Practice library
Problems
1Each diagonal of a quadrilateral \(ABCD\) divides it into two parts of equal area. Prove that \(ABCD\) is a parallelogram.Open2The 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 …3Let \(a\) and \(b\) be real numbers satisfying \[ a^3 - 3ab^2 = 8, \qquad b^3 - 3a^2 b = \sqrt{61} . \] Find \(a^2 + b^2\).4Find 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 \[ n \nmid 1 \cdot 2 \cdot 3 \cdots (n-1) . \] …5The polynomial \(P\) is given by \[ P(x) = x^{2000} - 2000x^{1999} + 2000x^{1998} - \cdots + 2000x^{2} - 2000x + 2000 . \] Compute \(P(1999)\).6Find the sum of all seven-digit numbers whose digits are \(1, 2, 3, 3, 4, 4, 4\) in some order.7Find all points \(P\) on the circle circumscribed about a triangle \(ABC\) for which the sum \[ PA + PB + PC \] is as small as possible.8Let \(x\) and \(y\) be integers such that \(90\) divides \(x^{2} + xy + y^{2}\). Prove that then \(900\) divides \(xy\).9Find the largest positive integer that is smaller than the sum of the squares of its decimal digits.10An 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 …11For 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.12A 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?13Determine 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.14Consider 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 \(p(m)\)?15Two 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 …16For a natural number \(n\), let \(P(n)\) denote the product of all digits of \(n\). Find every natural number \(n\) satisfying \[ n = P(n) + 18 . \]17In 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 …18The 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?19How 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 …20On 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 …21Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\), \[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \] …22The 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 …23The incircle of a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(D\) and \(E\) respectively. Let \(F\) be any point of the side \(AB\) lying between \(A\) and \(D\), and let \(G\) …24The real numbers \(a\) and \(b\) satisfy \[ \frac{3a}{a+b} + \frac{2b}{a+2b} = 1 . \] Determine every value that the expression \(\dfrac{2a-3b}{2a+b}\) can take.25Farmer Martin has made \(3\) identical bales of hay. The cross-section of each bale is a circle of radius \(r\). He stacks the bales into a pyramid, each bale touching the other two, and stretches a rope …26Find all three-digit numbers \(\overline{abc}\) that are divisible by \(9\) and satisfy \[ \overline{abc} = a^6 + b^2 + c^3, \] where \(a\), \(b\), \(c\) denote the digits of the number.27At 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 \(n - 1\) …28Vid 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 …29At 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 …30Janez 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 …31A 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 \(a\).32Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]33Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]34Find 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. …35Consider 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 …36Let \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …37Determine 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 \(n\).38Ana, 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, …39In 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 …40Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]41Can 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?42Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …43Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]44Four 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 …45Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).46The 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 \(CB\) …47Ana 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 …48Two 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, …49Two 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 …50At 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 …51Decide 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 …52In 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 …53Let \(n \geq 2\) 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 \(k_i\) …54Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]55Let \(a\) 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.56Two 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 …57An \(n \times n\) 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 …58Let \(k\) 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 …59Sixteen 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 …60Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …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 …62Let \(S\) 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 …63Let \(x, y \in \mathbb{R}\) 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, …64Find 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.65For 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 …66Ana 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 …67Jure 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 …68In 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 \(|CD| = 36\) …69Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.70An infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]71A quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).72In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).73For 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 \(n\).74Let \(ABC\) 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 \(P\). …75For 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 \[ x_{12} = 149162536496481100121144 . \] …76Determine 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 \((x, y, z)\) …77Three 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 …78Let \(ABC\) 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 …79How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?80Two 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 \(P\) …81Prove 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.82Let \(t_a\) 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 …83Which of the following two numbers is greater: \[ \frac{1.\underbrace{11\ldots1}_{2005 \text{ digits}}}{1.\underbrace{11\ldots1}_{2006 \text{ digits}}} \qquad \text{or} \qquad \frac{1.\underbrace{0101\ldots01}_{4010 \text{ digits}}}{1.\underbrace{0101\ldots01}_{4012 \text{ digits}}} \, ? \] …84Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).85Let \(ABC\) 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 \(\angle BNM = \angle ABC\). …86Circles \(k_1\) 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 …87The 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 …88Convex 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 …89Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]90The 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 …91Let \(ABC\) be a right triangle. Construct a point \(N\) inside \(\triangle ABC\) for which \[ \angle NBC = \angle NCA = \angle NAB. \]92Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.93On 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 …94Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]95Metka is standing \(60\) m east and \(80\) m south of the spot where Tine is standing. Both are the same distance from a linden tree in the town park, and the tree stands due east of Tine's spot. At the …96The value of the expression \(10^{2016} - 10^{15}\) is a natural number. What is the sum of the digits of that number? A \(1\) B \(17\) C \(2001\) D \(18\,000\) E \(18\,009\)97An airline charges a passenger nothing for baggage as long as its mass does not exceed a fixed permitted mass. Every kilogram above that must be paid for, always at the same price per kilogram. Mr and …98A real number \(a\) satisfies \(a^{2} - \tfrac{1}{2}a = \tfrac{1}{4}\). What is the value of \(a^{3} - \tfrac{1}{2}a\)? A \(-\tfrac{1}{4}\) B \(\tfrac{1}{4}\) C \(\tfrac{1}{2}\) D \(4\) E \(\tfrac{1}{8}\) …99Polona wants to draw three lines through one common point, as in the picture, so that the angles between them satisfy \(\beta = 2\alpha\) and \(\alpha = 3\gamma\). How many degrees must the angle \(\alpha\) …100The digits of a five-digit number add up to \(44\). What is the product of the digits of that number? A \(2^{3} \cdot 3^{8}\) B \(2^{3} \cdot 9^{3}\) C \(8 \cdot 4^{9}\) D \(8 \cdot 3^{4}\) E None of the …101Let \(x\), \(y\), \(z\) 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 \(6\) …102What is the value of \[ 2^{0^{2^{3}}} + 0^{2^{3^{2}}} + 2^{3^{2^{0}}} + 3^{2^{0^{2}}} \, ? \] A \(3\) B \(4\) C \(7\) D \(12\) E Greater than \(100\).103A goldsmith owns two alloys. The first is \(90\%\) gold, the second \(54\%\) gold. He melts together \(320\) g of the first alloy and \(160\) g of the second to obtain a new alloy. What percentage of gold …104For how many integers \(k\) is the number \(k + 6\) an integer multiple of the number \(k - 6\)? A \(0\) B \(4\) C \(6\) D \(8\) E \(12\)105Five semicircles, all of different sizes, stand side by side on one straight segment: each semicircle has its diameter on the segment, consecutive semicircles touch, and the five diameters together fill …106In 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 \(|CD| = |CE|\). …107Let \(H\) 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. …108In 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 …109Let \(a\), \(b\), \(c\), \(d\) be positive real numbers such that \[ \frac{5a+b}{5c+d} = \frac{6a+b}{6c+d} \qquad \text{and} \qquad \frac{7a+b}{7c+d} = 8 . \] Determine the value of \(\dfrac{9a+b}{9c+d}\). …110Let \(a\), \(b\), \(c\) be positive numbers with \(a > c\) and \(b > c\). Prove that \[ \sqrt{c(a-c)} + \sqrt{c(b-c)} \;\leq\; \sqrt{ab} . \]111Prove that among all triangles with one and the same perimeter, the equilateral triangle has the largest area.112The three presents in the picture are all boxes in the shape of a cuboid with edge lengths \(10\) cm, \(20\) cm and \(30\) cm. Reading from left to right, the ribbon tying them measures \(x\) cm, \(y\) …113What 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 \(440\)114Borut 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 …115Birds have gathered on a large pond. Exactly one third of them are swans and all the remaining birds are ducks. It turns out that \(70\%\) of all the birds on the pond are white, and every one of the swans …116Peter 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 …117Two 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 …118Find five real numbers whose pairwise sums are \[ 0,\; 2,\; 4,\; 5,\; 7,\; 9,\; 10,\; 12,\; 14,\; 17 . \] (Five numbers form exactly ten pairs, and ten values are listed; the sums may be listed in any …119(a) Factor the expression \(x^{4}+x^{2}y^{2}+y^{4}\) into a product of polynomials of lower degree. (b) Decide whether the number \(9^{1998}+3^{1998}+1\) is prime.120In a convex hexagon \(ABCDEF\), each of the two diagonals \(AD\) and \(BE\) divides the hexagon into two pieces of equal area. Prove that the quadrilateral \(BDEA\) is a trapezoid.121Let \(a\) and \(b\) be non-zero real numbers with \(a \ne -1\) and \(b \ne -1\), satisfying \[ \frac{a}{b+1} + \frac{b}{a+1} = 1 . \] Which of the following statements about the expression \(\dfrac{a}{b} + \dfrac{b}{a} - \dfrac{1}{ab}\) …122Five brothers - Jure, Klemen, Luka, Miha and Nace - bought a bar of chocolate. When they unwrapped it, they found that it had snapped into the seven pieces shown below, so they shared those seven pieces …123Let \(a\) 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 \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …124A 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 …125Let \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\), where \(n > 1\), be positive integers satisfying \[ \frac{a_1}{b_1} = \frac{a_2}{b_2} = \cdots = \frac{a_n}{b_n}. \] Prove that \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) …126Let \(a_1, a_2, \ldots, a_n\) be positive real numbers with \(a_1 + a_2 + \cdots + a_n = 1\), and let \[ S = \sum_{i=1}^{n} \sum_{j=1}^{n} \frac{a_i a_j}{a_i + a_j} \] be the sum of all \(n^2\) expressions …127Let \(N\) be the number whose decimal expansion consists of \(2025\) copies of the digit \(3\): \[ N = \underbrace{33\ldots3}_{2025} . \] What remainder does \(N\) leave when it is divided by \(12\)? A …128Squares \(BCDE\), \(ACFG\) 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 …129In 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 \[ \angle DCA = \angle EAC = 30^\circ . \] …130Miljan 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 …131The 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 \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …132Two 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 …133A 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 …134A 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 …135Let \(ABC\) 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 \(Q\) …136A 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 …137Let \(ABCD\) 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 …138Congruent 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 \(B\), …139In 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 …140A 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 …141Let \(T\) 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 \[ BA' = A''C . \] …142Determine 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 …143For 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 …144Two 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 …145A 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 …146Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.147An \(8 \times 8\) 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 …148Let \(k \geq 2\) 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 …149Let \(A\) 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 \(S\) …150Can the numbers \(1, 2, \dots, 100\) be distributed into three groups so that the sum of the numbers in the first group is divisible by \(102\), the sum of the numbers in the second group is divisible …151A polynomial \(p(x)\) has integer coefficients. Divided by \(x^2 - 12x + 11\), it leaves the remainder \(990x - 889\). Prove that no integer is a zero of \(p(x)\).152Let \(k\), \(m\) and \(n\) be natural numbers, none