KöMaL Problems in Mathematics, October 2024
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Problems with sign 'K'Deadline expired on November 11, 2024. |
K. 824. Starting from the string of letters ABABABABABAB, in each step we can swap two adjacent letters. Find the minimum number of steps needed to obtain the string of letters AAAAAABBBBBB.
(5 pont)
solution (in Hungarian), statistics
K. 825. 4202 points are given in the plane. Is it always possible to find a circle such that it contains none of the given points, and it contains exactly 2024 of them in its interior?
(5 pont)
solution (in Hungarian), statistics
K. 826. Prove that if the product of 7 consecutive positive integers is divisible by 1000, then it's possible to choose 3 of them such that their product is also divisible by 1000.
(5 pont)
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Problems with sign 'K/C'Deadline expired on November 11, 2024. |
K/C. 827. Each angle of a hexagon is \(\displaystyle 120^{\circ}\). Prove that the sum of the lengths of any two adjacent sides equals the sum of the lengths of the two adjacent sides opposite them.
(5 pont)
solution (in Hungarian), statistics
K/C. 828. Is it possible to find two positive integers such that the sum of their squares equals their smallest common multiple?
(5 pont)
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Problems with sign 'C'Deadline expired on November 11, 2024. |
C. 1823. Smart Pali once went to the market to sell \(\displaystyle 30\) apples. He planned to ask for one kreuzer for every three apples, i.e. he expected a total income of \(\displaystyle 10\) kreuzers. At the market, he met another person who was also selling apples. The other seller also had 30 apples for sale, but he gave two apples for one kreuzer, so he was expecting a total income of \(\displaystyle 15\) kreuzers. Smart Pali got tired of the hustle and bustle of the market and handed over his \(\displaystyle 30\) apples to the other person, instructing him to sell them so that five apples would cost two kreuzers. Pali said he would come back later for his share of the income.
\(\displaystyle a)\) If this person sold all 60 apples following Pali's instructions and kept the income he originally expected, how many kreuzers were left for Smart Pali?
\(\displaystyle b)\) For how many kreuzers should the 60 apples have been sold in order for both sellers to receive their originally planned income?
Based on a short story by Kálmán Mikszáth
(5 pont)
solution (in Hungarian), statistics
C. 1824. Next year, Boglárka wants to read ten different books; the thicker the book, the longer she plans to spend on it. She has decided to dedicate 5, 10, 15, ..., 45, and 50 days to reading each of the books. The remaining period of time will be divided into three equal parts, which she will use for active rest. She can use these rest periods for sports at any time during the year (even several of them consecutively). If she has to read two books without inserting a rest period in between, she will always start with the thicker one. In how many different ways can Boglárka allocate the \(\displaystyle 365\) days for reading the books?
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1825. Line \(\displaystyle e\) intersects circle \(\displaystyle k_1\) in two distinct points \(\displaystyle A\) and \(\displaystyle B\). Circle \(\displaystyle k_2\) is tangent to circle \(\displaystyle k_1\) at point \(\displaystyle C\) and to line \(\displaystyle e\) at point \(\displaystyle D\). Line \(\displaystyle CD\) and circle \(\displaystyle k_1\) intersect each other at point \(\displaystyle T\) for the second time. Prove that \(\displaystyle AT=TB\).
Swiss Olympiad Problem
(5 pont)
solution (in Hungarian), statistics
C. 1826. Prove that if \(\displaystyle 0<x\leq 1\), then \(\displaystyle \sqrt{1-x}+\sqrt{4-x} < 1+\sqrt{4-3x}\).
Proposed by Mihály Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1827. Prove that if two right triangles both have a perimeter of \(\displaystyle 1\) unit, then the difference between the lengths of their hypotenuses is at most \(\displaystyle 1.5-\sqrt{2}\).
Proposed by Norbert Csizmazia, Pécs
(5 pont)
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Problems with sign 'B'Deadline expired on November 11, 2024. |
B. 5406. Prove that in any base-\(\displaystyle n\) number system, equality \(\displaystyle \sqrt{\frac{123456787654321}{1234321}}=10001\) holds, where \(\displaystyle n\ge 9\).
Proposed by Mihály Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5407. Find positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) satisfying \(\displaystyle \frac{b}{a}=\frac{c}{b}=\frac{d}{c}\) and \(\displaystyle \frac{a+c}{2}=b+1\).
Proposed by Attila Sztranyák, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5408. In a triangle the length of one of the sides is the arithmetic mean of the lengths of the other two sides. Prove that the length of the angle bisector corresponding to the middle side is \(\displaystyle \frac{\sqrt{3}}{2}\) times the geometric mean of the lengths of the other two sides.
