Practice library
Problems
1Determine the remainder left by the polynomial \[ x^{2008} - x^{2007} - 3x + 4 \] on division by the polynomial \((x - 1)^{3}\).Open2A 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 …3The capital of a certain country is joined by a direct air route to each of the other \(2012\) cities. Moreover, every one of those \(2012\) cities is joined by an air route to at least one city other …4An operation \(\diamond\) on the set \(G = \{1, 2, 3, \dots, 2016\}\) is given by the table \[ \begin{array}{c|cccccc} \diamond & 1 & 2 & 3 & 4 & \cdots & 2016 \\ \hline 1 & 5 & 5 & 5 & 5 & \cdots & 5 \\ 2 & 1 & 2 & 5 & 5 & \cdots & 5 \\ 3 & 4 & 3 & 5 & 5 & \cdots & 5 \\ 4 & 5 & 5 & 5 & 5 & \cdots & 5 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 2016 & 5 & 5 & 5 & 5 & \cdots & 5 \end{array} \] …5Let \(a\), \(b\) and \(c\) be positive real numbers with \(a + b + c = 3\). Prove that \[ \frac{1}{\sqrt{a^2 + ab + bc}} + \frac{1}{\sqrt{b^2 + bc + ca}} + \frac{1}{\sqrt{c^2 + ca + ab}} \geq \sqrt{3} . \] …6Let \[ P(x) = a_n x^n + \dots + a_1 x + a_0 \] 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 …7Find 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 …8Each of the numbers \(x_1, x_2, \ldots, x_{2023}\) belongs to the set \(\{-1, 0, 1, 2\}\), and together they satisfy \[ x_1 + x_2 + \cdots + x_{2023} = 111, \qquad x_1^2 + x_2^2 + \cdots + x_{2023}^2 = 999 . \] …9Maksim 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 …10A 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 \[ BE = BD \qquad \text{and} \qquad CF = CD . \] …11Two 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 …12Let \(a\), \(b\), \(c\) 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 …13Let \(a\), \(b\), \(c\), \(d\), \(x\), \(y\) be positive real numbers such that \[ a + 2ay + y = b + 2bx + x \qquad \text{and} \qquad x + 2xd + d = y + 2yc + c . \] Prove that \[ a + 2ad + d = b + 2bc + c . \] …14Let \(x\) 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?15How 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?16Prove that for all real numbers \(a\) and \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]17A natural number \(n \geqslant 2\) is divided by each of the natural numbers \(1, 2, \ldots, n-1\) in turn, and all the remainders obtained are written down. Find every \(n\) for which the sum of the distinct …18How many functions \(f \colon \mathbb{R}^{+} \to \mathbb{R}\) are there such that \[ f\left(x + \frac{1}{x}\right) = x^{2} + \frac{1}{x^{2}} \] holds for every \(x \in \mathbb{R}^{+}\)?19For each \(n = 1, 2, 3, \ldots\) Perica looks for the smallest block of \(2n+1\) consecutive positive integers with the property that the sum of the squares of the smallest \(n+1\) of them equals the sum …20Two vertical poles stand on level ground, at a distance of \(9\) m from each other; one pole is \(11\) m high and the other is \(15\) m high. A rope of length \(15\) m is fastened to the top of one pole …21In the expression \[ *\,1 * 3 * 3^{2} * 3^{3} * \cdots * 3^{1997} * 3^{1998} \] Arkadije and Branislav take turns replacing one of the stars by \(+\) or by \(-\), one star per move, until no star is left. …22Let \(ABCD\) be a parallelogram whose interior angle at \(A\) is acute, and let \(E\) be a point of the plane such that \(EA \perp AB\) and \(EC \perp CB\). Prove that \[ \angle AED = \angle CEB . \]23Let \(ABC\) be a triangle and let \(M\), \(N\), \(P\) be points on its sides \(AB\), \(BC\), \(AC\) respectively, chosen so that \(AMNP\) is a parallelogram. Let \(k_1\) be the circle circumscribed about …24In a group of \(20\) people, every person chooses ten of the other nineteen and sends one letter to each of them. Prove that there are two people who sent a letter to each other.25How many isosceles trapezoids with integer side lengths have perimeter \(2005\)? (A trapezoid here means a quadrilateral with exactly two parallel sides, so a parallelogram is not one. Trapezoids with …26For a natural number \(n\), let \(S(n)\) be the sum of its decimal digits and \(P(n)\) the product of its decimal digits. Find all natural numbers \(n\) for which \[ S(n) + P(n) = n . \]27Let \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which \[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \] …28Find