KöMaL Problems in Mathematics, January 2026
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Problems with sign 'K'Deadline expired on February 10, 2026. |
K. 884. A number is called a zigzag number if, when reading its digits from left to right, the second digit is smaller than the first, the third digit is larger than the second, the fourth digit is smaller than the third, and so on. Create all the zigzag numbers from digits \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle 3\), \(\displaystyle 4\), \(\displaystyle 5\), i.e., all five-digit numbers \(\displaystyle \overline{abcde}\) with all different digits satisfying \(\displaystyle a>b\), \(\displaystyle b<c\), \(\displaystyle c>d\) and \(\displaystyle d<e\). How many such five-digit zigzag numbers are there?
(5 pont)
K. 885. Every morning, Peti drinks either cocoa or fruit juice (but only one of them). Drinking cocoa is a two-day activity in the sense that the cocoa-drinking periods consist of even number of days. How many ways are there to choose his morning drinks in the first ten days of February?
(5 pont)
K. 886. A tire of a car wears out if it runs 20,000 km when placed on the front wheel, or 30,000 km when placed on the rear wheel.
a) We have 4 fresh tires. What is the maximum number of kilometres we can travel if we can switch the tires between the front and the rear wheels? After how many kilometres should we switch tires between the front and the rear wheels to achieve the maximum distance?
b) We have 5 fresh tires. What is the maximum number of kilometres we can travel if we can switch the tires between the front and the rear wheels? How should we swap tires between the wheels to achieve the maximum distance?
(5 pont)
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Problems with sign 'K/C'Deadline expired on February 10, 2026. |
K/C. 887. Which three-digit number satisfies \(\displaystyle \overline{xyz}=x!+y!+z!\)? (\(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\) denote the digits of the three-digit numbers, and \(\displaystyle n!\) denotes the product of numbers between \(\displaystyle 1\) and \(\displaystyle n\), while \(\displaystyle 0!=1!=1\).)
(5 pont)
K/C. 888. The sum of the squares of three consecutive odd integers is a four-digit number with four equal digits. Find all such triples of positive integers.
(5 pont)
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Problems with sign 'C'Deadline expired on February 10, 2026. |
C. 1883. Find all natural numbers \(\displaystyle n\) satisfying \(\displaystyle n^3+25n\ge10n^2+16\).
Proposed by Mátyás Czett, Zalaegerszeg
(5 pont)
C. 1884. 30 students in a class did a test in mathematics. The teacher marked the tests, and sent a table of the marks to the students such that the marks appeared in a column: 15 of them were four, and 15 of them were five. Prove that it is always possible to find 14 consecutive rows such that the sum of the marks contained in them equals 63.
Problem of the competition ``Felvidéki Magyar Matematikaverseny''
(5 pont)
C. 1885. We inscribe reagular hexagon \(\displaystyle ABCDEF\) in equilateral triangle \(\displaystyle PQR\) such that points \(\displaystyle B\), \(\displaystyle D\) and \(\displaystyle F\) are midpoints of sides \(\displaystyle PQ\), \(\displaystyle QR\) and \(\displaystyle RP\). Find the area of triangle \(\displaystyle QPR\) if the area of pentagon \(\displaystyle ABQRF\) equals one unit.
Dutch competition problem
(5 pont)
C. 1886. The internal angles of hexagon \(\displaystyle ABCDEF\) are equal. Prove that the area of triangles \(\displaystyle ACE\) and \(\displaystyle BDF\) are equal.
Proposed by Márton Ujházy, Budapest
(5 pont)
C. 1887. How many numbers are there with a base nine representation of \(\displaystyle \overline{abcabc}_9\) and exactly 40 positive divisors?
Proposed by Márton Ujházy, Budapest
(5 pont)
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Problems with sign 'B'Deadline expired on February 10, 2026. |
B. 5502. Let \(\displaystyle A\) be a set of real numbers with \(\displaystyle n\) elements. Prove that at least \(\displaystyle 4n-3\) numbers can be written in the form \(\displaystyle a-2b+c\) such that \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\in A\) not necessarily distinct.
Proposed by Péter Pál Pach, Budapest
(3 pont)
B. 5503. Let \(\displaystyle P\) and \(\displaystyle Q\) be the third vertices of the equilateral triangles on side \(\displaystyle BC\) of triangle \(\displaystyle ABC\). Prove that \(\displaystyle AP^2+AQ^2=AB^2+BC^2+CA^2\).
Proposed by Géza Kiss, Csömör
(3 pont)
B. 5504. Let \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) be real numbers such that not all of them are equal. Prove that \(\displaystyle \frac{a+b+c}{3}>\sqrt[3]{abc}\) holds if and only if \(\displaystyle \sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}>0\).
Proposed by Mátyás Barczy, Szeged and Páles Zsolt, Debrecen
(4 pont)
B. 5505. Let the diagonal \(\displaystyle AC\) and \(\displaystyle BD\) of cyclic quadrilateral \(\displaystyle ABCD\) intersect each other in point \(\displaystyle P\). Let \(\displaystyle K\) be the circumcenter of triangle \(\displaystyle APB\), and let \(\displaystyle M\) be the orthocenter of triangle \(\displaystyle CPD\). Prove that points \(\displaystyle K\), \(\displaystyle M\) and \(\displaystyle P\) are collinear.
