Practice library
Problems
1In how many ways can three rooks be placed on a chessboard of dimensions \(6 \times 2006\) so that no two of them attack each other? (Two rooks attack each other when they stand in the same row or in the …Open2How many three-digit numbers written using the digits \(0, 1, 2, 3, 4, 5\) are divisible by \(15\), if (a) all digits must be distinct? (b) digits may repeat?3A delegation of \(6\) people is to be chosen from a group of \(16\), consisting of \(4\) people from Serbia, \(4\) from Romania, \(4\) from Bulgaria and \(4\) from Macedonia. (a) In how many ways can this …4How many equivalence relations on a set of six elements have the property that every equivalence class contains at least two elements?5A row contains \(2016\) chairs. Each chair is to be painted either red or blue. In how many ways can this be done so that the number of neighbouring pairs of chairs painted in the same colour is even?6A park has the shape of a square with side \(1\) km. Inside it grow \(4567\) trees, each of diameter at most \(50\) cm, and each tree lies entirely within the park. Prove that the park contains a \(10\) …7Eight players took part in a chess tournament, and each of them played exactly one game against every other participant. A win earns \(1\) point, a loss \(0\) points, and a draw \(0.5\) points for each …8A mosquito sits on the lower left cell of a rectangular board of format \(2003 \times 2004\). It travels above the board in the following manner: taking off from the cell it occupies, it flies over \(99\) …9Into 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 …10A car leaves town \(A\) and drives along a straight road at constant speed. Every \(15\) minutes it makes a turn of \(90\) degrees, to the left or to the right. Prove that the car can be back in \(A\) …11In how many ways can \(11\) birds be placed into \(3\) identical cages so that every cage contains at least three birds?12Ten teams took part in a volleyball tournament, and each team played exactly one match against each of the other nine. When the tournament ended, the first team had \(x_{1}\) wins and \(y_{1}\) losses, …13A snake starts in the upper-left cell of a \(2 \times n\) board, where \(n\) is a natural number. From one cell it may move to another whenever the two cells share an edge, but it may never visit a cell …14Aca and Branko play the following game on a \(2023 \times 2024\) board. First Aca chooses a square of the board and places a queen on it. Then the players move the queen alternately, following the rules …15Let \(A\) be a subset of the set \(\{1, 4, 7, \dots, 1996\}\) containing exactly \(335\) elements. Prove that \(A\) contains two distinct numbers whose sum equals \(2000\).16A class has \(30\) students, and every day exactly three of them are on duty in the school kitchen. Prove that the duty roster cannot be arranged so that every two students of the class are on duty together …17Let \(S = \{1, 2, \ldots, 20\}\). What is the largest possible number of elements of a subset \(A \subseteq S\) with the property that \(2x \notin A\) whenever \(x \in A\)?18Every cell of a \(3 \times 3\) board is to be painted in one of \(9\) colors so that all \(9\) colors are used. How many differently colored boards can be made? (Two boards are colored differently if they …19Two players alternately take balls from two boxes. On each turn, a player chooses one of the boxes and removes any number of balls from it (at least one). The player who takes the last ball wins. The first …20In how many ways can \(1000\) numbers be chosen from the set \(\{1, 2, \dots, 1999\}\) so that no two of the chosen numbers have sum \(1999\) or sum \(2000\)?21In how many ways can \(m\) distinct birds be placed into \(n\) distinct cages so that every cage contains at least one bird and at most two birds?22Several lines are drawn in the plane. Line \(a\) intersects exactly three of the other lines, and line \(b\) intersects exactly four of the other lines. Line \(c\) intersects exactly \(n\) of the other …23Words are built from the two letters \(A\) and \(B\) only. Is it possible to form a set of words containing \(3\) words of \(4\) letters, \(10\) words of \(5\) letters, \(30\) words of \(6\) letters and …24In a handball tournament every team played exactly one match against each of the other teams. A win is worth \(2\) points, a loss \(0\), and a drawn match gives \(1\) point to each of the two teams. The …25There are \(14\) books standing in a row on a shelf. In how many ways can \(5\) of them be chosen so that no two of the chosen books stand next to each other?26A table of dimensions \(2010 \times 2011\) is given. Determine the largest number of cells that can be colored so that every \(2 \times 2\) square of the table contains at most two colored cells.27A mathematical commission has \(2n\) members, where \(n \geqslant 3\). Every member of the commission is in a quarrel with exactly one other member (the relation is symmetric). In how many ways can the …28The numbers \(1, 2, 3, 4, 5, 6, 7, 8\) are split into three disjoint nonempty sets. Let \(P_{1}\), \(P_{2}\) and \(P_{3}\) be the products of the numbers in the first, the second and the third set, respectively, …29The caliph of Baghdad rewarded three wise men with ten purses: the first held \(0\) dinars, the second \(1\) dinar, the third \(2\) dinars, and so on up to the tenth, which held \(9\) dinars. The first …30The numbers \(1, 2, 3, 4, 5\) are divided into two groups so that each group contains at least one of them. Prove that one of the groups contains two numbers whose difference also belongs to that same …31Two operations \(F\) and \(G\) turn an ordered triple of real numbers into another triple by the following rules: \(F\) sends \((a, b, c)\) to \((a+1,\, b+c,\, c+1)\), and \(G\) sends \((a, b, c)\) to …32For natural numbers \(m\) and \(n\), consider a board of dimensions \(m \times n\) made up of \(mn\) unit squares. Call the skeleton of the board the set of all unit segments that are edges of at least …33What is the largest number of chips that can