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Found: 219

There are eight points inside a circle of radius 1. Show that there are at least two points with distance between them less then 1.

WizardLand Middle School offers a new elective this year: an astrology class. Fifteen students have registered for this course. Prove that at least 2 of these students were born under the same zodiac sign (there are 12 zodiac signs in total, one for each month).

Leo’s dad was making a pizza for lunch. He decided to place 7 pieces of pineapple on it. Assuming the pizza is a circle of a \(20\) cm radius, show that some two pieces of pineapple were placed closer than \(20\) cm apart.

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.

Prove that out of any 11 natural numbers, 2 can be found such that their difference is a multiple of 10.

Eight knights took part in a 3-contest tournament. They competed in archery, sword fighting, and lance throwing. For each contest, a knight was awarded 0, 1 or 2 points. Prove that at least two of these knights earned the same total number of points.

Leo’s grandma placed five empty plates on a square 1 metre\({}\times{}\)1 metre table for dinner. Show that some two of these plates were less than 75 cm apart.

London has more than eight million inhabitants. Show that nine of these people must have the same number of hairs on their heads if it is known that no person has more than one million hairs on his or her head.

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.

Each point on a circle was painted red or green. Show that there is an isosceles triangle whose vertices are on the circumference of the circle, such that all three vertices are red or all three are green.