Suppose that: \[\frac{x+y}{x-y}+\frac{x-y}{x+y} =3.\] Find the value of the following expression: \[\frac{x^2 +y^2}{x^2-y^2} + \frac{x^2 -y^2}{x^2+y^2}.\]
Pinocchio correctly solved a problem, but stained his notebook. \[(\bullet \bullet + \bullet \bullet+1)\times \bullet= \bullet \bullet \bullet\]
Under each blot lies the same number, which is not equal to zero. Find this number.
Seven coins are arranged in a circle. It is known that some four of them, lying in succession, are fake and that every counterfeit coin is lighter than a real one. Explain how to find two counterfeit coins from one weighing on scales without any weights. (All counterfeit coins weigh the same.)
Four people discussed the answer to a task.
Harry said: “This is the number 9”.
Ben: “This is a prime number.”
Katie: “This is an even number.”
And Natasha said that this number is divisible by 15.
One boy and one girl answered correctly, and the other two made a mistake. What is the actual answer to the question?
Peter recorded an example of an addition on a board, after which he replaced some digits with letters, with the same figures being replaced with the same letters, and different figures with different letters. He did it such that he was left with the sum: \(CROSS + 2011 = START\). Prove that Peter made a mistake.
Four numbers (from 1 to 9) have been used to create two numbers with four-digits each. These two numbers are the maximum and minimum numbers, respectively, possible. The sum of these two numbers is equal to 11990. What could the two numbers be?
Compute the following: \[\frac{(2001\times 2021 +100)(1991\times 2031 +400)}{2011^4}.\]
After a circus came back from its country-wide tour, relatives of the animal tamer asked him questions about which animals travelled with the circus.
“Where there tigers?”
“Yes, in fact, there were seven times more tigers than non-tigers.”
“What about monkeys?”
“Yes, there were seven times less monkeys than non-monkeys.”
“Where there any lions?”
What is the answer he gave to this last question?
There are 100 boxes numbered from 1 to 100. In one box there is a prize and the presenter knows where the prize is. The spectator can send the presented a pack of notes with questions that require a “yes” or “no” answer. The presenter mixes the notes in a bag and, without reading out the questions aloud, honestly answers all of them. What is the smallest number of notes you need to send to know for sure where the prize is?
Solve this equation: \[(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).\]