At what value of \(k\) is the quantity \(A_k = (19^k + 66^k)/k!\) at its maximum? You are given a number \(x\) that is greater than 1. Is the following inequality necessarily fulfilled \(\lfloor \sqrt{\!\sqrt{x}}\rfloor = \lfloor \sqrt{\!\sqrt{x}}\rfloor\)?
I am going to convince you that all people have the same eye color! How? Well, notice that if there were only one person in the world, then my claim would be true. Now we will explain that if the claim is true when there are \(n\) people in the world, then it will also be true when there are \(n+1\) people. Therefore, it will be true regardless of the amount of people! (This kind of proof is called a proof by induction)
Let’s imagine that the claim is true when there are \(n\) people in the world. Now take any group of \(n+1\) people, and label them \(a_1,a_2,\dots,a_{n+1}\). Remove \(a_1\). The remaining people \(a_2,a_3,\cdots,a_{n+1}\) form a group of \(n\) people, so they must all have the same eye colour.
On the other hand, let’s remove \(a_{n+1}\). The remaining people \(a_1,a_2,\dots,a_{n}\) also form a group of \(n\) people, so they must again all have the same eye colour.
Since these two smaller groups overlap (they both contain \(a_2,a_3,\dots,a_{n-1}\)), everyone in the full group \(a_1,a_2,\dots,a_n\) has the same eye colour.
Let’s look at triangular numbers, numbers which are a sum of the first \(n\) natural numbers: \[1+2+3+\dots +n\] Show using induction that the \(n\)-th triangular number is equal to \(\frac{n(n+1)}2\).
Show using induction that \(1+3+5+\dots+ (2n-1) = n^2\). That is, the sum of \(n\) first odd numbers is equal to \(n^2\).
Two convex polygons \(A_1A_2...A_n\) and \(B_1B_2...B_n\) have equal corresponding sides \(A_1A_2 = B_1B_2\), \(A_2A_3 = B_2B_3\), ... \(A_nA_1 = B_nB_1\). It is also known that \(n - 3\) angles of one polygons are equal to the corresponding angles of the other. Prove that the polygons \(A_1...A_n\) and \(B_1...B_n\) are congruent.
Show that \(2^{2n} - 1\) is always divisible by \(3\), if \(n\) is a positive natural number.
The famous Fibonacci sequence is a sequence of numbers, which starts
from two ones, and then each consecutive term is a sum of the previous
two. It describes many things in nature. In a symbolic form we can
write: \(F_0 = 1, F_1 = 1, F_n = F_{n-1} +
F_{n-2}\).
Show that \[F_0+F_1+ F_2 + \dots + F_n =
F_{n+2}-1\]
In a certain country, there are \(n\) cities. Some of them are connected by
roads, all of which go in both directions. It is possible to get from
any city to any other city using only roads. However, for any pair of
cities, there is always only one way to get from one of them to the
other and there are no alternative routes.
Show that there are exactly \(n-1\)
roads in this country.
If \(x\) is any positive real number and \(n \ge 2\) is a natural number, show that \[(1+x)^n > 1+nx\]
Anna and Bob play a game with the following rules: they both receive
a positive integer number. They do not know each other’s numbers, but
they do know that their numbers come one after another – they do not
know which one is larger. (If Anna gets \(n\), Bob gets either \(n-1\) or \(n+1\)). Anna then asks Bob – “do you know
what number I have?” If Bob does know, he has to say Anna’s number and
he wins the game. If he does not, he has to say that he does not. Then,
he asks Anna if she knows his number. If Anna does not know, she asks
Bob. This continues until one of them finds out what is the other’s
number. Assuming that both Anna and Bob know mathematics sufficiently
well to be able to solve this problem, find out who wins the game and
how.
For simplicity let’s assume Bob always gets the odd number and Anna
always gets the even number - two consecutive numbers have opposite
parity!