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 \(b\), \[ a(1 + b^2) + b(1 + a^2) \leq (1 + a^2)(1 + b^2) . \]17Tina 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 …18A 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 …19Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.20Determine 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 \(1, 2, 3, \ldots\).) …21Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).22Find 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.23The 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 …24Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]25For 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 \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …26Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).27The 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 \(x - y\).28A 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 …29Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]30Find 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.31Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]32Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).33Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.34Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]35Find every real number \(x\) for which \[ \left(2x^{2} + 7x + 6\right)^{3} - \left(x^{2} + 3x + 2\right)^{3} = \left(x^{2} + 4x + 4\right)\left(117x^{2} + 128x + 52\right) . \]36A 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 …37How 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 \(x \in \mathbb{R}^{+}\)?38For 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 …39Two 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 …40In 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. …41Let \(ABCD\) 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 \[ \angle AED = \angle CEB . \]42Let \(ABC\) 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 …43In 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.44How 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 …45Let \(E\) be a point on the side \(CD\) of a square \(ABCD\). The point \(F\) lies on the line \(AB\) but not on the segment \(AB\), and satisfies \(|BF| = |DE|\). Prove that the lines \(AC\) and \(EF\) …46Sixteen points of the integer lattice are marked, as in the picture: all points \((x,y)\) with \(x\) and \(y\) taken from \(\{1,2,3,4\}\). At most how many of these points can be coloured red so that no …47For 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 \[ S(n) + P(n) = n . \]48Let \(A_1, A_2, \dots, A_{501}\) be arbitrary pairwise distinct points of the plane. Prove that on every circle of radius \(4\) there is a point \(M\) for which \[ MA_1 + MA_2 + \dots + MA_{501} \geq 2004 . \] …49Find every triple of integers \(x, y, z\) with \[ 3x^2 + 3y^2 + 3z^2 + 2x + 2y + 2z = 2004 . \]50Ana picked the eight digits \(1, 2, 3, 4, 5, 6, 7\) and \(9\). She then forms groups of four two-digit primes, each group using all of her chosen digits. What is the sum of the four primes in one such …51In triangle \(ABC\) the side \(AB\) is twice as long as \(AC\), that is \(|AB| = 2|AC|\). The point \(D\) lies on the ray \(CA\) and satisfies \(|CD| = 3|AC|\). Prove that the perpendicular dropped from …52Prove that there is a natural number \(n\) for which the number \[ 2p^{n} + 3 \] is composite for every prime number \(p\).53On \(41\) 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 …54Two 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 …55A 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 …56Let \(ABCD\) 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 …57For 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 …58Does there exist a natural number which is a perfect square and whose sum of digits equals \(2008^{2009}\)?59Let \(n > 1\) 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 …60Let \(ABC\) 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 \[ \angle PBA = \angle PCA . \] …61Let \(k > 0\). 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 \[ \frac{A_1C_2}{C_2B_1} = \frac{B_1A_2}{A_2C_1} = \frac{C_1B_2}{B_2A_1} = k . \] …62Let \(a\), \(b\), \(c\) 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 …63Let \(a\), \(b\), \(c\) 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 …64In 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 …65For 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 \(f \colon \mathbb{N} \to \mathbb{N}_0\) …66Prove 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 \(n^2\).67Positive 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 \(a = b = c = d = e\).68A 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 …69Decide 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 …70Let \(n\) be a natural number and let \(d\) be a positive divisor of \(2n^2\). Can \(n^2 + d\) be a perfect square?71A triangle \(ABC\) is given. Find every point \(M\) of its plane for which the three triangles \(ABM\), \(BCM\) and \(CAM\) have equal areas.72Let \(k > 3\) and consider the number \(2^k\). Prove that no rearrangement of the decimal digits of \(2^k\) can produce the number \(2^n\) with \(n > k\).73The 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 …74Let \(AB\) be