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?
Solve this equation: \[(x+2010)(x+2011)(x+2012)=(x+2011)(x+2012)(x+2013).\]
The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and
for all \(n \geq 2\).
Prove that all members of the sequence are integers.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?
The graph of the function \(y=kx+b\) is shown on the diagram below. Compare \(|k|\) and \(|b|\).
Compare the numbers: \(A=2011\times 20122012\times 201320132013\) and \(B= 2013\times 20112011 \times 201220122012\).
You are given 12 different whole numbers. Prove that it is possible to choose two of these whose difference is divisible by 11.