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 …23Consider 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 …24Let \(a\) and \(b\) be real numbers with \(0 < b \leqslant a\). Prove that \[ \frac{1}{8} \cdot \frac{(a-b)^{2}}{a} \;\leqslant\; \frac{a+b}{2} - \sqrt{ab} \;\leqslant\; \frac{1}{8} \cdot \frac{(a-b)^{2}}{b} . \] …25Determine 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 \(n\).26Ana, 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, …27In 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 …28Let \(M\) be an interior point of a parallelogram \(ABCD\). Prove that \[ MA + MB + MC + MD < \text{the perimeter of } ABCD . \]29Can 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?30Find all real numbers \(a\), \(b\), \(c\), \(d\) for which \[ \begin{aligned} abc + ab + bc + ca + a + b + c &= 2, \\ bcd + bc + cd + db + b + c + d &= 5, \\ cda + cd + da + ac + c + d + a &= 7, \\ dab + da + ab + bd + d + a + b &= 11. \end{aligned} \] …31Determine every positive integer \(n\) for which \[ 5^n + 7^n + 11^n = 6^n + 8^n + 9^n . \]32Four 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 …33Prove that the number \[ \sqrt{1 + \sqrt{2 + \cdots + \sqrt{n}}} \] is irrational for every natural number \(n \geq 2\).34The 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 \(CB\) …35Ana 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 …36Two 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, …37Two 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 …38At 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 …39Decide 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 …40In 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 …41Let \(n \geq 2\) 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 \(k_i\) …42Determine all natural numbers \(k\), \(m\) and \(n\) for which \[ 2^k + 10^m - 10^n = 2014 . \]43Let \(a\) 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.44Two 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 …45An \(n \times n\) 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 …46Let \(k\) 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 …47Sixteen 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 …48Find all digits \(n\) and all \(2018\)-digit positive integers \(x = \overline{a_{2017} \ldots a_2 a_1 a_0}\) for which \[ n \cdot x = \overline{(a_{2017} + n) \ldots (a_2 + n)(a_1 + n)(a_0 + n)} . \] …49Every 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 …50Let \(S\) 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 …51Let \(x, y \in \mathbb{R}\) 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, …52Find 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.53For 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 …54An infinite set \(S\) of pairs of positive integers is given. Prove that \(S\) contains two different pairs \((a,b)\) and \((x,y)\) for which \[ a \leq x \quad \text{and} \quad b \leq y . \]55A quadrilateral \(ABCD\) satisfies \[ AD = BC \qquad \text{and} \qquad \angle DAB > \angle ABC . \] Prove that then \(\angle BCD > \angle CDA\).56In a pentagon \(ABCDE\) all five sides are congruent to one another, and \[ \angle BAE = 2 \angle CAD . \] Determine \(\angle BAE\).57For 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 \(n\).58Let \(ABC\) 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 \(P\). …59For 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 \[ x_{12} = 149162536496481100121144 . \] …60Determine 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 \((x, y, z)\) …61Let \(ABC\) 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 …62How many pairs \((x, y)\) of rational numbers satisfy \(2x^{2} + 5y^{2} = 1\)?63Two 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 \(P\) …64Prove 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.65Let \(t_a\) 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 …66Which of the following two numbers is greater: \[ \frac{1.\underbrace{11\ldots1}_{2005 \text{ digits}}}{1.\underbrace{11\ldots1}_{2006 \text{ digits}}} \qquad \text{or} \qquad \frac{1.\underbrace{0101\ldots01}_{4010 \text{ digits}}}{1.\underbrace{0101\ldots01}_{4012 \text{ digits}}} \, ? \] …67Integers \(x\), \(y\), \(z\) satisfy \[ x^{2}z + y^{2}x + z^{2}y = x^{2}y + y^{2}z + z^{2}x + x + y + z . \] Prove that \(27 \mid x + y + z\).68Let \(ABC\) 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 \(\angle BNM = \angle ABC\). …69Circles \(k_1\) 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 …70The 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 …71Convex 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 …72Find all points \(X\) inside the square \(ABCD\) for which \[ AX + CX = BX + DX . \]73The 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 …74Let \(ABC\) be a right triangle. Construct a point \(N\) inside \(\triangle ABC\) for which \[ \angle NBC = \angle NCA = \angle NAB. \]75Solve the equation \[ 20^x + 2^y = 2022^z \] in the set of natural numbers.76On 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 …77Let \(\mathbb{N}_{0}=\mathbb{N}\cup\{0\}\). Determine all pairs \((a,b)\in\mathbb{N}_{0}\times\mathbb{N}_{0}\) for