a segment and let \(P\) be any point of it other than \(A\) and \(B\). On the hypotenuses \(AP\) and \(PB\) erect isosceles right triangles \(APQ\) and \(PBR\), with the right angles at the …75A circle \(k\) has diameter \(AB\). A point \(M\) of \(k\) is chosen, different from \(A\) and from \(B\). Let \(k_1\) be the circle with centre \(M\) that touches the diameter \(AB\). The line \(AB\) …76Let \(E\) and \(F\) be the midpoints of the sides \(AD\) and \(DC\) of a rectangle \(ABCD\), and let \(G\) be the point where the segments \(AF\) and \(EC\) cross. Prove that \[ \angle CGF = \angle FBE . \] …77The integers \(x\) and \(y\) satisfy \[ x + xy + y^{2} = 1 \qquad\text{and}\qquad y(5 + x) \ge 0 . \] Which integer values can the expression \(x - y\) take?78A square \(ABCD\) is given, together with points \(E\) and \(F\) lying outside the square such that the triangles \(BEC\) and \(CFD\) are equilateral. Prove that the triangle \(AEF\) is equilateral as …79Find all pairs of coprime integers \(x\) and \(y\) that satisfy the equation \[ 4x^{3} + y^{3} = 3xy^{2} . \]80Let \(ABC\) be a triangle. Points \(D\) and \(E\) lie on the rays \(CA\) and \(CB\) respectively, but not on the sides of the triangle \(ABC\), and are chosen so that \[ |AD| = |BE| = |AB| . \] Let \(G\) …81Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \frac{3xy}{x-y} = 2, \qquad \frac{2yz}{y+2z} = 3, \qquad \frac{xz}{z-4x} = 3 . \]82A disc \(K\) of radius \(R\) is cut into three circular sectors in such a way that the areas of the two smaller sectors add up to the area of the largest sector, while the difference of the areas of the …83In how many ways can natural numbers \(a\), \(b\), \(c\) be chosen so that all of the following hold? \[ 1^{\circ} \quad a < b < c < 52 ; \] \[ 2^{\circ} \quad a \mid c ; \qquad 3^{\circ} \quad b \mid c ; \] …84Two 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 …85Stars are drawn in the cells of a \(4 \times 4\) table, at most one star per cell. What is the least number of stars for which the following holds: whichever \(2\) rows and whichever \(2\) columns are …86Ales, Brane and Cvetka made a large pile of cards, writing on each card one of the numbers \(2, 3, 4, 5, 6, 7, 8\); every one of these numbers appears on many cards. Maja, who arrived later, picked three …87In 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 …88Call 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 …89Circles \(k_1, k_2, \ldots, k_{1999}\) 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 …90A \(2004 \times 2004\) 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 …91A 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 …92Eighteen matches are laid out to form the grid shown below: an equilateral triangle whose side is three matches long, divided into nine small triangles. What is the smallest number of matches that must …93Three squares are inscribed in a right triangle \(ABC\) with the right angle at \(B\), arranged as in the picture: the largest one has two sides lying on the legs of the triangle, and each of the two smaller …94Let \(A\) be the set of all integers from \(-20\) to \(20\), that is \[ A = \{\, a \in \mathbb{Z} \;:\; -20 \le a \le 20 \,\} . \] Let \(n\) be a positive integer and let \(A_1, A_2, \ldots, A_n\) be pairwise …95A 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 …96Let \(ABCDEF\) 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: \[ \frac{AM}{AC} = \frac{CN}{CE} = \lambda . \] …97Find the smallest possible value of \(x + y + z\) for non-negative real numbers \(x, y, z\) subject to \[ \begin{aligned} xy(10x + 10y + 7z) &\geq 27, \\ yz(10y + 10z + 7x) &\geq 27, \\ zx(10z + 10x + 7y) &\geq 27. \end{aligned} \] …98Let \(P(x)\) 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 \(P(p+q) = 2010\).99It 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 \(a \neq 0\). …100Let \(p\) be a prime number. Suppose that for some \(k \in \mathbb{N}\) the number \[ k^3 + pk^2 \] is a perfect cube. Prove that \(3 \mid p - 1\).101The 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 \(m^{n^{k}}\) …102Determine all polynomials \(R(x)\) whose coefficients all belong to the set \(\{-1, 1\}\) and which satisfy \[ R(3) = 130 \qquad \text{and} \qquad R(-2) = -45 . \]103Let \(ABC\) 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 …104Determine 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.105Let \(k\) 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 \(|A \triangle D| = |B \triangle D| = |C \triangle D| = k .