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1In the right triangle \(ABC\) the right angle is at the vertex \(C\). Let \(S\) be the midpoint of the side \(AB\), and let \(V\) be the point where the altitude dropped from \(C\) meets \(AB\). Determine …Open2Find every natural number with the following property: when the sum of its digits is added to the number itself, the result is \(313\).3Find the smallest natural number \(n\) for which the sum \[ n + 2n + 3n + \cdots + 9n \] is a number whose decimal representation has all of its digits equal.4Find every real number \(x\) that satisfies the equation \[ \bigl|\,2006 - |206 - x|\,\bigr| = 26 . \]5Prove that the inequality \[ x^2 + y^2 + 1 \ge 2\bigl(xy - x + y\bigr) \] holds for every pair of real numbers \(x\) and \(y\). When does equality occur?6Let \(d\) denote the greatest common divisor and \(v\) the least common multiple of the natural numbers \(m\) and \(n\). Prove that if \[ 3m + n = 3v + d , \] then \(n\) divides \(m\).7A rectangle \(ABCD\) has \(|AB| = 2a\) and \(|AD| = a\). Let \(E\) be the midpoint of the side \(AB\), and choose an arbitrary point \(F\) on the side \(AD\). The area of the triangle \(ECF\) depends on …8An equilateral triangle \(ACG\), a regular pentagon \(CDEFG\) and a regular octagon \(BHIJKLDC\) all meet at the point \(C\), as shown in the figure. Determine the angles of the triangle \(ABC\).9Two concentric circles of radii \(7\) cm and \(11\) cm are drawn in the plane. The smaller circle cuts a chord of the larger circle into three pieces of equal length. How long is that chord?10Find the smallest three-digit number with the property that every digit of its triple is even.11Let \(t\) be a real number and let \(a\) and \(b\) be positive real numbers satisfying \[ 2a^2 - 3abt + b^2 \;=\; 2a^2 + abt - b^2 \;=\; 0 . \] Determine the value of \(t\).12Let \(n\) be the number \(100\ldots001\) whose decimal expansion consists of the digit \(1\), then \(2017\) digits \(0\), then the digit \(1\) again. Decide, with proof, whether \(n\) is divisible by (a) …13Show that the number \(7^{2018} + 9^{2020n}\) is divisible by \(5\) for every natural number \(n\).14Let \(x\) and \(y\) be real numbers satisfying \(y - x = 2\). Compute the value of the expression \[ 2x^3 + y^3 - 3y^2x + 6x^2 . \]15Find all pairs of real numbers \(x\) and \(y\) that solve the system \[ \frac{x}{6} + \frac{4}{y} = 2, \qquad \frac{18}{x} + \frac{y}{2} = 5 . \]16Determine every pair of integers \((x, y)\) for which \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} . \]17Let \(a\), \(b\), \(c\) be positive numbers with \(a^2 + c^2 = 2b^2\). Prove that \[ \frac{2}{a+c} = \frac{1}{a+b} + \frac{1}{b+c} . \]18Find all natural numbers \(m\) and \(n\) that satisfy \[ \frac{3}{m} + \frac{5}{n} = 1 . \]19The rhombus \(ABCD\) has an acute interior angle at the vertex \(A\). The perpendicular dropped from \(D\) to the side \(AB\) meets it at the point \(E\), which splits the side into the two pieces \[ |AE| = x , \qquad |EB| = y . \] …20Jana and Zana each wrote down her own age. Both ages turned out to be two-digit numbers written with the same two digits, only in the opposite order. Five years from now, Jana will be exactly twice as …21A competition paper had \(20\) problems. Each problem answered correctly was worth \(8\) points, each problem answered incorrectly cost \(5\) points, and a problem left blank scored \(0\). Tine handed …22Find every real number \(x\) that satisfies \[ \Bigl|\, \bigl|\, |2-x| - x \,\bigr| - 8 \,\Bigr| \le 2008 . \]23A competition paper consisted of \(24\) multiple-choice questions. A contestant who circled no answer to a question, or circled more than one, received \(0\) points for it; a circled correct answer was …24Find the smallest three-digit number with the following property: if its digits are written in the reverse order and the number so obtained is added to the original number, then every digit of the sum …25Find all pairs of integers \(x\) and \(y\) that satisfy the equation \[ x^2 + xy + y^2 = 1 . \]26Find all solutions of the equation \[ x = \bigl|\,2x - |60 - 2x|\,\bigr| . \]27In the right triangle \(ABC\) the right angle is at \(C\), and \(|AC| = 4\), \(|BC| = 8\). A point \(D\) is taken on the side \(BC\) so that \(|CD| = 5\). How far is \(D\) from the side \(AB\)?28The equilateral triangle \(ABC\) has sides of length \(4\) cm. Let \(D\) be the midpoint of the side \(AB\), and let \(E\) and \(F\) be the feet of the perpendiculars dropped from \(D\) to the sides \(BC\) …29Every school of a certain region sent exactly \(3\) students to a competition, and Andrej, Blaz and Zan all came from the same school. When the competitors lined up to collect their starting numbers, Andrej …30Jure wanted to prepare presents for nine friends, giving each friend two chocolate bars. In the shop he saw that a hazelnut chocolate bar costs \(6\) tolars more than a milk one, and that each of them …31Find all real numbers \(x\) that satisfy the inequality \[ \frac{|x-3| + x}{x+1} < 1 . \]32Find all pairs of natural numbers \(a\) and \(b\) that satisfy the equation \[ a^2 - 5ab + 24 = 0 . \]33The number whose cube equals \(2012^{12}\) was multiplied by the square of the number \(2012^{11}\). Which number was obtained? A \(2012^{58}\) B \(2012^{26}\) C \(2012^{88}\) D \(2012^{15}\) E \(2012^{12}\) …34A factory modernised its equipment, and its productivity then rose by \(25\%\). Some time later a number of workers were dismissed, and the productivity fell by \(20\%\). By what percentage has the productivity …35One digit of the seven-digit number \(2345678\) is to be deleted, so that the six-digit number left behind is divisible by \(9\). Which digit must it be? A \(8\) B \(7\) C \(6\) D \(5\) E \(4\)36Suppose the numbers \(a\) and \(b\) satisfy \(a + b = 1\) and \(ab \neq 0\). Prove that \[ \frac{a}{b^3 - 1} - \frac{b}{a^3 - 1} = \frac{2(b-a)}{a^2b^2 + 3} . \]37Let \(M\) be the midpoint of the side \(BC\) and \(N\) the midpoint of the side \(CD\) of a rectangle \(ABCD\). Determine the ratio of the side lengths of \(ABCD\), given that \(AMN\) is a right triangle …38Nika and Tim played a series of card games. A draw was impossible. They agreed in advance that after each game the winner receives more points than the loser and that the loser receives a positive number …39Andrej and Blaz drew a circle in the yard and marked two diametrically opposite points on it. Each of them stood on one of these points, and at the same instant they started walking around the circle in …40Let \(a\) and \(b\) be arbitrary nonnegative real numbers. Prove that \[ (ab+1)(a+b) \ge 4ab , \] and determine all pairs \((a,b)\) for which equality holds.41The figure shows two squares and two congruent circles whose centres lie on a diagonal of the larger square. The smaller square occupies a corner of the larger one, and the diagonal in question runs from …42A rectangle \(ABCD\) satisfies \(|AB| = 10\) and \(|AC| = 5\sqrt{7}\). Let \(M\) be the midpoint of the side \(AB\). Inside the rectangle we draw the semicircle with diameter \(AB\) and the triangle \(CDM\), …43The squares \(ABCD\) and \(EFGH\) both have side \(1\). The first is cut into nine congruent small squares and the second into sixteen congruent small squares; in every small square the inscribed circle …44Lili noticed that the digits of the year \(2015\) have average \(2\), because \(\frac{2+0+1+5}{4} = 2\). How many years of the 21st century after \(2015\) have the same digit average as \(2015\)? A \(1\) …45A circle of radius \(\sqrt{2}\) is given. A second circle, of radius \(2\), has its centre on the first circle. Find the area of the shaded region, that is, of the part of the smaller disc that lies outside …46Find all prime numbers \(p\), \(q\) and \(r\) for which \[ 15p + 7pq + qr = pqr . \]47Show that the number \[ 2^{\,2n+3} + 3^{\,n+2}\cdot 7^{\,n} \] is divisible by \(17\) for every natural number \(n\).48Find all integers \(n\) for which the number \[ \frac{9n+8}{n+7} \] is also an integer.49The numbers \(x\), \(y\) and \(a\) satisfy \(3^{x} = a\) and \(a^{y} = 81\). What is the value of the product \(x\cdot y\)? A \(4\) B \(3\) C \(12\) D \(0\) E \(1\)50Let \(a\) and \(b\) be natural numbers, both greater than \(1\), such that \[ \sqrt{a\sqrt{a\sqrt{a}}} = b . \] What is the smallest possible value of the sum \(a + b\)?51The real numbers \(a\) and \(b\) satisfy \[ a^3 = 3ab^2 + 11 , \qquad b^3 = 3a^2 b + 2 . \] Compute