A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.
Does the number of 1999 occur in the Pascal triangle?
How many times greater is the sum of the numbers in the hundred and first line of the Pascal triangle than the sum of the numbers in the hundredth line?
Let’s put plus and minus signs in the 99th line of Pascal’s triangle. Between the first and second number there is a minus sign, between the second and the third there is a plus sign, between the third and the fourth there is a minus sign, then again a plus sign, and so on. Find the value of the resulting expression.
a) Can 4 points be placed on a plane so that each of them is connected by segments with three points (without intersections)?
b) Can 6 points be placed on a plane and connected by non-intersecting segments so that exactly 4 segments emerge from each point?
There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.
9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.
A spherical planet is surrounded by 25 point asteroids. Prove, that at any given moment there will be a point on the surface of the planet from which an astronomer will not be able to see more than 11 asteroids.
In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside or on the boundary the quadrilateral.