Problems

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It is known that \(a + b + c = 5\) and \(ab + bc + ac = 5\). What are the possible values of \(a^2 + b^2 + c^2\)?

Find all the prime numbers \(p\) such that there exist natural numbers \(x\) and \(y\) for which \(p^x = y^3 + 1\).

Determine all prime numbers \(p\) such that \(5p+1\) is also prime.

When we write 137 in decimal, we mean \(1 \cdot 10^2 + 3 \cdot 10 + 7 \cdot 1\). If we write it instead using powers of 2, we have \(137 = 1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\). To tell apart binary representation from decimals, we can use the following notation: \(137 = (10001001)_2\).

What is the number 273 in binary?