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1The 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 \(G\) …Open2The 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.3Farmer 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 …4Find 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.5At 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 \(n - 1\) …6Vid 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 …7At 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 …8Janez 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 …9A 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 \(a\).10Find all integers \(x\) and \(y\) that satisfy \[ 3xy + 2x + y = 12 . \]11Positive real numbers \(a\) and \(b\) have product \(1\), and the sum of their squares equals \(4\). Determine the exact value of \[ a^{-3} + b^{-3} . \]12Find 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. …13Ana 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 …14Jure 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 …15In 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 \(|CD| = 36\) …16Find all natural numbers \(n\) and all primes \(p\) for which \[ \sqrt{\,n + \frac{p}{n}\,} \] is a natural number.17Three 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 …18Metka 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 …19The 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 \(18\,009\)20An 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 …21A 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 \(\tfrac{1}{8}\) …22Polona 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 \(\alpha\) …23The 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 …24Let \(x\), \(y\), \(z\) 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 \(6\) …25What 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 \(100\).26A 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 …27For 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 \(12\)28Five 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 …29In 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 \(|CD| = |CE|\). …30The 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, \(y\) …31What 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 \(440\)32Borut 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 …33Birds 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 …34Peter 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 …35Let \(a\) 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 \(\dfrac{a}{b} + \dfrac{b}{a} - \dfrac{1}{ab}\) …36Five 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 …37Let \(a\) 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 \[ (a+b)\left( \frac{a}{1+a^2} + \frac{b}{1+b^2} \right) . \] …38Let \(N\) 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 …39Let \(k\), \(m\) 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 \(5\). …40The 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?41A prime number \(p\) is given. Find all pairs of natural numbers \(x\) and \(y\) (positive integers) that satisfy \[ p\,(x - 5) = x\,y . \]42Is 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 \(3\)?43The real numbers \(x\), \(y\), \(z\) satisfy \(xyz = 1\). Compute the value of \[ \frac{x+1}{xy+x+1} + \frac{y+1}{yz+y+1} + \frac{z+1}{zx+z+1}. \]44Let \(a\) and \(b\) be real numbers for which \[ \frac{a}{1+a} + \frac{b}{1+b} = 1 . \] Prove that \[ \frac{a}{1+b^{2}} - \frac{b}{1+a^{2}} = a - b . \]45Find all real numbers \(x\) for which \[ \Biggl|\, \Bigl|\, \bigl|\, |x| - 2 \,\bigr| - 20 \,\Bigr| - 200 \,\Biggr| = 2007 . \]46Find all pairs of prime numbers \(p\) and \(q\) for which the number \[ 2p^{2}q + 45pq^{2} \] is a perfect square.47Find all real numbers \(x\), \(y\), \(z\) that satisfy the system \[ \begin{aligned} x + y + 2z &= 0, \\ xy - z^2 &= 0, \\ y^2 + 5z + 6 &= 0. \end{aligned} \]48The 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?49Determine all prime numbers \(p\) and \(q\) for which the number \[ 2^{2} + p^{2} + q^{2} \] is also prime.50Find 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 \(n\).51Find all triples of prime numbers \(p\), \(q\), \(r\) for which \[ p + q^{2} = r^{4} . \]52Real 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 \(\dfrac{b}{a}\). …53Eva, 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; …54Jure 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 …55In 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 \(|BD| = 3\). …56The 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 …57What is the value of the expression \[ 2019^{3} - 3 \cdot 2019 \cdot 2018 - 2018^{3} \, ? \] A \(-1\) B \(0\) C \(1\) D \(2018\) E \(2019\)58Each 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 …59Peter 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 \(30 \ \mathrm{cm}\). …60In 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 \[ |BD| = |BE| \qquad\text{and}\qquad |CF| = |CE| , \] …61Lara 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, …62Tine 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 …63Oliver 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 …64Benjamin 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 \[ A = 1 + 2 + 3 + \dots + 2013 , \] …65Let \(ABC\) 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 \(ACD\) …66Let \(a\), \(b\), \(c\) 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 …67Find all primes \(p\), \(q\), \(r\) and \(s\) for which \[ p + q = r \qquad\text{and}\qquad q + r = s^{2} . \]68A 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 …69Tina 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 …70A 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 …71Find every prime \(p\) for which the number \(p^2 + 11\) has fewer than \(11\) positive divisors.72Determine 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 \(1, 2, 3, \ldots\).) …73Find all pairs of natural numbers \(m\) and \(n\) whose sum equals \(2007\) and whose product is divisible by \(2007\).74Find 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.75The 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 …76Find all pairs of real numbers \(x\) and \(y\) satisfying \[ x + y^{2} = xy + 1 \qquad\text{and}\qquad xy = 4 + y . \]77For 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 \[ \left[\frac{x+23}{8}\right] = \left[\sqrt{x}\,\right] = y . \] …78Let \(m\) and \(n\) be positive integers such that \(5m+n\) divides \(5n+m\). Prove that \(m\) divides \(n\).79The 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 \(x - y\).80A 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 …81Find all prime numbers \(p\), \(q\) and \(r\) that satisfy \[ r^{4} = pq + 4 . \]82Find 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.83Prove that there are no natural numbers \(a\) and \(b\) satisfying \[ \sqrt{a} + \sqrt{b} = \sqrt{2021} . \]84Find all pairs of natural numbers \(a\) and \(b\) for which \[ v = ab - 2a - 4b , \] where \(v\) denotes the least common multiple of \(a\) and \(b\).85Determine the smallest possible value of \[ \left|25^{m} - 36^{n}\right| \] where \(m\) and \(n\) are positive integers.86Find all real numbers \(x\) that satisfy the equation \[ \left(\frac{x^{2}-13}{2x+1}\right)^{2} \;-\; 8\cdot\frac{x^{2}-13}{2x+1} \;=\; 48 . \]87Find 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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