every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]29Prove that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).30On \(41\) squares of a chessboard - the ordinary \(8 \times 8\) board - a king is placed, one king on each of those squares. Prove that among these kings one can find three pairwise disjoint sets, each …31Two circles \(R_1\) and \(R_2\) meet at points \(A\) and \(B\). A line through \(A\) is allowed to vary; it meets \(R_1\) again at \(P\) and \(R_2\) again at \(Q\). Prove that all the resulting perpendicular …32A circle \(k\) has radius \(31\,\mathrm{mm}\), and \(\ell\) is a broken line of length \(61\,\mathrm{mm}\) whose two endpoints both lie on \(k\). Prove that there is a line \(p\) passing through the centre …33Let \(ABCD\) be a trapezoid with \(AB \parallel CD\), and let \(P\) be a point on the extension of the diagonal \(AC\) beyond \(C\), so that \(C\) lies between \(A\) and \(P\). Let \(X\) and \(Y\) be the …34For a natural number \(n\), let \(x_n\) be the number obtained by writing the natural numbers from \(1\) to \(n\) one after another, for example \[ x_{15} = 123456789101112131415 . \] Find all natural …35Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?36Let \(n > 1\) be a natural number. How many \(n\)-digit numbers are palindromes and divisible by \(9\)? (A number is a palindrome when its decimal representation is symmetric, that is, it reads the same …37Let \(ABC\) be an acute triangle and let \(D\) be the foot of the altitude from \(A\), so that \(D\) lies on the side \(BC\). A point \(P\) is chosen on the segment \(AD\) in such a way that \[ \angle PBA = \angle PCA . \] …38Let \(k > 0\). On the sides \(A_1B_1\), \(B_1C_1\) and \(C_1A_1\) of a triangle \(A_1B_1C_1\), points \(C_2\), \(A_2\) and \(B_2\) are chosen, respectively, so that \[ \frac{A_1C_2}{C_2B_1} = \frac{B_1A_2}{A_2C_1} = \frac{C_1B_2}{B_2A_1} = k . \] …39Let \(a\), \(b\), \(c\) be the side lengths of a triangle \(ABC\), let \(S\) be its area and \(R\) the radius of its circumscribed circle, and let \(M\) be a point in the interior of the triangle. Write …40Let \(a\), \(b\), \(c\) be arbitrary positive integers. Prove the inequality \[ \gcd(a,\,b-1)\cdot\gcd(b,\,c-1)\cdot\gcd(c,\,a-1) \;\le\; ab+bc+ca-a-b-c+1 , \] and prove that equality is attained for infinitely …41In every cell of a table with \(2017\) rows and \(2017\) columns one of the numbers \(1, 2, 3, \dots, 2017\) is written. Is it possible to do this so that in every row, in every column and along every …42For a positive integer \(n\), let \(x_n\) be the number obtained by writing the decimal representations of \(1, 2, \dots, n\) one after another; for example \(x_{14} = 1234567891011121314\). Define \(f \colon \mathbb{N} \to \mathbb{N}_0\) …43Prove that there exist infinitely many pairs \((m, n)\) of distinct positive integers such that the sum of all positive divisors of \(m^2\) is equal to the sum of all positive divisors of \(n^2\).44Positive real numbers \(a, b, c, d, e\) satisfy \[ a(b+c) = b(c+d) = c(d+e) = d(e+a) = e(a+b) . \] Prove that \(a = b = c = d = e\).45A square board of size \(n \times n\) is given, where \(n \geq 2\). The numbers \(1, 2, \dots, n^2\) are written into the \(n^2\) unit cells of the board, one number in each cell, each number used exactly …46Decide whether there exists a polynomial \(P(x)\) with integer coefficients such that, for some four pairwise distinct integers \(a\), \(b\), \(c\), \(d\), \[ P(a) = P(b) = P(c) = P(d) = 2024 , \] and …47Let \(n\) be a natural number and let \(d\) be a positive divisor of \(2n^2\). Can \(n^2 + d\) be a perfect square?48A triangle \(ABC\) is given. Find every point \(M\) of its plane for which the three triangles \(ABM\), \(BCM\) and \(CAM\) have equal areas.49Let \(k > 3\) and consider the number \(2^k\). Prove that no rearrangement of the decimal digits of \(2^k\) can produce the number \(2^n\) with \(n > k\).50The number \(1\) is written on a board \(2005\) times. A move consists of erasing two of the numbers written on the board and writing, in their place, one quarter of their sum. The move is repeated until …51In how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …52Two fixed points \(A\) and \(B\) are given in the plane. A point \(M\) is