Opštinsko takmičenje (Municipal round) 2022 · Lemma

1GeometryProofExactly \(2021\) points are chosen on the line \(AB\), and none of them lies on the segment \(AB\). Prove that the sum of the distances from these \(2021\) points to \(A\) can never be equal to the sum …2AlgebraFree answerFind all pairs of real numbers \((x, y)\) satisfying \[ \frac{|x+y|}{1+|x+y|} = \frac{|x|}{1+|x|} + \frac{|y|}{1+|y|}. \]3Number theoryFree answerDetermine all pairs of prime numbers \(p\) and \(q\) for which \((p^3 + 1)^q\) is the square of a natural number.4AlgebraFree answerLet \(f\) and \(g\) be linear functions with the following three properties: the graph of \(x \mapsto f(g(x))\) passes through the point \((2021, 2022)\), the graph of \(x \mapsto g(f(x))\) passes through …5GeometryFree answerIn a triangle \(ABC\) write \(a = BC\), \(b = CA\), \(c = AB\), and let \(S\) be the centre and \(r\) the radius of its inscribed circle. Consider the line joining the midpoints of the sides \(BC\) and …

Work through the paper in order - each problem opens in the workspace with this paper as its trail.