be placed on the cells of a \(7 \times 7\) board so that no rectangle of area \(6\), with sides running along the grid lines, contains more than one chip?34Every 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 …35Ten teams took part in a volleyball tournament, and every team played exactly one match against each of the others. When the tournament ended, the first team had \(x_1\) wins and \(y_1\) losses, the second …36Prove or disprove the following assertion. Among any six positive integers it is always possible to choose three of them that are pairwise coprime, or three of them that have a common divisor greater than …37Each unit cell of a \(3 \times 3\) table is coloured with one of three colours. How many such colourings are there in which every two cells sharing a side are coloured differently?38A bishop on a chessboard attacks every square lying on one of the two diagonals through it. Call a square covered if a bishop stands on it or a bishop attacks it. Prove that seven bishops can never be …39Baron Munchausen lives in a country \(Z\) which has \(2018\) cities, some pairs of them joined by roads (every road can be travelled in both directions). The Baron has established that there is a city …40In the game Minesweeper, mines are placed on some cells of an \(a \times b\) board (\(a, b \in \mathbb{N}\)), and on every remaining cell one writes the number of neighbouring cells that contain a mine. …41The cells of an \(n \times n\) table are to be coloured with \(n\) different colours in such a way that every row and every column contains cells of all \(n\) colours. Determine the smallest and the largest …42How many three-element subsets \(\{a, b, c\}\) does the set \[ A = \{19, 20, 21, \ldots, 98\} \] have with the property that \(a + b + c\) is divisible by \(3\)?43(a) In how many ways can one choose two two-digit numbers that are not neighbours, that is, whose difference is not equal to \(1\)? (b) How many five-digit numbers are there in which the digit \(5\) occurs …44At a volleyball tournament \(n > 1\) teams took part, and every two of them played exactly one match against each other. Prove that the teams can be numbered \(1, 2, \ldots, n\) in such a way that for …45Let \(n\) be a positive integer. Let \(A_n\) be the set of all \(n\)-digit numbers whose decimal digits add up to \(4\), and let \(B_n\) be the set of all \(n\)-digit numbers whose decimal digits multiply …46In how many ways can a king, a queen, two rooks, two bishops and two knights be placed on the eight squares of the first rank of a chessboard so that the two rooks stand on opposite sides of the king, …47Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.48An entry of a permutation is called right-minimal if it is smaller than every entry standing to its right. For example, in the permutation \[ (2,\; 1,\; 4,\; 6,\; 3,\; 7,\; 8,\; 5) \] the right-minimal …49A cinema row has \(20\) seats. In how many ways can six couples take their seats in this row if every couple wants to sit on two adjacent seats?50The number \(2025\) is written on a board. Ana and Bojan play the following game, moving alternately. A move consists of erasing the number currently on the board and writing in its place the difference …51Vid 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 …52At 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 …53Can the plane be tiled by squares - that is, covered completely, with no two squares overlapping - in such a way that no side length is used by more than two of the squares?54Four vertices of a given regular octagon are to be coloured blue and the remaining four red. Two colourings are called equivalent if one of them is carried onto the other by a rotation of the octagon about …55Ana and Branko placed a number of tokens on the squares of an \(8 \times 8\) board, no square carrying more than one token. Ana then wrote down the number of tokens in each of the eight rows, and Branko …56Two roads run from Novi Sad to Belgrade, an old one and a new one, and they are joined by \(7\) connecting roads. In how many different ways can one travel from Novi Sad to Belgrade along these roads, …57At most how many rooks can be placed on a chessboard of dimensions \(5 \times 4\) (five rows and four columns) so that every rook attacks at most one of the remaining ones? Here a rook attacks every rook …58Let \(n \geq 2\) be a natural number. Every cell of a square table \(A\) of size \(n \times n\) is filled with one of the numbers \(1\) and \(-1\). For each \(i \in \{1, 2, \ldots, n\}\) write \(k_i\) …59An \(n \times n\) table is to be filled with zeros and ones so that for every index \(i \in \{1, 2, \dots, n\}\) the number of ones in the \(i\)-th row and the number of ones in the \(i\)-th column differ …60Sixteen teams take part in a basketball tournament played as a double round robin: every two teams meet exactly twice. The eight best-placed teams qualify for the next tournament. Teams are ranked by the …61Every positive integer is painted in one of two colours, one of which is called red. The painting is periodic with period \(d\): the numbers \(x\) and \(x + d\) always receive the same colour. Suppose …62An infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]63The cells of a \(4 \times 4\) table are to be coloured with several colours so that in every figure congruent to the one shown below, all four cells have different colours. The figure may be rotated or …64In the Mad Forest there lived \(6\) werewolves, \(17\) unicorns and \(55\) spiders. A werewolf can eat a spider or a unicorn, but not another werewolf; a spider can eat a unicorn, but neither a werewolf …65Borut 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 …66Peter 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 …67Two teams, each consisting of \(6\) footballers, have at their disposal \(4\) pairs of shorts and \(4\) jerseys in each of the colours red, blue and white. In how many ways can the footballers dress for …68A merchant has to ferry seven goods across a river: a piece of cheese, a mouse, a rat, a cat, a dog, a wolf