Problems

Jane wrote another number on the board. This time it was a two-digit number and again it did not include digit 5. Jane then decided to include it, but the number was written too close to the edge, so she decided to t the 5 in between the two digits. She noticed that the resulting number is 11 times larger than the original. What is the sum of digits of the new number?

a) Find the biggest 6-digit integer number such that each digit, except for the two on the left, is equal to the sum of its two left neighbours.

b) Find the biggest integer number such that each digit, except for the rst two, is equal to the sum of its two left neighbours. (Compared to part (a), we removed the 6-digit number restriction.)

Sarah believes that two watermelons are heavier than three melons, Anna believes that three watermelons are heavier than four melons. It is known that one of the girls is right, and the other is mistaken. Is it true that 12 watermelons are heavier than 18 melons? (It is believed that all watermelons weigh the same and all melons weigh the same.)

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:

Alex is 1 year older than V,

Beatrice is 2 years older than W,

Victor is 3 years older than X,

Gregory is 4 years older than Y.

Who is older and by how much: Deborah or Z?

Four aliens – Dopey, Sleepy, Happy, Moody from the planet of liars and truth tellers had a conversation: Dopey to Sleepy: “you are a liar”; Happy to Sleepy: “you are a liar”; Moody to Happy: “Yes, they are both liars,” (after a moment’s thought), “however, so are you.” Which of them is telling the truth?

An adventurer is travelling to the planet of liars and truth tellers with an official guide and is introduced to a local. “Are you a truth teller?” asked the adventurer. The alien answers “Yrrg,” which means either “yes” or “no”. The adventurer asks the guide for a translation. The guide says “"yrrg" means "yes". I will add that the local is actully a liar.” Is the local alien liar or truth teller?

On the selection to the government of the planet of liars and truth tellers $12$ candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the $12$th, who said: “And now $12$ times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?

The planet has $n$ residents, some are liars and some are truth tellers. Each resident said: “Among the remaining residents of the island, more than half are liars.” How many liars are on the island?

In the government of the planet of liars and truth tellers there are $101$ ministers. In order to reduce the budget, it was decided to reduce the number of ministers by $1.$ But each of the ministers said that if they were to be removed from the government, then the majority of the remaining ministers would be liars. How many truth tellers and how many liars are there in the government?

Let $x$ be a natural number. Among the statements:

$2x$ is more than 70;

$x$ is less than 100;

$3x$ is greater than 25;

$x$ is not less than 10;

$x$ is greater than 5;

three are true and two are false. What is $x$?