4$ puddings that form a square with size $1$. A pudding $A$ could jump over a pudding $B$, or equivalently, $A$ moves to the symmetric point with respect to $B$. Is it possible that after finite times of jumping, the puddings form a square with size $2$?
2001 Argentina National Olympiad, 6
Given a rectangle $\mathcal{R}$ of area $100000 $, Pancho must completely cover the rectangle $\mathcal{R}$ with a finite number of rectangles with sides parallel to the sides of $\mathcal{R}$ . Next, Martín colors some rectangles of Pancho's cover red so that no two red rectangles have interior points in common. If the red area is greater than $0.00001$, Martin wins. Otherwise, Pancho wins. Prove that Pancho can cover to ensure victory,
1997 ITAMO, 3
The positive quadrant of a coordinate plane is divided into unit squares by lattice lines. Is it possible to color the squares in black and white so that:
(i) In every square of side $n$ ($n \in N$) with a vertex at the origin and sides are parallel to the axes, there are more black than white squares;
(ii) Every diagonal parallel to the line $y = x$ intersects only finitely many black squares?
1998 Singapore MO Open, 2
Let $N$ be the set of natural numbers, and let $f: N \to N$ be a function satisfying $f(x) + f(x + 2) < 2 f(x + 1)$ for any $x \in N$. Prove that there exists a straight line in the $xy$-plane which contains infinitely many points with coordinates $(n,f(n))$.
2016 Tournament Of Towns, 3
Rectangle $p*q,$ where $p,q$ are relatively coprime positive integers with $p <q$ is divided into squares $1*1$.Diagonal which goes from lowest left vertice to highest right cuts triangles from some squares.Find sum of perimeters of all such triangles.
2011 Tournament of Towns, 3
(a) Does there exist an in nite triangular beam such that two of its cross-sections are similar but not congruent triangles?
(b) Does there exist an in nite triangular beam such that two of its cross-sections are equilateral triangles of sides $1$ and $2$ respectively?
2014 Abels Math Contest (Norwegian MO) Final, 3b
Nine points are placed on a circle. Show that it is possible to colour the $36$ chords connecting them using four colours so that for any set of four points, each of the four colours is used for at least one of the six chords connecting the given points
2017 Brazil Undergrad MO, 3
Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1.
a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\].
Where $\sigma(n+1) = \sigma(1)$.
b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$,
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\].
Where $\sigma(n+1) = \sigma(1)$.
2000 All-Russian Olympiad Regional Round, 10.4
For what smallest $n$ can a $n \times n$ square be cut into squares $40 \times 40$ and $49 \times 49$ so that squares of both types are present?
1999 Tournament Of Towns, 4
(a) On each of the $1 \times 1$ squares of the top row of an $8 \times 8$ chessboard there is a black pawn, and on each of the $1 \times 1$ squares of the bottom row of this chessboard there is a white pawn. On each move one can shift any pawn vertically or horizontally to any adjacent empty $1 \times 1$ square. What is the smallest number of moves that are needed to move all white pawns to the top row and all black pawns to the bottom one?
(b) The same question for a $7 \times 7$ board.
(A Shapovalov_
1966 IMO Longlists, 52
A figure with area $1$ is cut out of paper. We divide this figure into $10$ parts and color them in $10$ different colors. Now, we turn around the piece of paper, divide the same figure on the other side of the paper in $10$ parts again (in some different way). Show that we can color these new parts in the same $10$ colors again (hereby, different parts should have different colors) such that the sum of the areas of all parts of the figure colored with the same color on both sides is $\geq \frac{1}{10}.$
Gheorghe Țițeica 2025, P4
Consider $4n$ points in the plane such that no three of them are collinear ($n\geq 1$). Show that the set of centroids of all the triangles formed by any three of these points contains at least $4n$ elements.
[i]Radu Bumbăcea[/i]
1959 Poland - Second Round, 5
In the plane, $ n \geq 3 $ segments are placed in such a way that every $ 3 $ of them have a common point. Prove that there is a common point for all the segments.
2013 Dutch IMO TST, 4
Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.)
What is the minimal value of $N$ for which this is possible?
Kvant 2021, M2660
4 tokens are placed in the plane. If the tokens are now at the vertices of a convex quadrilateral $P$, then the following move could be performed: choose one of the tokens and shift it in the direction perpendicular to the diagonal of $P$ not containing this token; while shifting tokens it is prohibited to get three collinear tokens.
Suppose that initially tokens were at the vertices of a rectangle $\Pi$, and after a number of moves tokens were at the vertices of one another rectangle $\Pi'$ such that $\Pi'$ is similar to $\Pi$ but not equal to $\Pi $. Prove that $\Pi$ is a square.
1999 Singapore Team Selection Test, 2
Is it possible to use $2 \times 1$ dominoes to cover a $2k \times 2k$ checkerboard which has $2$ squares, one of each colour, removed ?
2001 China Second Round Olympiad, 3
An $m\times n(m,n\in \mathbb{N}^*)$ rectangle is divided into some smaller squares. The sides of each square are all parallel to the corresponding sides of the rectangle, and the length of each side is integer. Determine the minimum of the sum of the sides of these squares.
2008 Abels Math Contest (Norwegian MO) Final, 2a
We wish to lay down boards on a floor with width $B$ in the direction across the boards. We have $n$ boards of width $b$, and $B/b$ is an integer, and $nb \le B$. There are enough boards to cover the floor, but the boards may have different lengths. Show that we can cut the boards in such a way that every board length on the floor has at most one join where two boards meet end to end.
[img]https://cdn.artofproblemsolving.com/attachments/f/f/24ce8ae05d85fd522da0e18c0bb8017ca3c8e8.png[/img]
1954 Moscow Mathematical Olympiad, 278
A $17 \times 17$ square is cut out of a sheet of graph paper. Each cell of this square has one of thenumbers from $1$ to $70$. Prove that there are $4$ distinct squares whose centers $A, B, C, D$ are the vertices of a parallelogramsuch that $AB // CD$, moreover, the sum of the numbers in the squares with centers $A$ and $C$ is equal to that in the squares with centers $B$ and $D$.
2014 Turkey Team Selection Test, 3
At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.