Opštinsko takmičenje (Municipal round) 1999 · Lemma

1CombinatoricsFree answerHow many equivalence relations on a set of six elements have the property that every equivalence class contains at least two elements?2Number theoryProofLet \(x\) and \(y\) be integers. Prove that if \(6x + 11y\) is divisible by \(31\), then \(x + 7y\) is divisible by \(31\) as well.3Number theoryProofProve that a natural number of the form \(4n + 1\) can be represented as a sum of two squares if and only if the number \(8n + 2\) can be represented as a sum of two squares.4CombinatoricsFree answerInto a box, \(k\) smaller boxes are placed. Then \(k\) still smaller boxes are placed into some of the smaller boxes, each, and this procedure is repeated several times. If, at the end, \(m\) of all these …5AlgebraFree answerFind all values \(a \in \mathbb{R}\) for which the equation \[ |x - a| + |a - 1| = 1 \] has two solutions, and determine those solutions.

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