the value of \(a^2 + b^2\).52In the right triangle \(ABC\) the right angle is at \(C\), the hypotenuse \(AB\) measures \(1\) dm, and \(\angle BAC = 30^\circ\). A point \(D\) inside the triangle satisfies \[ \angle BDC = 90^\circ \qquad\text{and}\qquad \angle ACD = \angle DBA . \] …53Nonzero real numbers \(a\), \(b\), \(c\) and a real number \(d\) satisfy \[ a = b + 2c, \qquad a + c = b + d, \qquad b = d + c . \] Which of the following equalities is certainly true? A \(d = 2c\) B \(a = 3c\) …54Peter keeps horses and cows on his farm. To begin with he had exactly as many horses as cows, and that common number was greater than \(0\). He then bought some more cows, so that the number of cows rose …55There are \(17\) girls and \(12\) boys on a playground. At least how many more children must arrive so that everybody present can then be divided into two groups of the same size, in such a way that each …56Every edge of a certain pyramid has length \(5\) cm, and the pyramid has five vertices in all, so its base is a square. All five vertices are sliced off, each by a single plane, and the cuts are made so …57Find all pairs of integers \(a\) and \(b\) satisfying \[ 4a - 2b + 22ab^2 - 11b^3 = 2024 . \]58Grandmother cut a round pizza into \(6\) equilateral triangles and \(6\) circular segments, as in the picture. Each of her \(6\) grandchildren ate one triangle. The grandchildren do not like the crust, …59For which values of the parameter \(a\) does the system of equations \[ |x - 1| + |y - a| = 1 , \qquad y = -2|x - 1| - 1 \] have exactly three solutions \((x, y)\)?60The edge of a wooden cube is a natural number \(a > 2\). The cube is painted all over and then cut into unit cubes. It turns out that the number of unit cubes with exactly two painted faces divides the …61Let \(ABCD\) be a quadrilateral and let \(K\) be a point inside triangle \(ABD\) such that triangles \(ABD\) and \(KCD\) are similar, the vertices corresponding in the written order. Prove that triangles …62In triangle \(ABC\) the sides satisfy \(|AB| = 2|AC|\). A point \(D\) is chosen on side \(AB\) and a point \(E\) on side \(BC\) so that \(\angle BAE = \angle ACD\). The segments \(AE\) and \(CD\) intersect …63Find all pairs of real numbers \(x\), \(y\) that satisfy the system \[ \begin{aligned} \frac{3}{x-4y} + \frac{2}{x+y-5} &= 0, \\ \frac{2}{x^{2}-4y^{2}} + \frac{1}{x^{2}+y^{2}-5} &= 0. \end{aligned} \]64During the first lesson of the day, the number of boys and the number of girls in a class were in the ratio \(3 : 4\). Before the second lesson another \(4\) girls joined the class and \(4\) boys left …65Alenka and Barbara order a pizza. Two straight cuts, perpendicular to each other and neither of them passing through the centre of the pizza, divide it into four pieces. Alenka takes one piece first, then …66In the isosceles right triangle \(ABC\) the right angle is at \(C\) and each leg has length \(2\). A circular arc \(\ell\) centred at \(A\) divides the triangle into two parts of equal area, and a circular …67Find all rational numbers \(r\) and all integers \(k\) that satisfy \[ r\bigl(5k - 7r\bigr) = 3 . \]68Let \(ABC\) be an isosceles triangle with apex \(C\). Points \(D\) and \(E\) lie on the sides \(AC\) and \(BC\) respectively, and the bisector of the angle \(\angle DEB\) and the bisector of the angle …69Find every integer \(n\) that can be written in the form \[ n = \frac{m+2021}{2021-m}, \] where \(m\) is an integer.70The natural numbers \(m\) and \(n\) satisfy \(19 \leq m \leq 49\) and \(51 \leq n \leq 101\). What is the largest value that \[ \frac{n + m}{n - m} \] can take? A \(20\) B \(30\) C \(40\) D \(50\) E \(60\) …71The figure shows the route a hare ran while a wolf was chasing it through the fog. The hare first ran east; then it turned right, after a while it turned left, and shortly afterwards it turned left once …72The 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 \(G\) …73The 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.74Farmer 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 …75Find 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.76At 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 \(n - 1\) …
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