Can you find
a) in the 100th line of Pascal’s triangle, the number \(1 + 2 + 3 + \dots + 98 + 99\)?
b) in the 200th line the sum of the squares of the numbers in the 100th line?
Prove there are no integer solutions for the equation \(3x^2 + 2 = y^2\).
Can seven phones be connected with wires so that each phone is connected to exactly three others?
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?
Write out in a row the numbers from \(1\) to \(9\) (every number once) so that every two consecutive numbers give a two-digit number that is divisible by \(7\) or by \(13\).
Several Top Secret Objects are connected by an underground railway in such a way that each Object is directly connected to no more than three others and from each Object one can reach any other Object by going and by changing no more than once. What is the maximum number of Top Secret Objects?
Prove that the sum of
a) any number of even numbers is even;
b) an even number of odd numbers is even;
c) an odd number of odd numbers is odd.
Prove that the product of
a) two odd numbers is odd;
b) an even number with any integer is even.
A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?
Several guests are sitting at a round table. Some of them are familiar with each other; mutually acquainted. All the acquaintances of any guest (counting himself) sit around the table at regular intervals. (For another person, these gaps may be different.) It is known that any two have at least one common acquaintance. Prove that all guests are familiar with each other.