chosen and then travels along the straight segment from \(M\) to \(A\). Determine all positions of \(M\) for which the distance …53In a triangle \(ABC\) we have \(\angle ABC = 45^\circ\) and \(\angle CAB = 15^\circ\). Let \(M\) be the point of the ray \(BC\) for which \[ \overrightarrow{BM} = 3 \cdot \overrightarrow{BC} . \] Determine …54Call a positive integer symmetric if its decimal representation reads the same from left to right as from right to left. Prove that there are infinitely many positive integers \(n\) for which the numbers …55Circles \(k_1, k_2, \ldots, k_{1999}\) lie in the plane and touch one another externally in a closed chain: \(k_1\) touches \(k_2\) at \(A_1\), \(k_2\) touches \(k_3\) at \(A_2\), and so on, and finally …56A \(2004 \times 2004\) board is completely tiled by pieces of size \(1 \times 4\); each piece covers four cells of a single row (call it horizontal) or four cells of a single column (vertical). Can the …57A pentagon \(ABCDE\) is inscribed in a circle of radius \(r\), and three of its sides have length \(r\): \[ AB = BC = DE = r . \] Let \(G\) and \(F\) be the midpoints of the sides \(CD\) and \(EA\). Prove …58A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way …59Let \(ABCDEF\) be a regular hexagon. A point \(M\) is taken on the diagonal \(AC\) and a point \(N\) on the diagonal \(CE\) so that both are placed at the same relative position: \[ \frac{AM}{AC} = \frac{CN}{CE} = \lambda . \] …60Find the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …61Let \(P(x)\) be a polynomial with integer coefficients for which there exist prime numbers \(p < q < r\) with \[ \{P(p),\, P(q),\, P(r)\} = \{20,\, 3,\, 2010\} . \] Prove that \(P(p+q) = 2010\).62It is known that for some positive integers \(x\) and \(y\), \[ 23^{x} \cdot 111^{y} = \overline{aab3dc6902b2c74d456b} , \] where \(a, b, c, d\) are digits, not necessarily different, and \(a \neq 0\). …63Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).64The numbers \(a, b, c, x, y, z\) satisfy \[ \{a, b, c\} = \{x, y, z\} = \{15, 3, 2014\} . \] Must the number \[ a^{b^{c}} + x^{y^{z}} \] be composite? (For \(m, n, k \in \mathbb{N}\), the symbol \(m^{n^{k}}\) …65Determine all polynomials \(R(x)\) whose coefficients all belong to the set \(\{-1, 1\}\) and which satisfy \[ R(3) = 130 \qquad \text{and} \qquad R(-2) = -45 . \]66Let \(ABC\) be an acute triangle with \(AB < AC\), and let \(D\) be the midpoint of its side \(BC\). Let \(p\) be the image of the line \(AD\) under reflection in the bisector of the angle \(BAC\), and …67Determine all pairs of positive integers \(a\) and \(b\) for which the number \[ a^4 b + 3b - 2a^2 b^2 - a^2 - 3b^3 \] is a power of two.68Let \(k\) be a positive integer and let \(A\), \(B\), \(C\) be sets such that \(|A \triangle B| = |B \triangle C| = |C \triangle A| = 2k .\) Prove that there is exactly one set \(D\) for which \(|A \triangle D| = |B \triangle D| = |C \triangle D| = k .\) …69Two players play the following game. Taking turns, each player writes down one digit, the digits appearing in a row from left to right in the order in which they are written, and no player is allowed to …70Determine every natural number \(n\) for which there exist real numbers \(a\), \(b\), \(c\) satisfying \[ \bigl\{\, a + b + c,\; ab + bc + ca,\; abc \,\bigr\} = \{\, n,\; n+1,\; n+2 \,\} . \]71Let \(ABC\) be a triangle with \(\angle CAB = 60^\circ\). Denote by \(O\) and \(I\) the centres of the circle circumscribed about it and of the circle inscribed in it, respectively, and let \(A'\) be the …72A point \(P\) inside a triangle \(ABC\) satisfies \[ \angle BPC = \angle BAC + 60^\circ, \qquad \angle CPA = \angle CBA + 60^\circ, \qquad \angle APB = \angle ACB + 60^\circ. \] The lines \(AP\), \(BP\), …73An octagon has all of its interior angles equal, and the lengths of all of its sides are rational numbers. Prove that it has a centre of symmetry.74A triangle \(ABC\) is given. Consider all lines which cut the side \(AC\) at a point \(M\) and the side \(BC\) at a point \(N\) in such a way that \(MN = AM + BN\). Prove that there is a circle \(k\) which …75On the sides \(BC\), \(CA\) and \(AB\) of a triangle \(ABC\) points \(A_1\), \(B_1\) and \(C_1\) are marked, respectively. Let \(T\) be the centroid of the