\) …106Two 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 …107Determine every natural number \(n\) for which there exist real numbers \(a\), \(b\), \(c\) satisfying \[ \bigl\{\, a + b + c,\; ab + bc + ca,\; abc \,\bigr\} = \{\, n,\; n+1,\; n+2 \,\} . \]108Let \(ABC\) 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 …109A 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 \(AP\), \(BP\), …110An 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.111A 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 …112On 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 …113How 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 \(a\)?114Let \(ABC\) 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 \(\angle APC = 90^\circ\). …115Andraz and Breda cut two long strips out of a newspaper, of lengths \(a\) and \(b\), to play a game with. A move consists of choosing one of the strips and cutting a piece of length \(d\) off it, so that …116A spider has spun the web shown below: five regular octagons nested one inside the other, with each vertex of an octagon joined by a thread to the corresponding vertex of the neighbouring octagons. The …117Find the smallest natural number \(n\) for which an \(n \times n\) board of unit cells can be covered completely and without overlaps by equally many tiles of the two shapes below: an \(L\)-shaped tile …118The triangle \(ABC\) has side lengths \(|AB| = 15\) cm, \(|BC| = 14\) cm and \(|CA| = 13\) cm. Let \(D\) be the foot of the altitude drawn from \(A\) to the side \(BC\), and let \(E\) be the point of that …119Let \(ABCD\) be a convex quadrilateral and let \(E\) and \(F\) be points on the sides \(AB\) and \(AD\) respectively, chosen so that \(EF \parallel BD\). The segment \(CE\) meets the diagonal \(BD\) at …120In a rectangle \(ABCD\) with \(|AB| > |BC|\), the perpendicular bisector of the diagonal \(AC\) meets the side \(CD\) at the point \(E\). The circle with center \(E\) and radius \(|AE|\) meets the side …121A \(4 \times 4\) table is divided into \(16\) unit cells. On this table we place tiles of the shape drawn alongside: two unit squares that meet at a single corner. A tile may be rotated, and each tile …122A triangle \(ABC\) has a point \(D\) on side \(AB\) and a point \(E\) on side \(AC\) such that \[ |AE| = |ED| = |DB| \qquad\text{and}\qquad |AD| = |DC| = |CB| . \] Determine the angles of triangle \(ABC\). …123A parallelogram \(ABCD\) satisfies \(|AB| = |BD|\). Let \(K\) be the point of line \(AB\), different from \(A\), with \(|KD| = |AD|\). Let \(M\) be the image of \(C\) under the half-turn about \(K\) (so …124Let \(n \ge 2\) be a natural number. Maja and Peter want to colour every cell of an \(n \times n\) table either black or blue, subject to one rule: among any four cells that can be covered by a square …125The lines containing the two diagonals of a quadrilateral \(ABCD\) meet at an angle of \(60^{\circ}\). Each vertex of the quadrilateral is reflected in the line containing the diagonal that joins its two …126Natasa glued a square and an equilateral triangle of the same side length into a pentagon. Out of seven copies of that pentagon she built the figure shown on the right, which sits inside a large square. …127Three curves are drawn inside a square \(ABCD\): the quarter circle \(\mathcal{Q}\) centred at the vertex \(A\) and passing through \(B\) and \(D\); the semicircle \(\mathcal{P}\) centred at the midpoint …128Determine all quintuples of primes \(p_{1} \le p_{2} \le p_{3} \le p_{4} \le p_{5}\) with the property that each of the five primes divides the sum of the remaining four.129The 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 …130Let \(H\) 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. …131Let \(P(x)\) 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.132Let \(a\), \(b\), \(c\) 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 \(ab\), \(ac\), …133An 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 …134Let \(ABC\) 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 …135Let \(M\) be the midpoint of the base \(AB\) of a trapezoid \(ABCD\), so that \(AB \parallel CD\). A point \(E\) is taken in the interior of the segment \(AC\) in such a way that the lines \(BC\) and \(ME\) …136Let \(I\) be the incenter of a triangle \(ABC\), and suppose that \[ |CA| + |AI| = |BC| . \] Determine the ratio of the sizes of the angles \(\angle BAC\) and \(\angle CBA\).137For some integer \(n > 3\), all the digits of the number \[ 1 + 2 + \cdots + n \] are equal to one another. Which digit can this be?138Level 1 of the computer game Zakladnica takes place in an underground treasury built from \(13\) octagonal and \(12\) square rooms, arranged as in the figure. The only way into the treasury, and the only …139Three 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 …140In 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 …141In 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 \(AF = BD\); …142Can 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 …143Let \(ABC\) 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 \(\angle BPQ = \angle PQA = 90^\circ\). …144Does there exist a polynomial \(P(x)\) whose coefficients are not all integers, such that \[ P(0)=0 \qquad\text{and}\qquad \frac{P(a)-P(b)}{a-b}\ \text{ is an integer for every pair of distinct integers } a, b\ ? \] …145The product of the binomial coefficient \(\binom{64}{21}\) and an unknown odd number equals \[ 5{\ast}\,6{\ast}0\,{\ast}8{\ast}\,862\,{\ast}1{\ast}\,7{\ast}7\,{\ast}4{\ast}\,4{\ast}5\,12{\ast}\,9{\ast}{\ast} \, , \] …146Let \(ABC\) 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 …147Prove 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 …148Let \(n\) 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 …149All 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 …150For 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 \[ f(n) < f(n+1) < f(n+2) < f(n+3) . \]151Suppose