and a bear. His boat has room for only \(k\) of the seven at a time. If they are left without …69Miljan and Mladen play the following game. They take turns naming divisors of \(200\), with one restriction: the number a player names must not be a divisor of any number named earlier in the game. A player …70A number is written in every cell of an \(8 \times 8\) table. A move consists of choosing any \(3 \times 3\) square of the table (nine cells) or any \(4 \times 4\) square (sixteen cells) and increasing …71A plane figure of area greater than \(1006\) can be placed inside a rectangle with side lengths \(2011\) and \(1\). Prove that the figure contains two points, on its boundary or in its interior, whose …72A cinema hall has \(2015\) seats, and \(2014\) viewers, one of whom is Mika, walk in. They all sit down on arbitrary seats, paying no attention to the seat assigned to them by their ticket, so exactly …73Two players alternately write one of the numbers \[ 473, \quad 523, \quad 573, \quad 623, \quad 673, \quad 723, \quad 773, \quad 823, \quad 873 \] into a free cell of a \(3 \times 3\) table, where each …74An \(8 \times 8\) board is tiled with copies of the three figures below. The figures may be rotated and reflected, and any number of copies of each of the three shapes may be used; the board counts as …75Let \(k \geq 2\) be a natural number. Margita has written on the board the first \(2k - 1\) natural numbers \(1, 2, \ldots, 2k - 1\). In one move she may erase any two numbers from the board and write …76Let \(A\) be a set of \(10\) numbers chosen from \(\{1, 2, \ldots, 100\}\). Prove that \(A\) has two nonempty subsets \(S\) and \(T\) with no element in common such that the sum of the elements of \(S\) …77Jure 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 …78Lara 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, …79An island is inhabited by \(45\) chameleons: \(17\) yellow, \(15\) grey and \(13\) blue. They wander about and meet from time to time, never more than two at a time. When two chameleons of the same colour …80On each of \(n > 4\) cards, one of the numbers \(+1\) and \(-1\) is written. A single question consists of naming exactly three of the cards, after which we are told the product of the numbers written …81The pupils of a school went to the theatre on two occasions, and every pupil of the school attended at least one of the two performances. Boys formed \(60\%\) of the audience at the first performance and …82The points of a plane \(\alpha\) are split between two nonempty sets \(A\) and \(B\): no point belongs to both, and every point belongs to one of them. Prove that some isosceles right triangle has all …83The 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 …84Maksim and Mina play the following game. Maksim starts by drawing a line in the plane; Mina then draws a line different from it; Maksim then draws a line different from both lines already drawn, and so …85How many integers \(n\) with \(10 \le n < 100000\) are divisible by \(4\), contain no digit \(0\) in their decimal representation, and have no two adjacent digits equal?86In the expression \[ *\,1 * 3 * 3^{2} * 3^{3} * \cdots * 3^{1997} * 3^{1998} \] Arkadije and Branislav take turns replacing one of the stars by \(+\) or by \(-\), one star per move, until no star is left. …87In a group of \(20\) people, every person chooses ten of the other nineteen and sends one letter to each of them. Prove that there are two people who sent a letter to each other.88How many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …89Sixteen 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 …90On \(41\) 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 …91Let \(n > 1\) 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 …92In 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 …93A 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 …94The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …95Stars 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 …96A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …97Eighteen 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 …98Let \(A\) 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 …99A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way …100Two 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 …101Andraz 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 …102A 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 …103Find 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 …104A \(4 \times 4\) 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 …105Let \(n \ge 2\) 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 …106Level 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 …107In 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 …108Can 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 …109Prove 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 …110All 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 …111A 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. …112A 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 …113We 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 …114Three 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 …115A 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 …116Timotej 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 …117Finitely 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 …118A 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 …119Anja 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 …120Every 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 …121For 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? …122A 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 …123A 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 …124A 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 …125Every 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 …126Every 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 \(r\) …127Let \(n\) 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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