which \[ 1+3^{a}+2025^{b}=2027^{b}. \]78Let \(H\) 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. …79In 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 …80Let \(a\), \(b\), \(c\), \(d\) 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 \(\dfrac{9a+b}{9c+d}\). …81Let \(a\), \(b\), \(c\) be positive numbers with \(a > c\) and \(b > c\). Prove that \[ \sqrt{c(a-c)} + \sqrt{c(b-c)} \;\leq\; \sqrt{ab} . \]82Prove that among all triangles with one and the same perimeter, the equilateral triangle has the largest area.83Two 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 …84Find 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 …85(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.86In 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.87A 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 …88Let \(a_1, a_2, \dots, a_n\) 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 \(a_1 + a_2 + \cdots + a_n + b_1 + b_2 + \cdots + b_n\) …89Let \(a_1, a_2, \ldots, a_n\) 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 …90Squares \(BCDE\), \(ACFG\) 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 …91In 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 \[ \angle DCA = \angle EAC = 30^\circ . \] …92Miljan 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 …93The 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 \[ \angle ABC = \angle BCD = \angle CDE = 45^\circ , \] …94Two 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 …95A 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 …96A 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 …97Let \(ABC\) 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 \(Q\) …98A 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 …99Let \(ABCD\) 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 …100Congruent 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 \(B\), …101In 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 …102A 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 …103Let \(T\) 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 \[ BA' = A''C . \] …104Determine 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 …105For 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 …106Two 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 …107A 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 …108Find all natural numbers \(n\) for which the number \[ n \cdot 2^{n} + 4 \] is the square of an integer.109An \(8 \times 8\) 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 …110Let \(k \geq 2\) 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 …111Let \(A\) 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 \(S\) …112Can 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 …113A 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 \(p(x)\).114On 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 …115An 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 …116On 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 …117The 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 …118Ali-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, …119Let \(a = 123456789\) and \(b = 987654321\). (1) Find \(\gcd(a, b)\). (2) Find the remainder left by \(\operatorname{lcm}(a, b)\) on division by \(11\).120Real numbers \(a, b, c, d\) satisfy \[ a^{2} + b^{2} = c^{2} + d^{2} , \qquad ab + cd > 0 , \qquad ac + bd > 0 . \] Prove that \(ad + bc > 0\).121The 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 …122Let \(m\) be an arbitrary integer. Prove that there is at least one pair \((x, y)\) of integers for which \[ 2x^2 + 11xy + 12y^2 + 4x + 5y + 6 = 2m . \]123Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).124A 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 …125The 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 …126An operation \(\diamond\) on the set \(G = \{1, 2, 3, \dots, 2016\}\) is given by the table \[ \begin{array}{c|cccccc} \diamond & 1 & 2 & 3 & 4 & \cdots & 2016 \\ \hline 1 & 5 & 5 & 5 & 5 & \cdots & 5 \\ 2 & 1 & 2 & 5 & 5 & \cdots & 5 \\ 3 & 4 & 3 & 5 & 5 & \cdots & 5 \\ 4 & 5 & 5 & 5 & 5 & \cdots & 5 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 2016 & 5 & 5 & 5 & 5 & \cdots & 5 \end{array} \] …127Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …128Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] 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 …129Find 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 …130Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …131Maksim 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 …132A 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 \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …133Two 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 …134Let \(a\), \(b\), \(c\) 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 …135Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …136Let \(x\) 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?137How 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?138Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]
Showing 138 of 651 - problem statements are free for everyone.