Drzavno tekmovanje (national round) 2022 · Lemma

1Number theoryMultiple choiceLet \(x\), \(y\), \(z\) and \(w\) be natural numbers. At most how many of the six sums \[ x+y, \quad x+z, \quad x+w, \quad y+z, \quad y+w, \quad z+w \] can be odd? A \(2\) B \(3\) C \(4\) D \(5\) E \(6\) …2Number theoryMultiple choiceA large rectangle is cut into seven smaller rectangles, each of which has both of its side lengths equal to a whole number of metres. Five of the seven pieces have their areas written in them, as shown …3Number theoryMultiple choiceWhat is the largest natural number \(n\) with the property that, when \(n\) is divided by \(20\), the remainder is equal to the quotient? A \(21\) B \(92\) C \(231\) D \(399\) E \(440\)4Number theoryFree answerFind all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).5GeometryProofThree curves are drawn inside a square \(ABCD\): the quarter circle \(\mathcal{Q}\) centred at the vertex \(A\) and passing through \(B\) and \(D\); the semicircle \(\mathcal{P}\) centred at the midpoint …6CombinatoricsFree answerA strip of \(1 \times n\) cells is given, where \(n > 10\) is a natural number, and its cells are numbered \(1, 2, \dots, n\) from left to right. Cell number \(10\) is black and carries a token; every …

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