triangle \(ABC\) and \(T_1\) the centroid of …76How many triples \((a, b, c)\) of positive integers are there such that \(2a + 1\) is divisible by \(b\), \(2b + 1\) is divisible by \(c\), and \(2c + 1\) is divisible by \(a\)?77Let \(ABC\) be an isosceles triangle with \(AB = AC\). Let \(D\) be the point of the side \(AC\) for which \(CD = 2 \, AD\), and let \(P\) be a point of the segment \(BD\) with \(\angle APC = 90^\circ\). …78The figure below consists of a triangle \(ABC\) together with the circular segment erected on the side \(BC\), on the opposite side of \(BC\) from \(A\). Construct at least one straight line that divides …79Let \(H\) and \(O\) be the orthocentre and the circumcentre of a triangle \(ABC\) with \(AB \neq AC\). The lines \(AH\) and \(AO\) meet the circumcircle of \(ABC\) a second time at \(M\) and \(N\) respectively. …80Let \(P(x)\) be a polynomial with integer coefficients such that, for every positive integer \(n\), dividing \(P(P(n))\) by \(n\) leaves remainder \(n - 1\). Prove that \(P(x)\) has no integer root.81Let \(a\), \(b\), \(c\) and \(d\) be real numbers with \(abcd = 1\) and \[ a + b + c + d = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} . \] Prove that some two of the numbers \(ab\), \(ac\), …82An acute triangle \(ABC\) is given. Construct, with ruler and compass, a point \(P\) inside the triangle such that the rays \(AP\), \(BP\) and \(CP\) meet the circumcircle of \(ABC\) again in the three …83Let \(ABC\) be an acute triangle. The bisector of its interior angle at \(B\) meets \(AC\) at \(K\), and \(CD\) is the altitude from \(C\), with \(D\) on \(AB\). Let \(N\) be the point of \(CD\) for which …84For some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?85Three points \(A\), \(B\), \(C\), not lying on one line, are given. Construct a point \(D\) for which the quadrilateral \(ABCD\) is at the same time cyclic and tangential, that is, admits both a circumscribed …86In the plane of a triangle \(ABC\) one draws \(n\) lines, each of them parallel to one of the three sides of the triangle. Determine the smallest \(n\) for which these \(n\) lines can cut the plane into …87In a triangle \(ABC\) the angle at \(B\) equals \(80^\circ\). Three further points are marked: the point \(D\) on the side \(BC\) with \(AB = AD = CD\); the point \(F\) on the side \(AB\) with \(AF = BD\); …88Can nine points, no three of them collinear, be placed inside the cross-shaped figure below (its boundary included) in such a way that whenever three of them span a triangle lying inside the figure, that …89Let \(ABC\) be an isosceles triangle with \(AB = AC\). A point \(P\) is taken inside the triangle so that \[ \angle BPC = 90^\circ + \tfrac{1}{2}\angle BAC , \] and a point \(Q\) is taken so that \(\angle BPQ = \angle PQA = 90^\circ\). …90Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that \[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \] …91The product of the binomial coefficient \(\binom{64}{21}\) and an unknown odd number equals \[ 5{\ast}\,6{\ast}0\,{\ast}8{\ast}\,862\,{\ast}1{\ast}\,7{\ast}7\,{\ast}4{\ast}\,4{\ast}5\,12{\ast}\,9{\ast}{\ast} \, , \] …92Let \(ABC\) be a triangle. Prove that the following three lines all pass through one point: the bisector of the angle at \(A\); the line through the midpoints of the sides \(CA\) and \(CB\); and the line …93Prove that the disk of radius \(100\) centred at the origin contains fewer than \(31600\) points whose two coordinates are both integers. (A point counts as contained in the disk if it lies inside it or …94Let \(n\) be a natural number divisible by \(6\). Prove that there exists a convex \(n\)-gon whose interior angles are all equal and which can be cut into finitely many pieces, each piece being of one …95All powers of two are written on a board in increasing order: \(1, 2, 4, \ldots\). Aca and Braca now take turns, Aca first. A move consists of choosing two numbers that stand next to each other on the …96For a positive integer \(n\), let \(f(n)\) denote the least common multiple of the numbers \(1, 2, \ldots, n\). Find all positive integers \(n\) for which \[ f(n) < f(n+1) < f(n+2) < f(n+3) . \]97Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy \[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \] …98Let \(ABC\) be a triangle and let \(k\) be its circumcircle, with centre \(O\). Construct a point \(D\) on \(k\) such that the centroids of the triangles \(ABC\) and \(ABD\) are collinear with the point …99For a positive integer \(x\), let \(S(x)\) denote the sum of the decimal digits of \(x\). (a) Determine the smallest element of the set \(\{\, S(11n^2 + n + 1) \mid n \text{ a positive integer} \,\}\). …100The circle inscribed in a triangle \(ABC\) touches the sides \(AB\) and \(AC\) at the points \(M\) and \(N\) respectively. Let \(P\) be the point in which the bisector of the angle \(ABC\) meets the line …101In an equilateral triangle \(ABC\) the side has length \(|AB| = 2\). Let \(M\) and \(N\) be interior points of the side \(AB\) with \(|MN| = 1\). Prove that \[ \angle MCN > 30^\circ . \]102Finitely many arcs are marked on a circle. The length of each of them is smaller than half of the circumference, and any three of the marked arcs have a common point. Prove that there is a point of the …103Let \(C\) be a point of the segment \(AB\) other than \(A\) and \(B\), and let \(k_{0}, k_{01}, k_{02}\) be the circles with diameters \(AB, AC\) and \(CB\) respectively. Let \(D\) be a point in which …104For which positive integers \(m\) and \(n\) can an \(m \times n\) rectangle be covered completely and without overlaps by copies of the three figures shown below, each of them built from unit squares? …105A pile of \(n\) tokens lies on a table. Two players, \(A\) and \(B\), move alternately, and \(A\) moves first. In one move a player must do one of the following: remove one token from one of the piles …106Let \(ABC\) be a right triangle with hypotenuse \(AB\), and let \(D\) be the midpoint of \(AB\). Let \(k\) be the circumcircle of the triangle \(BCD\), and let \(E\) be an arbitrary point of the shorter …107In a triangle \(ABC\), let \(O\) and \(I\) be the centres of the circumscribed and of the inscribed circle, respectively. Let \(O_1\) be the image of \(O\) under the reflection in the point \(I\), so that …108Determine all nonnegative integers \(n\) and all digits \(a\), \(b\), \(c\) for which the number \[ M = \overline{1\,\underbrace{0 \dots 0}_{n}\,a\,\underbrace{0 \dots 0}_{n}\,b\,\underbrace{0 \dots 0}_{n}\,c} \] …109Determine the largest possible value of \(n\) for which there exists a convex \(n\)-gon that can be decomposed into a disjoint union of triangles, each of which is either right isosceles or right-angled …110Let \(ABC\) be a triangle, let \(D\) be the foot of the altitude from \(A\) to the line \(BC\), and let \(\omega\) be the circle whose diameter is the segment \(AD\). Denote by \(G\) the second common …111A finite sequence of \(n\) numbers is given, each of them equal to \(0\) or \(1\), where \(n\) is a positive integer. One move consists of choosing two adjacent terms \(x\) and \(y\), deleting both of …112Circumscribe about a given triangle \(ABC\) an equilateral triangle \(PQR\) whose side is as long as possible. (Here \(\triangle PQR\) is called circumscribed about \(\triangle ABC\) when \(A \in QR\), …113A bank guard looks after \(n\) safes. Every safe has its own key, no key fits two safes, and all the keys look alike. He is given as many identical circular metal rings as he wants. On any ring he may …114Every point of three-dimensional space is coloured with one of two colours, red or blue, in such a way that whenever three points \(A\), \(B\), \(C\) have the same colour and \(AB = AC\), the midpoint …115Every point of space is painted in one of three colours. Prove that one of the three colours can be chosen in such a way that for every positive real number \(r\) there exists a triangle of area \(r\) …116Let \(n\) be a positive integer. What is the largest number of rooks that can be placed on an \(n \times n\) board so that every rook attacks at most \(3\) of the other rooks? Attacks are the usual chess …117Determine the smallest positive integer \(n\) with the following property: for some real numbers \(a_0, a_1, \dots, a_n\) the function \(f : \mathbb{R} \to \mathbb{R}\) defined by \[ f(x) = \bigl| \, \cdots \, \bigl| \bigl| \, |x - a_0| - a_1 \bigr| - a_2 \bigr| - \cdots - a_{n-1} \bigr| - a_n, \qquad x \in \mathbb{R}, \] …
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