Prove that the number of all arrangements of the largest possible amount of peaceful bishops (figures that move on diagonals and don’t threaten each other) on the \(8\times 8\) chessboard is an exact square.
In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?
We call a number \(x\) rational if
it can be represented as \(x=\frac{p}{q}\) for coprime integers \(p\) and \(q\). Otherwise we call the number
irrational.
Non-zero numbers \(a\) and \(b\) satisfy the equality \(a^2b^2 (a^2b^2 + 4) = 2(a^6 + b^6)\). Prove
that at least one of them is irrational.
The real numbers \(x\) and \(y\) are such that for any distinct prime odd \(p\) and \(q\) the number \(x^p + y^q\) is rational. Prove that \(x\) and \(y\) are rational numbers.
Determine all prime numbers \(p\) and \(q\) such that \(p^2 - 2q^2 = 1\) holds.
Burbot-Liman. Find the numbers that, when substituted for letters instead of the letters in the expression \(NALIM \times 4 = LIMAN\), fulfill the given equality (different letters correspond to different numbers, but identical letters correspond to identical numbers)
Sage thought of the sum of some three natural numbers, and the Patricia thought about their product.
“If I knew,” said Sage, “that your number is greater than mine, then I would immediately name the three numbers that are needed.”
“My number is smaller than yours,” Patricia answered, “and the numbers you want are ..., ... and ....”
What numbers did Patricia name?
A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number. The result was three times greater.
Find these numbers.
a) Could an additional \(6\) digits be added to any \(6\)-digit number starting with a \(5\), so that the \(12\)-digit number obtained is a complete square?
b) The same question but for a number starting with a \(1\).
c) Find for each \(n\) the smallest \(k = k (n)\) such that to each \(n\)-digit number you can assign \(k\) more digits so that the resulting \((n + k)\)-digit number is a complete square.
Janine and Zahara each thought of a natural number and said them to Alex. Alex wrote the sum of the thought of numbers onto one sheet of paper, and on the other – their product, after which one of the sheets was hidden, and the other (on it was written the number of 2002) was shown to Janine and Zahara. Seeing this number, Janine said that she did not know what number Zahara had thought of. Hearing this, Zahara said that she did not know what number Janine had thought of. What was the number which Zahara had thought of?