Proposed by Mihály Hujter, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5409. Each card in a deck of german suited playing cards has a suit and a value. The suit can be Hearts, Bells, Leaves or Acorns, and the value can be VII, VIII, IX, X, Unter, Ober, King or Ace. The deck contains all possible combinations of the suits and the values. We arrange the 32 cards of a deck in 4 rows and 8 coloumns. Let \(\displaystyle A\) denote the random event that no coloumn contains two cards with the same suit, and let \(\displaystyle B\) denote the random event that no row contains two cards with the same value. Which of the two events has the larger probability?
Proposed by Zoltán Bertalan, Békéscsaba
(5 pont)
solution (in Hungarian), statistics
B. 5410. We draw three circles centered at each of the three vertices of a right triangle such that each two of these circles are externally tangent to each other. Determine the center and the radius of the circle that is internally tangent to all three of these circles.
Proposed by László Németh, Fonyód
(5 pont)
solution (in Hungarian), statistics
B. 5411. Prove the following statement for every \(\displaystyle n\ge 2\) positive integer: \(\displaystyle \sum_{k=1}^{n^2-1} \lfloor \sqrt{k} \rfloor]^2=\frac{(n-1)n(3n^2-n-1)}{6}\). \(\displaystyle \lfloor x \rfloor\) denotes the floor of the real number \(\displaystyle x\).
Proposed by Mihály Bencze, Brassó
(4 pont)
solution (in Hungarian), statistics
B. 5412. A fugitive at a vertex of a finite connected graph is being chased by three policemen at three other vertices. First the fugitive makes a move along an edge, then each of the three policemen can also make a move along an edge. The game continues by alternating turns between the fugitive and the policemen. Can the policemen always catch the fugitive in a finite number of steps, i.e., can they always guarantee that one of them will be on the same vertex as the fugitive after a finite number of steps?
Proposed by Péter Pál Pach, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5413. Let \(\displaystyle M\) be the orthocenter, \(\displaystyle S\) be the centroid and \(\displaystyle I\) be the incenter of the non-equilateral triangle \(\displaystyle ABC\). Prove that \(\displaystyle \angle MIS>90^\circ\).
Proposed by Viktor Vígh, Sándorfalva
(6 pont)
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Problems with sign 'A'Deadline expired on November 11, 2024. |
A. 887. A non self-intersecting polygon is given in a Cartesian coordinate system such that its perimeter contains no lattice points, and its vertices have no integer coordinates. A point is called semi-integer if exactly one of its coordinates is an integer. Let \(\displaystyle P_1\), \(\displaystyle P_2\), \(\displaystyle \ldots\), \(\displaystyle P_k\) denote the semi-integer points on the perimeter of the polygon. Let \(\displaystyle n_i\) denote the floor of the non-integer coordinate of \(\displaystyle P_i\). Prove that integers \(\displaystyle n_1, n_2, \ldots , n_k\) can be divided into two groups with the same sum.
Proposed by Áron Bán-Szabó, Budapest
(7 pont)
A. 888. Let \(\displaystyle n\) be a given positive integer. Find the smallest positive integer \(\displaystyle k\) for which the following statement is true: for any given simple connected graph \(\displaystyle G\) and minimal cuts \(\displaystyle V_1\), \(\displaystyle V_2\), \(\displaystyle \ldots\), \(\displaystyle V_n\), at most \(\displaystyle k\) vertices can be chosen with the property that picking any two of the chosen vertices there exists an integer \(\displaystyle 1\leq i\leq n\) such that \(\displaystyle V_i\) separates the two vertices. A partition of the vertices of \(\displaystyle G\) into two disjoint non-empty sets is called a minimal cut if the number of edges crossing the partition is minimal.
Proposed by András Imolay, Budapest
(7 pont)
A. 889. Let \(\displaystyle W, A, B\) be fixed real numbers with \(\displaystyle W>0\). Prove that the following statements are equivalent.
- For all \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\geq0\) satisfying \(\displaystyle x+y\leq z+W\), \(\displaystyle x+z\leq y+W\), \(\displaystyle y+z\leq x+W\) we have \(\displaystyle Axyz+B\geq x^{2}+y^{2}+z^{2}\).
- \(\displaystyle B\geq W^{2}\) and \(\displaystyle AW^{3}+B\geq3W^{2}\).
Proposed by Ákos Somogyi, London
(7 pont)
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