that pairwise different positive integers \(a_1, a_2, \dots, a_{2024}\) satisfy \[ [a_1, a_2] + (a_2, a_3) + [a_3, a_4] + (a_4, a_5) + \dots + [a_{2023}, a_{2024}] + (a_{2024}, a_1) = a_1 + a_2 + \dots + a_{2024} . \] …152Let \(ABC\) 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 …153For a positive integer \(x\), let \(S(x)\) denote the sum of the decimal digits of \(x\). (a) Determine the smallest element of the set \(\{\, S(11n^2 + n + 1) \mid n \text{ a positive integer} \,\}\). …154The 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 …155In 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 \[ \angle MCN > 30^\circ . \]156A mole has dug a number of underground rooms and joined them by tunnels, in such a way that from every room exactly \(3\) tunnels lead out, to \(3\) different rooms. Tunnels meet one another only at rooms. …157Let \(I\) be the centre of the inscribed circle of a triangle \(ABC\), and let \(A_1\), \(B_1\), \(C_1\) be the feet of the perpendiculars dropped from \(I\) to the sides \(BC\), \(AC\) and \(AB\). The …158Let \(D\) be the midpoint of side \(AB\) of an acute triangle \(ABC\). Points \(A'\) and \(B'\) are chosen on the segments \(AC\) and \(BC\) so that the triangles \(ADA'\) and \(DBB'\) are both isosceles …159A natural number is written in every cell of a square table. Call the table interesting if the sum of all the numbers in it is odd and, in addition, the sum of the four numbers covered by any placement …160A teacher handed Matej four sheets of paper, each carrying one nonzero digit. Matej laid the sheets in a row and so formed a four-digit number. He then interchanged two of the sheets, without flipping …161In a hexagon \(ABCDEF\) the following hold: \(\angle BAF = 150^{\circ}\), \(\angle ACB = \angle ADC = 90^{\circ}\), \(|AC| = |BC|\), triangle \(ABC\) is similar to triangle \(ADE\), and triangle \(BCD\) …162We want to cover a \(4 \times 4\) board with tiles of the shape shown below, rotations and reflections being allowed. The tiles are permitted to overlap one another and to stick out beyond the edge of …163Three piles of tokens lie on a table, holding \(a\), \(b\) and \(c\) tokens, where \(a \ge b \ge c > 0\). Players \(A\) and \(B\) move tokens alternately, and \(A\) starts. In one move a player first selects …164A rectangular grid of size \(7 \times 9\) is given: seven rows of cells and nine columns of cells, as in the figure. At the bottom-left node of the grid sits a colony of ants, and at the top-right node …165Find all pairs of real numbers \(x\) and \(y\) that satisfy the system \[ \frac{y^{3} + 15x^{2}}{y^{4} - x^{3}} = \frac{y^{2} + 15x}{y^{3} - x^{2}} , \] \[ \frac{1500y^{3} + 4x^{2}}{9y^{4} - 4} = \frac{1500y^{2} + 4x}{9y^{3} - 4} . \] …166Timotej had a sheet of squared paper measuring \(8 \times 8\) little squares. He folded it a few times, each fold running along one of the lines of the grid, until he was left with a square piece measuring …167Finitely 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 …168A strip of \(1 \times n\) cells is given, where \(n > 10\) is a natural number, and its cells are numbered \(1, 2, \dots, n\) from left to right. Cell number \(10\) is black and carries a token; every …169We want to choose a set \(P\) of \(k\) primes and a set \(N\) of \(n\) consecutive positive integers in such a way that every number \(a \in N\) is divisible by at least one prime \(p \in P\). (a) Determine …170Anja owns tiles shaped like a single unit square, Bojan tiles shaped like an L-tromino: three unit squares forming an L, as drawn below. The two players alternately place one tile of their own onto a rectangular …171Every cell of an \(n \times n\) table contains the number \(0\). One step consists of choosing three cells that form the shape and adding \(1\) to each of the three numbers standing in them. Can we, after …172Let \(C\) 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 …173For 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? …174A 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 …175Let \(ABC\) 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 …176In 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 …177Determine all nonnegative integers \(n\) and all digits \(a\), \(b\), \(c\) for which the number \[ M = \overline{1\,\underbrace{0 \dots 0}_{n}\,a\,\underbrace{0 \dots 0}_{n}\,b\,\underbrace{0 \dots 0}_{n}\,c} \] …178Determine 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 …179Let \(ABC\) 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 …180A 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 …181Circumscribe 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 \(A \in QR\), …182A 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 …183Every 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 …184Every 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 \(r\) …185Let \(n\) 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 …186Determine 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 \[ f(x) = \bigl| \, \cdots \, \bigl| \bigl| \, |x - a_0| - a_1 \bigr| - a_2 \bigr| - \cdots - a_{n-1} \bigr| - a_n, \qquad x \in \mathbb{R}, \] …
Showing 186 of 651 - problem statements are free for everyone.