Filter Problems

Showing 1 to 20 of 55 entries

Alice took a red marker and marked 5 points with integer coordinates on a coordinate plane. Miriam took a blue marker and marked a midpoint for each pair of red points. Prove that at least 1 of the blue points has integer coordinates.

With a red marker, Margaret marked three points with integer coordinates on a number line. With a blue marker, Angelina marked a midpoint for every pair of red points. Prove that at least 1 of the blue points has an integer coordinate.

In a regular shape with 25 vertices, all the diagonals are drawn.

Prove that there are no nine diagonals passing through one interior point of the shape.

Some points with integer co-ordinates are marked on a Cartesian plane. It is known that no four points lie on the same circle. Prove that there will be a circle of radius 1995 in the plane, which does not contain a single marked point.

The smell of a flowering lavender plant diffuses through a radius of 20m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.

15 points are placed inside a $4 \times 4$ square. Prove that it is possible to cut a unit square out of the $4 \times 4$ square that does not contain any points.

A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon $($see the drawing$)$. Is it possible to reconstruct the original square using just this information?

A carpet of size 4m by 4m has had 15 holes made in it by a moth. Is it always possible to cut out a 1m x 1m area of carpet that doesn’t contain any holes? The holes are considered to be points.

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive l there exists a line segment of length l with both ends of the same colour.

A square $ABCD$ contains 5 points. Prove that the distance between some pair of these points does not exceed $ \frac{1}{2} AC$

A regular 1981-gon has 64 vertices. Prove that there exists a trapezium with vertices at the marked points.

There are several squares on a rectangular sheet of chequered paper of size $m \times n$ cells, the sides of which run along the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square contains another square within itself. What is the largest number of such squares?

On a circle of radius 1, the point O is marked and from this point, to the right, a notch is marked using a compass of radius l. From the obtained notch $O_1$, a new notch is marked, in the same direction with the same radius and this is process is repeated 1968 times. After this, the circle is cut at all 1968 notches, and we get 1968 arcs. How many different lengths of arcs can this result in?

Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

In draughts, the king attacks by jumping over another draughts-piece. What is the maximum number of draughts kings we can place on the black squares of a standard 8×8 draughts board, so that each king is attacking at least one other?

1956 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

There are 40 identical cords. If you set any cord on fire on one side, it burns, and if you set it alight on the other side, it will not burn. Ahmed arranges the cords in the form of a square $($see the figure below, each cord makes up a side of a cell$)$. Then, Helen arranges 12 fuses. Will Ahmed be able to lay out the cords in such a way that Helen will not be able to burn all of them?

What is the minimum number of 1×1 squares that need to be drawn in order to get an image of a 25×25 square divided into 625 smaller 1×1 squares?

A castle is surrounded by a circular wall with nine towers, at which there are knights on duty. At the end of each hour, they all move to the neighbouring towers, each knight moving either clockwise or counter-clockwise. During the night, each knight stands for some time at each tower. It is known that there was an hour when at least two knights were on duty at each tower, and there was an hour when there was precisely one knight on duty on each of exactly five towers. Prove that there was an hour when there were no knights on duty on one of the towers.

What is the minimum number of points necessary to mark inside a convex $n$-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?