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1Borut 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 …Open2Jure 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 …3The capital of a certain country is joined by a direct air route to each of the other \(2012\) cities. Moreover, every one of those \(2012\) cities is joined by an air route to at least one city other …4Sixteen 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 …5On \(41\) squares of a chessboard - the ordinary \(8 \times 8\) board - a king is placed, one king on each of those squares. Prove that among these kings one can find three pairwise disjoint sets, each …6Let \(n > 1\) be a natural number. How many \(n\)-digit numbers are palindromes and divisible by \(9\)? (A number is a palindrome when its decimal representation is symmetric, that is, it reads the same …7In every cell of a table with \(2017\) rows and \(2017\) columns one of the numbers \(1, 2, 3, \dots, 2017\) is written. Is it possible to do this so that in every row, in every column and along every …8A square board of size \(n \times n\) is given, where \(n \geq 2\). The numbers \(1, 2, \dots, n^2\) are written into the \(n^2\) unit cells of the board, one number in each cell, each number used exactly …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 …12Two players play the following game. Taking turns, each player writes down one digit, the digits appearing in a row from left to right in the order in which they are written, and no player is allowed to …13Andraz 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 …14A 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 …15Find 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 …16A \(4 \times 4\) 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 …17Let \(n \ge 2\) 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 …18Level 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 …19In the plane of a triangle \(ABC\) one draws \(n\) lines, each of them parallel to one of the three sides of the triangle. Determine the smallest \(n\) for which these \(n\) lines can cut the plane into …20Can nine points, no three of them collinear, be placed inside the cross-shaped figure below (its boundary included) in such a way that whenever three of them span a triangle lying inside the figure, that …21Prove that the disk of radius \(100\) centred at the origin contains fewer than \(31600\) points whose two coordinates are both integers. (A point counts as contained in the disk if it lies inside it or …22All powers of two are written on a board in increasing order: \(1, 2, 4, \ldots\). Aca and Braca now take turns, Aca first. A move consists of choosing two numbers that stand next to each other on the …23A 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. …24A 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 …25We 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 …26Three 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 …27A 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 …28Timotej 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 …29Finitely many arcs are marked on a circle. The length of each of them is smaller than half of the circumference, and any three of the marked arcs have a common point. Prove that there is a point of the …30A 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 …31Anja 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 …32Every 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 …33For which positive integers \(m\) and \(n\) can an \(m \times n\) rectangle be covered completely and without overlaps by copies of the three figures shown below, each of them built from unit squares? …34A pile of \(n\) tokens lies on a table. Two players, \(A\) and \(B\), move alternately, and \(A\) moves first. In one move a player must do one of the following: remove one token from one of the piles …35A finite sequence of \(n\) numbers is given, each of them equal to \(0\) or \(1\), where \(n\) is a positive integer. One move consists of choosing two adjacent terms \(x\) and \(y\), deleting both of …36A bank guard looks after \(n\) safes. Every safe has its own key, no key fits two safes, and all the keys look alike. He is given as many identical circular metal rings as he wants. On any ring he may …37Every point of three-dimensional space is coloured with one of two colours, red or blue, in such a way that whenever three points \(A\), \(B\), \(C\) have the same colour and \(AB = AC\), the midpoint …38Every point of space is painted in one of three colours. Prove that one of the three colours can be chosen in such a way that for every positive real number \(r\) there exists a triangle of area \(r\) …39Let \(n\) be a positive integer. What is the largest number of rooks that can be placed on an \(n \times n\) board so that every rook attacks at most \(3\) of the other rooks? Attacks are the usual chess …
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