Proposed by Mihály Bence, Brassó
(4 pont)
B. 5506. Solve the following equation on the set of positive integers: \(\displaystyle x^5-xy^2+y^2=1\).
Proposed by István Molnár, Békéscsaba
(5 pont)
solution (in Hungarian), statistics
B. 5507. Let us choose two different integers, \(\displaystyle a\) and \(\displaystyle b\) from interval \(\displaystyle (n^2,n^2+n)\), for a given \(\displaystyle n>2\) positive integer. Prove that it is not possible to find an integer number different from \(\displaystyle a\) and \(\displaystyle b\) that divides product \(\displaystyle ab\).
Proposed by Sándor Róka, Nyíregyháza
(5 pont)
B. 5508. Let \(\displaystyle ABCD\) be a convex quadrilateral.
a) Prove that if \(\displaystyle \tan\angle BAC\cdot\tan\angle DCA =\tan\angle CAD\cdot\tan\angle ACB\), then \(\displaystyle \tan\angle CBD\cdot\tan\angle ADB=\tan\angle DBA\cdot\tan\angle BDC\).
b) Prove that if \(\displaystyle \tan\frac{\angle BAC}{2}\cdot\tan\frac{DCA}{2}={\tan\frac{\angle CAD}{2}\cdot\tan\frac{\angle ACB}{2}}\), then \(\displaystyle {\tan\frac{\angle CBD}{2}\cdot\tan\frac{\angle ADB}{2}}={\tan\frac{\angle DBA}{2}\cdot\tan\frac{\angle BDC}{2}}\).
Proposed by Bálint Hujter, Budapest and Géza Kós, Budapest
(6 pont)
B. 5509. Let \(\displaystyle V=\{v_1,v_2,\ldots,v_n\}\) be the set of vertices of a graph on \(\displaystyle n\) vertices, and let \(\displaystyle d_i\) denote the degree of vertex \(\displaystyle v_i\). We call function \(\displaystyle f\) Verőcese, if it takes non-negative real values and for every edge \(\displaystyle v_iv_j\) of the graph \(\displaystyle f(v_i)+f(v_j)\geq d_i+d_j\) holds. Find the largest real number \(\displaystyle \lambda\) such that for every \(\displaystyle n\), every simple graph on \(\displaystyle n\) vertices and every Verőcese function \(\displaystyle f\) \(\displaystyle f(v_1)+f(v_2)+\ldots+f(v_n)\geq\lambda(d_1+d_2+\ldots+d_n)\) holds true.
Proposed by Boldizsár Varga, Verőce
(6 pont)
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Problems with sign 'A'Deadline expired on February 10, 2026. |
A. 923. \(\displaystyle 2026\) people visited an exhibition where \(\displaystyle 1000\) paintings were displayed. Prove that it is possible to send some of the visitors to two rooms, with at least one visitor in each room, such that there is no painting that was liked by someone in one room but by nobody in the other, and there is no painting whose painter is personally known by someone in one room but by nobody in the other.
(7 pont)
A. 924. For which pairs \(\displaystyle (k,l)\) of positive integers is the following statement true: for every positive integer \(\displaystyle n\) there exists an integer \(\displaystyle i\) with \(\displaystyle 0\le i\le l-1\) such that \(\displaystyle \binom{n}{i}+{(-1)}^l\binom{n}{i+l}+{(-1)}^{2l}\binom{n}{i+2l}+\dots\) is not divisible by \(\displaystyle k\)?
(Based on a Kvant problem)
(7 pont)
A. 925. Call four points to be in general position if they are pairwise distinct, no three of them are collinear, and no two of the six lines determined by them are parallel. Let \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\) be points in general position lying on a circle. Let \(\displaystyle E\) be the intersection point of lines \(\displaystyle AB\) and \(\displaystyle CD\), let \(\displaystyle F\) be the intersection point of lines \(\displaystyle AC\) and \(\displaystyle BD\), and let \(\displaystyle G\) be the intersection point of lines \(\displaystyle AD\) and \(\displaystyle BC\). In triangle \(\displaystyle EFG\), denote by \(\displaystyle P\), \(\displaystyle Q\), and \(\displaystyle R\) the feet of the altitudes corresponding to the vertices \(\displaystyle E\), \(\displaystyle F\), and \(\displaystyle G\), respectively.
a) Show that the triples of lines \(\displaystyle AR\)–\(\displaystyle BQ\)–\(\displaystyle CP\), \(\displaystyle AQ\)–\(\displaystyle BR\)–\(\displaystyle DP\), \(\displaystyle AP\)–\(\displaystyle CR\)–\(\displaystyle DQ\), and \(\displaystyle BP\)–\(\displaystyle CQ\)–\(\displaystyle DR\) are either concurrent or parallel.
b) Suppose that each of the four triples of lines is concurrent, and denote their points of concurrency by \(\displaystyle X\), \(\displaystyle Y\), \(\displaystyle Z\), and \(\displaystyle W\), respectively. Assume further that these points are in general position. Prove that the Miquel points of quadrilaterals \(\displaystyle XWYZ\), \(\displaystyle XYZW\), and \(\displaystyle XZWY\) are precisely the points \(\displaystyle P\), \(\displaystyle Q\), and \(\displaystyle R\).
Proposed by Boldizsár Varga, Verőce and Áron Bán-Szabó, Palaiseau
(7 pont)
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