of which is divisible by \(5\). Prove that at least one of the three numbers \(k^2 - m^2, \qquad m^2 - n^2, \qquad n^2 - k^2\) is divisible by \(5\). …153The smallest natural number whose square ends in three fours is \(38\), because \(38^2 = 1444\). Which natural number is the next smallest one with this property?154A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]155Is there a natural number \(n\) with the following property: if \(n\) is multiplied by the sum of its own digits, then the digits of the resulting product add up to \(3\)?156The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]157Let \(a\) and \(b\) be real numbers for which \[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \] Prove that \[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]158Find all real numbers \(x\) for which \[ \Biggl|\, \Bigl|\, \bigl|\, |x| - 2 \,\bigr| - 20 \,\Bigr| - 200 \,\Biggr| = 2007 . \]159Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.160Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]161The integers \(a\), \(b\), \(c\), \(d\) satisfy \(a > b > c > d\) and \[ (1 - a)(1 - b)(1 - c)(1 - d) = 10 . \] Which values can the expression \(a + b - c - d\) take?162Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.163Find every natural number \(n \ge 10\) all of whose decimal digits are nonzero and which has the following property: deleting any single digit of \(n\) leaves a number that divides \(n\).164Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]165Real numbers \(a\) and \(b\) satisfy \(|a| \ne |b|\) and \(a \ne 0\), together with \[ \frac{a-b}{a^{2}+ab} + \frac{a+b}{a^{2}-ab} = \frac{3a-b}{a^{2}-b^{2}} . \] Determine the value of \(\dfrac{b}{a}\). …166Eva, Igor, Marko and Marusa each wrote one natural number on a sheet of paper. Deleting the last digit of Eva's number gives Igor's number; deleting the last digit of Igor's number gives Marko's number; …167Jure 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 …168In the convex quadrilateral \(ABCD\) the line through \(A\) and \(D\) and the line through \(B\) and \(C\) meet at a right angle, as the figure shows. Furthermore \(|CD| = 1\), \(|AC| = 2\) and \(|BD| = 3\). …169The horizontal segment in the picture is divided into six parts of equal length, and every triangle appearing in the picture is equilateral. The whole figure is shaded in two colours, light grey and dark …170What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)171Each of the five figures drawn below is bounded entirely by semicircular arcs, and the largest arc is the same in all five figures. Among the figures with the smallest perimeter, which one has the largest …172Peter covered a wound with two rectangular plasters, laid across each other as in the picture. The region covered by both plasters at once has area \(40 \ \mathrm{cm}^{2}\) and perimeter \(30 \ \mathrm{cm}\). …173In triangle \(ABC\) the angle at \(A\) is a right angle. Points \(D\), \(E\) and \(F\) are chosen on the sides \(AB\), \(BC\) and \(CA\) respectively, so that \[ |BD| = |BE| \qquad\text{and}\qquad |CF| = |CE| , \] …174Lara 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, …175On 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 …176An 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 …177Tine collects stamps. For his birthday he received a new album with room for plenty of them, so he took \(2002\) tolars out of his money box and decided to spend all of it on stamps. A friend offered him …178Oliver rolled an ordinary die \(100\) times and multiplied together all \(100\) numbers that came up on top. The product he obtained was \(6^{70}\). What is the smallest possible number of rolls on which …179Benjamin 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 \[ A = 1 + 2 + 3 + \dots + 2013 , \] …180Let \(ABC\) 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 \(ACD\) …181On 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 …182The 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 …183Ali-Baba is standing in a cave full of gold and diamonds. A kilogram of gold is worth \(20\) dinars and a kilogram of diamonds is worth \(60\) dinars. He has a single chest with him. Filled with gold, …184Let \(a = 123456789\) and \(b = 987654321\). (1) Find \(\gcd(a, b)\). (2) Find the remainder left by \(\operatorname{lcm}(a, b)\) on division by \(11\).185Real numbers \(a, b, c, d\) satisfy \[ a^{2} + b^{2} = c^{2} + d^{2} , \qquad ab + cd > 0 , \qquad ac + bd > 0 . \] Prove that \(ad + bc > 0\).186The 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 …187Let \(a\), \(b\), \(c\) be nonzero real numbers such that \[ a(b + c) + b(c + a) + c(a + b) = ab + bc + ca . \] Prove that the value of \[ \frac{a^{2}(b + c) + b^{2}(a + c) + c^{2}(a + b)}{abc} \] is an …188Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]189A 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 …190Let \(m\) be an arbitrary integer. Prove that there is at least one pair \((x, y)\) of integers for which \[ 2x^2 + 11xy + 12y^2 + 4x + 5y + 6 = 2m . \]191Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).192A circle can be inscribed in the trapezoid \(ABCD\), whose parallel sides are \(AB\) and \(CD\). Prove that the circle having \(BC\) as a diameter and the circle having \(AD\) as a diameter touch each …193The 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 …194An operation \(\diamond\) on the set \(G = \{1, 2, 3, \dots, 2016\}\) is given by the table \[ \begin{array}{c|cccccc} \diamond & 1 & 2 & 3 & 4 & \cdots & 2016 \\ \hline 1 & 5 & 5 & 5 & 5 & \cdots & 5 \\ 2 & 1 & 2 & 5 & 5 & \cdots & 5 \\ 3 & 4 & 3 & 5 & 5 & \cdots & 5 \\ 4 & 5 & 5 & 5 & 5 & \cdots & 5 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 2016 & 5 & 5 & 5 & 5 & \cdots & 5 \end{array} \] …195Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …196Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] be a polynomial with integer coefficients. Suppose that \(P\) has two distinct integer zeros, neither of which is positive (\(P\) may have further zeros besides …197Find every three-digit number \(\overline{abc}\) whose digits \(a\), \(b\), \(c\) are all nonzero and which satisfies \[ \overline{abc} = 3 \cdot a! + 2 \cdot b! + c! . \] Here \(\overline{abc}\) is the …198Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …199Maksim 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 …200A 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 \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …201Two concentric circles \(k_1\) and \(k_2\) have radii \(a\) and \(b\). Consider all rectangles that have two vertices on \(k_1\) and the remaining two vertices on \(k_2\). Determine the rectangle of largest …202Let \(a\), \(b\), \(c\) be three distinct nonzero real numbers, and for real \(x, y \neq a\) set \[ V(x,y)=\frac{1}{(a-x)^{2}(a-y)^{2}}\Bigl((a-b)^{2}(c-x)(c-y)-(c-a)^{2}(b-x)(b-y)\Bigr). \] Prove that …203Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …204Let \(x\) and \(y\) be nonnegative real numbers with \(x + y = 2\). Prove that \[ x^2 y^2 \left( x^2 + y^2 \right) \leq 2 . \] When does equality hold?205How 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?206Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]207Tina wrote one natural number on each of five slips of paper and refused to say which numbers they were. Sharp-witted Zan talked her into revealing instead all the sums that can be formed from two of the …208A natural number is written on each face of a cube. At every vertex of the cube one writes the product of the numbers on the three faces that meet at that vertex. The eight numbers at the vertices add …209Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.210Determine every pair of coprime natural numbers \(m\) and \(n\) for which \[ \frac{5m - n}{m + n} \] is itself a natural number. (Here the natural numbers are the positive integers \(1, 2, 3, \ldots\).) …211Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).212Find the smallest natural number \(n\) that is divisible by \(20\) and for which \(n^2\) is a perfect cube and \(n^3\) is a perfect square.213The nonzero real numbers \(x\), \(y\), \(z\) satisfy \[ 3x + 2y = z \qquad \text{and} \qquad \frac{3}{x} + \frac{1}{y} = \frac{2}{z} . \] Prove that the value of \(5x^{2} - 4y^{2} - z^{2}\) is always an …214Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]215For a real number \(a\), let \([a]\) denote the largest integer that is not greater than \(a\). Find all integers \(y\) for which there exists a real number \(x\) satisfying \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …216Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).217The real numbers \(x\) and \(y\) satisfy \[ x^{3} + x^{2} + xy + x + y + 2 = 0 \qquad\text{and}\qquad y^{3} - y^{2} + 3y - x = 0 . \] Determine the value of \(x - y\).218A kangaroo called Pythagoras likes exactly those natural numbers that are divisible by \(4\), have digit sum \(3\), and have exactly five digits equal to \(0\) in their decimal representation. How many …219Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]220Find all natural numbers \(n\) whose cube equals the sum of the squares of three divisors of \(n\), where the three divisors need not be different from one another.221Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]222Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).223Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.224Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]225Find every real number \(x\) for which \[ \left(2x^{2} + 7x + 6\right)^{3} - \left(x^{2} + 3x + 2\right)^{3} = \left(x^{2} + 4x + 4\right)\left(117x^{2} + 128x + 52\right) . \]
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