Practice library
Problems
1Ten 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 …Open2Two circles that do not intersect are given. Construct all of their common tangent lines.3Find all integer solutions of the equation \[ 6x^{3} + 7y^{2} + 8z^{3} = 66\,677\,888. \]4Find every real value of the parameter \(a\) for which the polynomial \[ P(x) = x^{2021} - 2x^{2} + x + a^{3} - a \] is divisible by the polynomial \[ Q(x) = x^{2} - (a+1)x + a. \]5In a school there are \(2023\) pupils and \(2023\) lockers, the lockers bearing the numbers \(1, 2, \ldots, 2023\). At the start every locker is closed. The pupils file past the lockers one after another …6A binary relation \(\varrho\) on the set of real numbers is defined by \[ (\forall x, y \in \mathbb{R}) \quad x \varrho y \iff x^2 - 3xy + 2y^2 = 0 . \] (a) Determine whether \(\varrho\) is reflexive, …7On the set \[ A=\left\{0,\;1,\;-1,\;2,\;\tfrac12,\;-2,\;-\tfrac12,\;3,\;\tfrac13\right\} \] define the relation \[ \rho=\left\{(a,b)\in A\times A \;:\; \left(a^{2}-b^{2}\right)(ab-1)=0\right\}. \] (a) …8The number whose cube equals \(2012^{12}\) was multiplied by the square of the number \(2012^{11}\). Which number was obtained? A \(2012^{58}\) B \(2012^{26}\) C \(2012^{88}\) D \(2012^{15}\) E \(2012^{12}\) …9A factory modernised its equipment, and its productivity then rose by \(25\%\). Some time later a number of workers were dismissed, and the productivity fell by \(20\%\). By what percentage has the productivity …10One digit of the seven-digit number \(2345678\) is to be deleted, so that the six-digit number left behind is divisible by \(9\). Which digit must it be? A \(8\) B \(7\) C \(6\) D \(5\) E \(4\)11Prove that a positive integer whose decimal representation uses no digits other than \(2\) and \(6\) cannot be written as a difference of the squares of two integers.12Find all natural numbers \(n\) for which the fraction \[ \frac{2n+3}{5n+7} \] can be reduced, that is, for which its numerator and denominator have a common divisor greater than \(1\).13Let \(a, b, c, d\) be numbers satisfying \[ a^2 + b^2 + (a+b)^2 = c^2 + d^2 + (c+d)^2 . \] Prove that then \[ a^4 + b^4 + (a+b)^4 = c^4 + d^4 + (c+d)^4 . \]14Let \(k\) be a positive integer. Prove that the number \[ 2^{2k-1} + 2^{k} + 1 \] is never divisible by \(7\).15The lengths of the sides of a triangle \(ABC\) are three consecutive natural numbers. The median drawn from \(A\) is perpendicular to the bisector of the angle \(\angle ABC\). Determine the lengths of …16Determine the remainder left by the number \[ 3^{1000} + 4^{1000} \] when it is divided by \(13\).17The circles \(k_1\) and \(k_2\) meet at two points \(A\) and \(B\). Through \(A\) and through \(B\) two parallel lines are drawn. They meet the circle \(k_1\) for a second time at the points \(C\) and …18Let \(p\), \(q\), \(r\) be real numbers such that \[ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0 \qquad \text{and} \qquad p + q + r = 1 . \] Prove that for all real numbers \(a\), \(b\), \(c\), \[ a^{2} + b^{2} + c^{2} = (pa + qb + rc)^{2} + (qa + rb + pc)^{2} + (ra + pb + qc)^{2} . \] …19Let \(x\), \(y\) and \(z\) be positive integers satisfying both \[ x^{3} - y^{3} - z^{3} = 3xyz \qquad \text{and} \qquad x^{2} = 2(y + z). \] Determine the value of \(x + y + z\).20Aca, Branka, Vera and Goran were each given the same kind of task by their mathematics teacher: divide one positive real number by another. Aca computed \(a_1 : a_2\), Branka computed \(b_1 : b_2\), Vera …21The squares \(ABCD\) and \(EFGH\) both have side \(1\). The first is cut into nine congruent small squares and the second into sixteen congruent small squares; in every small square the inscribed circle …22Lili noticed that the digits of the year \(2015\) have average \(2\), because \(\frac{2+0+1+5}{4} = 2\). How many years of the 21st century after \(2015\) have the same digit average as \(2015\)? A \(1\) …23Find every pair of integers \(x\) and \(y\) satisfying \[ 2\left(x^2 + y^2\right) = 5\left(xy + 1\right) . \]24Let \(n\) be a natural number. Prove that \[ (n+1)^{3n} - n^{2n}(n+3)^n \] is divisible by \(3n+1\).25The numbers \(x\), \(y\) and \(a\) satisfy \(3^{x} = a\) and \(a^{y} = 81\). What is the value of the product \(x\cdot y\)? A \(4\) B \(3\) C \(12\) D \(0\) E \(1\)26Does there exist a polynomial \(P\) with integer coefficients for which \[ \textbf{a)}\quad P(7) = 8 \ \text{ and } \ P(15) = 12; \qquad\qquad \textbf{b)}\quad P(8) = 7 \ \text{ and } \ P(12) = 15\,? \] …27Nonzero real numbers \(a\), \(b\), \(c\) and a real number \(d\) satisfy \[ a = b + 2c, \qquad a + c = b + d, \qquad b = d + c . \] Which of the following equalities is certainly true? A \(d = 2c\) B \(a = 3c\) …28Peter keeps horses and cows on his farm. To begin with he had exactly as many horses as cows, and that common number was greater than \(0\). He then bought some more cows, so that the number of cows rose …29There are \(17\) girls and \(12\) boys on a playground. At least how many more children must arrive so that everybody present can then be divided into two groups of the same size, in such a way that each …30Every edge of a certain pyramid has length \(5\) cm, and the pyramid has five vertices in all, so its base is a square. All five vertices are sliced off, each by a single plane, and the cuts are made so …31How 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 \(3\)?32A convex pentagon \(A_1A_2A_3A_4A_5\) is given. Let \(B_1\), \(B_2\), \(B_3\), \(B_4\) be the midpoints of the sides \(A_1A_2\), \(A_2A_3\), \(A_3A_4\), \(A_4A_5\), in that order, and let \(M\) be the …33A set \(A\) is given. Among its subsets a relation \(\sim\) is defined by \[ X \subseteq A, \quad Y \subseteq A, \qquad X \sim Y \iff X \cap Y \neq \varnothing . \] Determine whether \(\sim\) is reflexive, …34In a triangle \(ABC\) the angle at \(A\) measures \(60^\circ\). Write \(a\), \(b\), \(c\) for the lengths of the sides \(BC\), \(CA\), \(AB\). Prove that the area of the triangle equals \[ \frac{\sqrt{3}}{4}\left(a^2 - (b-c)^2\right) . \] …35Find every triple \((x, y, z)\) of positive integers for which \[ xyz + xy + xz + yz + x + y + z = 2000 . \]36Let \(x\), \(y\) and \(z\) be real numbers such that \[ x^{2} + y^{2} + z^{2} = 18 \qquad \text{and} \qquad xy + yz + zx = 9 . \] Determine the value of \(|x| + |y| + |z|\).37A circle is drawn through two vertices of a triangle and through the orthocentre of that triangle. Prove that this circle has the same radius as the circle circumscribed about the triangle.38(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 …39Do there exist positive integers \(a\), \(b\), \(c\) such that \[ 2010 = (a + b) \cdot (b + c) \cdot (c + a) \, ? \]40Let \(H\) be the orthocenter of a triangle \(ABC\), and let \(K\) be the point symmetric to \(H\) with respect to the midpoint of the side \(BC\). Prove that \(AK\) is a diameter of the circumcircle of …41At 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 …42Solve the equation \[ x! + 76 = y^2 \] in the set of natural numbers.43A positive real number \(x\) is written on a board. In one move you are allowed to do the following: if a number \(a\) is already on the board, you may write down one of the numbers \(a+1\) or \(\frac{1}{a}\); …44Find every natural number \(n\) for which \[ 7 \cdot 2^n + 1 \] is a perfect square, that is, the square of an integer.45Let \(A\), \(B\), \(C\) and \(D\) be finite sets such that \(D \subseteq A \cup B\), \(D \subseteq C\) and \[ |A \triangle B| + |B \setminus C| + |C \setminus D| + |B \cap D| = |A| . \] (a) Prove that …46Let \(a\), \(b\) and \(c\) be positive integers for which both of the numbers \[ 24^{a} + 2^{b} + 2018^{c} \qquad \text{and} \qquad 10^{c} + 3^{a} + 2018^{b} \] are divisible by \(7\). Prove that the number …47A pentagon \(ABCDE\) is inscribed in a circle. Let \(F\), \(G\), \(H\) and \(I\) be the midpoints of the segments \(BC\), \(CD\), \(DE\) and \(EA\) respectively. The lines \(FG\) and \(HI\) meet at the …48A relation \(\diamond\) is defined on the set \(\mathbb{R}\) of real numbers by \[ a \diamond b \quad \text{if and only if} \quad |a - 1| + |b - 2| \leqslant 1 . \] Suppose the real numbers \(x\) and \(y\) …49Find the smallest natural number \(n\) for which there exist natural numbers \(a\) and \(b\) whose digit sums are \(28\) and \(21\) respectively, and \[ a + b = \underbrace{11\ldots1}_{n}. \]50Let \(n\) 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 …51Let \(\varrho\) be a binary relation on the set \(\mathbb{N}\) defined, for all \(x, y \in \mathbb{N}\), by \[ x \varrho y \iff (\exists k \in A)\ x + 2y = 3k \cdot x. \] In each of the cases \[ \text{(a) } A = \mathbb{N}, \qquad \text{(b) } A = \mathbb{T}, \qquad \text{(c) } A = \mathbb{Q}, \] …52Find all real numbers \(r\) for which there exists exactly one real number \(a\) such that the polynomial \[ p(x) = (x + a)\left(x^2 + rx + 1\right) \] has all of its coefficients nonnegative.53Let \(ABC\) be a triangle and let \(X\) be a point of its plane. Denote by \(A'\), \(B'\), \(C'\) the images of \(A\), \(B\), \(C\) under the reflection in the point \(X\). Let \(M\), \(N\), \(P\) be the …54Each of the numbers \(1, 2, \dots, 1995\) is to be given a sign \(+\) or \(-\). How should the signs be chosen so that the value of \[ \pm 1 \pm 2 \pm \cdots \pm 1995 \] is as close to zero as possible? …55It is known that \[ 35! = 10333147966386144929\,ab\,6651337523200000000 , \] where the letters \(a\) and \(b\) stand for two unknown decimal digits. Determine these two digits.56During the first lesson of the day, the number of boys and the number of girls in a class were in the ratio \(3 : 4\). Before the second lesson another \(4\) girls joined the class and \(4\) boys left …57In 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, …58Over each side of a convex quadrilateral, as a diameter, a circle is constructed. Prove that these four circles cover the quadrilateral.59Find all pairs of integers \(m\) and \(n\) satisfying \[ 2m^{2} + n^{2} = 2mn + 3n . \]60Let \(M\) and \(N\) be two distinct points, neither of which lies on a given line \(p\). Construct a triangle \(ABC\) whose side \(AB\) lies on \(p\) and for which \(M\) and \(N\) are the feet of the altitudes …61The natural numbers \(m\) and \(n\) satisfy \(19 \leq m \leq 49\) and \(51 \leq n \leq 101\). What is the largest value that \[ \frac{n + m}{n - m} \] can take? A \(20\) B \(30\) C \(40\) D \(50\) E \(60\) …62The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …63Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.64The lengths of the sides of a triangle are three consecutive natural numbers, each greater than \(3\). The altitude drawn to the middle side splits that side into two segments. Prove that the lengths of …65Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).66Find all composite natural numbers \(n\) which do not divide the product of all natural numbers smaller than \(n\), that is, all composite \(n\) for which \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …67The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).68Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.69Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.70Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).71Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.72An 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 …73For every point of the first quadrant, determine the line through that point which, together with the positive parts of the coordinate axes, bounds a triangle of the smallest possible area.74A 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?75Determine all positive integers \(n\) for which the number \[ \left| n - \sqrt{6 + \sqrt{6 + \sqrt{6}}} \right| + \left| 3 - \sqrt{6 + \sqrt{6 + \sqrt{6}}} \right| \] is rational.76Consider the polynomials \[ p(x) = x^3 + x^2 + x + 2 , \qquad q(x) = x^3 - x + 3 . \] Does