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1Every school of a certain region sent exactly \(3\) students to a competition, and Andrej, Blaz and Zan all came from the same school. When the competitors lined up to collect their starting numbers, Andrej …Open2Vid cut a square \(ABCD\) of side length \(20\) units into \(400\) unit squares. Eva then picked four vertices of unit squares, all lying in the interior of \(ABCD\), that are the vertices of a rectangle …3At a national competition the students worked on \(4\) problems. Each problem was marked with a whole number of points, at least \(0\) and at most \(7\). Altogether \(42\) students competed. Exactly half …4Borut drew the table of size \(2 \times 7\) shown in the picture. He now wants to colour some of its cells so that every cell he leaves uncoloured shares a side with at least one coloured cell. What is …5Peter keeps \(111\) red and \(111\) blue marbles at home; they are made by his uncle. Every day Peter may visit his uncle and carry out one exchange: either he hands over \(11\) red marbles and receives …6Jure drew four distinct lines in the plane, one arrangement after another, and for each arrangement he wrote down the number \(n\) of points at which at least two of his lines cross. Which of the sets …7Lara and Sara draw \(n\) straight lines on a rectangular sheet of paper, taking turns and drawing one line each time. Every line is parallel to one of the edges of the sheet and runs from edge to edge, …8Sixteen points of the integer lattice are marked, as in the picture: all points \((x,y)\) with \(x\) and \(y\) taken from \(\{1,2,3,4\}\). At most how many of these points can be coloured red so that no …9Stars are drawn in the cells of a \(4 \times 4\) table, at most one star per cell. What is the least number of stars for which the following holds: whichever \(2\) rows and whichever \(2\) columns are …10Eighteen matches are laid out to form the grid shown below: an equilateral triangle whose side is three matches long, divided into nine small triangles. What is the smallest number of matches that must …11Let \(A\) be the set of all integers from \(-20\) to \(20\), that is \[ A = \{\, a \in \mathbb{Z} \;:\; -20 \le a \le 20 \,\} . \] Let \(n\) be a positive integer and let \(A_1, A_2, \ldots, A_n\) be pairwise …12Andraz and Breda cut two long strips out of a newspaper, of lengths \(a\) and \(b\), to play a game with. A move consists of choosing one of the strips and cutting a piece of length \(d\) off it, so that …13A spider has spun the web shown below: five regular octagons nested one inside the other, with each vertex of an octagon joined by a thread to the corresponding vertex of the neighbouring octagons. The …14Find the smallest natural number \(n\) for which an \(n \times n\) board of unit cells can be covered completely and without overlaps by equally many tiles of the two shapes below: an \(L\)-shaped tile …15A \(4 \times 4\) table is divided into \(16\) unit cells. On this table we place tiles of the shape drawn alongside: two unit squares that meet at a single corner. A tile may be rotated, and each tile …16Let \(n \ge 2\) be a natural number. Maja and Peter want to colour every cell of an \(n \times n\) table either black or blue, subject to one rule: among any four cells that can be covered by a square …17Level 1 of the computer game Zakladnica takes place in an underground treasury built from \(13\) octagonal and \(12\) square rooms, arranged as in the figure. The only way into the treasury, and the only …18A mole has dug a number of underground rooms and joined them by tunnels, in such a way that from every room exactly \(3\) tunnels lead out, to \(3\) different rooms. Tunnels meet one another only at rooms. …19A natural number is written in every cell of a square table. Call the table interesting if the sum of all the numbers in it is odd and, in addition, the sum of the four numbers covered by any placement …20We want to cover a \(4 \times 4\) board with tiles of the shape shown below, rotations and reflections being allowed. The tiles are permitted to overlap one another and to stick out beyond the edge of …21Three piles of tokens lie on a table, holding \(a\), \(b\) and \(c\) tokens, where \(a \ge b \ge c > 0\). Players \(A\) and \(B\) move tokens alternately, and \(A\) starts. In one move a player first selects …22A rectangular grid of size \(7 \times 9\) is given: seven rows of cells and nine columns of cells, as in the figure. At the bottom-left node of the grid sits a colony of ants, and at the top-right node …23Timotej had a sheet of squared paper measuring \(8 \times 8\) little squares. He folded it a few times, each fold running along one of the lines of the grid, until he was left with a square piece measuring …24A 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 …25Anja owns tiles shaped like a single unit square, Bojan tiles shaped like an L-tromino: three unit squares forming an L, as drawn below. The two players alternately place one tile of their own onto a rectangular …26Every cell of an \(n \times n\) table contains the number \(0\). One step consists of choosing three cells that form the shape and adding \(1\) to each of the three numbers standing in them. Can we, after …
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