Prove that if the irreducible rational fraction \(p/q\) is a root of the polynomial \(P (x)\) with integer coefficients, then \(P (x) = (qx - p) Q (x)\), where the polynomial \(Q (x)\) also has integer coefficients.
100 fare evaders want to take a train, consisting of 12 coaches, from the first to the 76th station. They know that at the first station two ticket inspectors will board two coaches. After the 4th station, in the time between each station, one of the ticket inspectors will cross to a neighbouring coach. The ticket inspectors take turns to do this. A fare evader can see a ticket inspector only if the ticket inspector is in the next coach or the next but one coach. At each station each fare evader has time to run along the platform the length of no more than three coaches – for example at a station a fare evader in the 7th coach can run to any coach between the 4th and 10th inclusive and board it. What is the largest number of fare evaders that can travel their entire journey without ever ending up in the same coach as one of the ticket inspectors, no matter how the ticket inspectors choose to move? The fare evaders have no information about the ticket inspectors beyond that which is given here, and they agree their strategy before boarding.
Find a natural number greater than one that occurs in the Pascal triangle a) more than three times; b) more than four times.
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.
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?