there exist an integer \(m\) such that \(q(m)\) divides \(p(m)\)?77Two segments of lengths \(a\) and \(b\) are given. Construct a triangle \(ABC\) having these two segments as sides, in such a way that the angle opposite one of them is three times as large as the angle …78For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]79In a quadrilateral \(ABCD\) the sides \(AD\) and \(BC\) are equal, and the interior angles at \(A\) and \(B\) satisfy \[ \angle DAB + \angle ABC = 120^\circ . \] Prove that the midpoint of the diagonal …80The side lengths of a certain triangle are mutually distinct natural numbers, and its area is a natural number as well. Must that triangle be right-angled?81How many solutions does the equation \[ x - 2019\{x\} = 2019 \] have in the set of real numbers? Here, for a real number \(x\), the symbol \(\lfloor x \rfloor\) denotes the greatest integer not exceeding …82On the sides \(AB\) and \(BC\) of an equilateral triangle \(ABC\), points \(Z\) and \(X\) are chosen so that \[ AZ : ZB = BX : XC = 2021 : 2020. \] The perpendicular bisector of the segment \(XZ\) meets …83Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …84The 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 …85At most how many interior angles of a polygon with \(n\) vertices can be greater than \(180^\circ\)? (The polygon is simple: its sides meet only at the shared endpoints of neighbouring sides.) A \(n - 1\) …86Janez drew a pattern on a sheet of paper, made up of congruent squares and congruent hexagons. On top of the pattern he then drew two dashed lines perpendicular to each other, as in the figure. What is …87A regular octagon is inscribed in a square of side length \(a\) so that four sides of the octagon lie on the four sides of the square. Express the side length of the octagon in terms of \(a\).88Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]89Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]90Find every natural number \(n\) with the following property: there is a rectangle whose side lengths are natural numbers, whose perimeter equals \(n\), and whose area is numerically equal to \(n\) as well. …91Consider the finite sequence of \(2003\) numbers given by \[ a_n = \left\lfloor \frac{n^2}{2004} \right\rfloor, \qquad n = 1, 2, \ldots, 2003, \] where \(\lfloor x \rfloor\) denotes the greatest integer …92Let \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …93Determine all natural numbers \(n\) for which the following assertion is true: a natural number \(x\) is divisible by \(n\) if and only if the sum of the digits of \(x\) is divisible by \(n\).94Ana, Biljana, Vesna and Gordana crossed a river in a canoe in the following way. There were three trips from the left bank to the right bank, and on each of them the canoe carried exactly two of the girls, …95In triangle \(ABC\) the bisector of the angle \(CAB\) meets the side \(BC\) at the point \(N\), and the bisector of the angle \(CBA\) meets the side \(AC\) at the point \(P\), where \[ PN = a . \] Let …96Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]97Can 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?98Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …99Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]100Four 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 …101Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).102The line through the circumcentre and the orthocentre of a triangle \(ABC\) (the Euler line of the triangle) crosses the interior of the side \(CA\) at a point \(M\) and the interior of the side \(CB\) …103Ana 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 …104Two 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, …105Two circles touch each other internally at a point \(A\). Let \(AB\) be a diameter of the larger circle. Through the other endpoint \(B\) of this diameter a line is drawn which touches the smaller circle …106At 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 …107Decide whether the following claim is true, and prove your answer. For every positive integer \(n\) there exists a positive integer \(x\) such that all three of the following hold: \(x\) is divisible by …108In the plane, two circles \(k_1\) and \(k_2\) and a line \(p\) are given. The line \(p\) cuts \(k_1\) at the points \(A\) and \(B\), and it cuts \(k_2\) at the points \(C\) and \(D\). Each of the two tangents …109Let \(n \geq 2\) 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 \(k_i\) …110Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]111Let \(a\) and \(b\) be natural numbers. Prove that natural numbers \(c\) and \(d\) with \[ a^2 + b^2 + c^2 = d^2 \] exist if and only if at least one of the numbers \(a\) and \(b\) is even.112Two hundred real numbers are written around a circle. Their total sum equals \(200\), and the sum of any three numbers standing next to one another on the circle is at most \(3\). Is it possible for all …113An \(n \times n\) 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 …114Let \(k\) be a circle with centre \(O\), and let \(T\) be a point outside it. The two tangents drawn from \(T\) touch \(k\) at the points \(A\) and \(B\). Let \(k'\) be the circle with centre \(T\) that …115Sixteen 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 …116Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …117Every 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 …118Let \(S\) be a finite set of natural numbers with the property that for every two elements \(x\) and \(y\) of \(S\) there exists an element \(z \in S\) such that \(z \mid x - y\). Prove that \(S\) contains …119Let \(x, y \in \mathbb{R}\) be such that \(x + y\) and \(x^2 + y\) are rational numbers. (a) If \(x + y^2\) is rational as well, must \(x\) and \(y\) be rational? (b) If \(x^3 + y\) is rational as well, …120Find all prime numbers \(p\), \(q\) and \(r\) for which the number \[ p^{\,q+r} + q^{\,p+r} + r^{\,q+p} \] is the square of an odd natural number.121For every natural number, Perica computed the remainder that this number leaves on division by the sum of its digits in the decimal system, and wrote that remainder on the board. Has Perica in this way …122Ana and Beno cycle clockwise along the rectangular track shown in the figure, on which the length of every section is marked. Ana rides the shorter of the two loops, Beno the longer one, and their speeds …123Jure wrote all the natural numbers from \(1\) to \(2015\) on a board. Urska then went through the numbers on the board from the smallest to the largest and rubbed out every one that was not divisible by …124In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at the point \(D\). The triangle \(ADC\) is isosceles with apex \(D\), that is, \(|DA| = |DC|\). Given \(|CD| = 36\) …125Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.126An infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]127A quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).128In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).129For a natural number \(n\), determine the greatest common divisor of the two numbers \[ n^{2} + 1 \qquad \text{and} \qquad (n+1)^{2} + 1 , \] expressed in terms of \(n\).130Let \(ABC\) be an acute triangle. The circle \(k\) with diameter \(AB\) meets the side \(AC\) at \(M\) and the side \(BC\) at \(N\). The tangents to \(k\) at \(M\) and at \(N\) meet at the point \(P\). …131For every natural number \(n\), let \(x_n\) be the number obtained by writing the squares of the first \(n\) natural numbers one after another, in increasing order; for example \[ x_{12} = 149162536496481100121144 . \] …132Determine the smallest possible value of the expression \[ F = \max\{x,\, 1 - y\} + \max\{y,\, 2 - z\} + \max\{z,\, 3 - x\}, \] where \(x\), \(y\), \(z\) are real numbers, and find all triples \((x, y, z)\) …133Three straight cuts divide a rectangle into four pieces, as shown in the figure; the cut meeting the top edge is perpendicular to it. The four pieces are then rearranged, without gaps or overlaps, into …134Let \(ABC\) be a triangle and let \(a\), \(b\), \(c\) denote the lengths of the sides opposite the vertices \(A\), \(B\), \(C\) respectively. Prove that a point \(S\) is the centre of the inscribed circle …135How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?136Two equilateral triangles \(ABC\) and \(PQR\) lie in the plane so that \(R\) is an interior point of the segment \(AB\) and \(C\) is an interior point of the segment \(PQ\), the points \(A\) and \(P\) …137Prove that for every integer \(n \geqslant 2\) one can find \(n\) pairwise distinct positive integers whose squares add up to the square of a positive integer.138Let \(t_a\) and \(t_b\) be the medians of a triangle \(ABC\) drawn to the sides \(BC\) and \(CA\), and let \(P\) be the area of the triangle. Prove that \[ t_a \cdot t_b \geqslant \tfrac{3}{2} P , \] and …139Which of the following two numbers is greater: \[ \frac{1.\underbrace{11\ldots1}_{2005 \text{ digits}}}{1.\underbrace{11\ldots1}_{2006 \text{ digits}}} \qquad \text{or} \qquad \frac{1.\underbrace{0101\ldots01}_{4010 \text{ digits}}}{1.\underbrace{0101\ldots01}_{4012 \text{ digits}}} \, ? \] …140Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).141Let \(ABC\) be an isosceles triangle with \(AB = BC\). A point \(M\) is chosen inside it so that \[ \angle AMC = 2 \angle ABC , \] and a point \(N\) on the segment \(AM\) satisfies \(\angle BNM = \angle ABC\). …142Circles \(k_1\) and \(k_2\) intersect at points \(P\) and \(Q\), and \(k_1\) passes through the centre of \(k_2\). Distinct points \(A\) and \(B\) lie on the arc of \(k_1\) that runs inside \(k_2\), and …143The circle inscribed in triangle \(ABC\) touches the sides \(BC\), \(CA\) and \(AB\) at the points \(D\), \(E\) and \(F\) respectively. A point \(K\) lies on the same side of the line \(EF\) as the vertex …144Convex quadrilaterals \(ABCD\) and \(PQRS\) are given, where the vertices of the quadrilateral \(PQRS\) lie on the sides or in the interior of the quadrilateral \(ABCD\). Can the sum of the diagonals of …145Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]146The 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 …147Let \(ABC\) be a right triangle. Construct a point \(N\) inside \(\triangle ABC\) for which \[ \angle NBC = \angle NCA = \angle NAB. \]148Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.149On the sides of an acute triangle \(ABC\) points \(A_1 \in BC\), \(B_1 \in CA\) and \(C_1 \in AB\) are chosen so that \[ \angle CC_1B = \angle AA_1C = \angle BB_1A = \varphi, \] where \(\varphi\) is an …150Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]151In a triangle \(ABC\), the bisector of the angle \(\angle BAC\) meets the side \(BC\) at \(D\), and the bisector of the angle \(\angle CBA\) meets the side \(AC\) at \(E\). Suppose that \(|CD| = |CE|\). …152Let \(H\) be the orthocentre of an acute triangle \(ABC\), and let \(A_1\), \(B_1\) and \(C_1\) be the centres of the circles circumscribed about the triangles \(BHC\), \(CHA\) and \(AHB\) respectively. …153In 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 …154Peter 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 …155Two 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 …156Let \(a\) and \(b\) be real numbers satisfying \[ \frac{a^2}{1+a^2} + \frac{b^2}{1+b^2} = 1 . \] Determine all possible values of the expression \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …157A 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 …158Squares \(BCDE\), \(ACFG\) and \(BAHK\) are erected outwards on the sides of a triangle \(ABC\). After that, the parallelograms \(BKPE\) and \(CDQF\) are drawn. Prove that the triangle \(PAQ\) is right …159In triangle \(ABC\) the angles at \(A\) and \(B\) measure \(\angle A = 50^\circ\) and \(\angle B = 60^\circ\). Points \(D\) and \(E\) are taken on the sides \(AB\) and \(BC\) respectively, so that \[ \angle DCA = \angle EAC = 30^\circ . \] …160Miljan 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 …161The points \(A, B, C, D, E\) lie on one circle in such a way that \(A\) and \(D\) are on opposite sides of the line \(BC\), and \(B\) and \(E\) are on opposite sides of the line \(CD\). Given that \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …162Two points \(A_1\), \(B_1\) and a line \(p\) are given in the plane. Construct a triangle \(ABC\) in which \(A_1\) is the midpoint of the side \(BC\), \(B_1\) is the midpoint of the side \(CA\), and the …163A 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 …164A triangle \(ABC\) has \(AB = 2\), \(BC = 3\) and \(CA = 4\). Find a polygonal line \(XYZ\) whose endpoints \(X\) and \(Z\) lie on the boundary of the triangle \(ABC\), such that \[ XY = YZ = 1 \] and …165Let \(ABC\) be a triangle with \(BC \neq CA\), and let \(H\), \(T\) and \(O\) be its orthocentre, centroid and circumcentre. Let \(P\) be the point symmetric to \(T\) with respect to \(O\), and let \(Q\) …166A 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 …167Let \(ABCD\) be a convex quadrilateral which is not a trapezoid. The perpendicular bisectors of the sides \(AD\) and \(BC\) meet at a point \(P\), and the perpendicular bisectors of the sides \(AB\) and …168Congruent circles \(k_1\), \(k_2\) and \(k\) all pass through a point \(P\), and each pair of them meets in one further point: \(k\) and \(k_1\) meet again at \(A\), \(k\) and \(k_2\) meet again at \(B\), …169In a triangle \(ABC\) the bisector of the angle at the vertex \(A\) meets the side \(BC\) at the point \(D\). The perpendicular dropped from \(B\) to the line \(AD\) meets the circumcircle of the triangle …170A 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 …171Let \(T\) be the centroid of an acute triangle \(ABC\). Let \(A'\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(A''\) be the point of the segment \(BC\) for which \[ BA' = A''C . \] …172Determine all natural numbers \(n\) with all of the following properties: \(n\) is divisible by \(2\) but not by \(4\); the sum of the digits of \(n\) equals \(6\); the number of divisors of \(n\) equals …173For a set \(X\) let \(\mathcal{P}(X) = \{Y : Y \subseteq X\}\) denote its power set. For example \(\mathcal{P}(\{1\}) = \{\varnothing, \{1\}\}\), since the subsets of \(\{1\}\) are \(\varnothing\) and …174Two 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 …175A collection of \(2020\) consecutive positive integers is split into two subsets of \(1010\) numbers each. Can the least common multiple of all the numbers in the first subset be equal to the least common …176Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.177An \(8 \times 8\) 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 …178Let \(k \geq 2\) 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 …179Lara 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, …180On the sides of a triangle \(ABC\), equilateral triangles \(ADB\), \(BEC\) and \(CFA\) are constructed outwardly, so that \(D\), \(E\), \(F\) are the apexes over \(AB\), \(BC\), \(CA\) respectively. Prove …181Benjamin was working out the sum \(1 + 2 + 3 + \dots + 2012\). He left out a few of the terms, and the wrong total he ended up with was divisible by \(2011\). Anika was working out the sum \[ A = 1 + 2 + 3 + \dots + 2013 , \] …182Let \(ABC\) be a right triangle with its right angle at \(C\), and write \(|BC| = a\), \(|AC| = b\). Let \(D\) be a point on the opposite side of the line \(AC\) from \(B\) for which the triangle \(ACD\) …183On 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 …184Maksim 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 …185A point \(D\) is chosen on the side \(BC\) of a triangle \(ABC\). Points \(E\) and \(F\), both different from \(D\), are chosen on the line \(BC\) so that \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …
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