KöMaL Problems in Mathematics, December 2025
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Problems with sign 'K'Deadline expired on January 12, 2026. |
K. 879. A bunny is carrying carrots in its basket. If it meets another bunny, it gives it one carrot. If it meets a fox, the fox takes half of its carrots.
It meets 4 animals in total (each one is either a bunny or a fox) during which, at some point, it is left with no carrots. How many carrots could it have started with?
(5 pont)
solution (in Hungarian), statistics
K. 880. Eight three-legged dragons formed a choir. As we all know, a three-legged dragon can have one, three, seven, or nine heads. The three-headed ones are sopranos, the seven-headed ones are mezzos, the nine-headed ones are all altos, and the one-headed ones do not sing, they can only be choir leaders. The dragons have altogether twice as many heads as legs. How many heads could have sung each voice of the famous three-voice choral piece: “I wish I were a rosebud”.
(5 pont)
K. 881. Andris, Bori, Csaba, Dóri, Egon, Fanni and Gábor took part in a summer camp together. Andris knows everybody except Fanni, Bori knows three of the kids, Csaba knows one kid, and Dóri knows half as much of the kids as Egon. Fanni knows one less kid than Gábor. (Acquaintances are considered mutual.) Find the possible number of acquaintances of Fanni.
(5 pont)
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Problems with sign 'K/C'Deadline expired on January 12, 2026. |
K/C. 882. Inside regular dodecagon \(\displaystyle ABCDEFGHIJKL\) we draw squares on diagonals \(\displaystyle AC\) and \(\displaystyle FH\). Prove that the two squares share a common vertex.
(5 pont)
K/C. 883. A sequence is obtained with the following recursive method: if member \(\displaystyle t\) is a positive odd number, then the next member is \(\displaystyle 3t-9\), if it's a positive even number, then the next member is \(\displaystyle 2t-7\). Suppose that the sequence alternates between two positive values: \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle \ldots\) Find the possible values of \(\displaystyle a\) and \(\displaystyle b\).
(5 pont)
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Problems with sign 'C'Deadline expired on January 12, 2026. |
C. 1878. How many four-digit positive integers in base ten are there in which at most one of the digits is divisible by 3, the sum of the digits is \(\displaystyle 15\), and the sum of the mod \(\displaystyle 3\) remainders of the digits is \(\displaystyle 6\)?
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1879. A convex \(\displaystyle n\)-gon is given on the table, the sides of which have length 2. We place a disk with unit radius on each vertex of the polygon such that the adjacent disks are tangent to each other. We roll another circular token with unit radius around this arrangement such that it returns to its initial position. Determine, as a function of \(\displaystyle n\), the total angle through which this disk rotates during the rolling process.
German competition problem
(5 pont)
solution (in Hungarian), statistics
C. 1880. Solve equation \(\displaystyle (x+1)!=x^3-x\) on the set of natural numbers.
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1881. Given an \(\displaystyle n\times n\) chessboard, we want to cover it without overlaps using L-shaped pieces consisting of 3 squares, in such a way that at most one square remains uncovered. Prove that this is possible for any positive integer \(\displaystyle n\neq 3\). (All the sides of the L-shape should follow the grid lines.)
Proposed by Mátyás Czett, Zalaegerszeg
(5 pont)
C. 1882. Let \(\displaystyle ABCD\) be a convex quadrilateral, and let \(\displaystyle I\) be the incenter of triangle \(\displaystyle ABD\). Prove that if \(\displaystyle CB=CD=CI\), then \(\displaystyle ABCD\) is a cyclic quadrilateral.
Proposed by Viktor Vígh, Sándorfalva
(5 pont)
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Problems with sign 'B'Deadline expired on January 12, 2026. |
B. 5494. Anna and Béla plays the following game on a \(\displaystyle 2\times 2026\) board. They take turns to place dominoes on the board until no new domino can be placed on the board. If the board is fully covered, the winner is Anna, otherwise Béla is the winner. Who has a winning strategy, if Anna is the first player? (A domino covers two adjacent squares on the board. The dominoes cannot extend beyond the edges of the board and cannot overlap each other.)
Proposed by Márton Lovas, Budakalász
(3 pont)
solution (in Hungarian), statistics
B. 5495. The legs of a right triangle are \(\displaystyle a\) and \(\displaystyle b\), its hypotenuse is \(\displaystyle c\), and its inradius is \(\displaystyle r\). Prove that
\(\displaystyle 2r^2=(c-a)(c-b). \)
Proposed by Géza Kiss, Csömör
(3 pont)
solution (in Hungarian), statistics
B. 5496. Let \(\displaystyle p(r)\) denote the product of the divisors of positive integer \(\displaystyle r\). For a given positive integer \(\displaystyle n\) determine those positive integers \(\displaystyle k\) for which \(\displaystyle p(k^n)\) is the \(\displaystyle n^{\text{th}}\) power of an integer.
Proposed by Áron Hotváth, Nemesbőd
(4 pont)
solution (in Hungarian), statistics
B. 5497. Circular arc \(\displaystyle k\) with endpoints \(\displaystyle AB\) is given. Let \(\displaystyle F\) be the midpoint of the arc. A circle is tangent internally to \(\displaystyle k\) at point \(\displaystyle P\), and tangent internally to line segment \(\displaystyle AB\) at point \(\displaystyle Q\). Prove that the sum of the radii of circle \(\displaystyle APQ\) and \(\displaystyle BPQ\) equals the length of line segment \(\displaystyle AF\).
Proposed by László Németh, Fonyód
(4 pont)
solution (in Hungarian), statistics
B. 5498. Let \(\displaystyle n\geq 2\) be a given positive integer. For which permutation \(\displaystyle a_1,a_2,\dots,a_n\) of numbers \(\displaystyle 1,2,\dots,n\) will the value of
\(\displaystyle a_1a_2+a_2a_3+\dots+a_{n-1}a_n\)
be maximal?
Proposed by Márton Lovas, Budakalász
(5 pont)
solution (in Hungarian), statistics
B. 5499. Each of the vertices of a regular 45-gon are colored red, yellow or green, and each color is used exactly 15 times. A triangle is red (yellow, green), if all three of its vertices is red (yellow, green). Prove that there exists a red, a yellow and a green triangle, all three of which are congruent to each other.
Proposed by Sándor Róka, Nyíregyháza
(6 pont)
solution (in Hungarian), statistics
B. 5500. Find the smallest possible value of \(\displaystyle |a|+|b|+|c|+|d|+|e|\), if equation \(\displaystyle x^{6}+ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+1=0\) has a real root.
Proposed by Ákos Somogyi, London
(5 pont)
solution (in Hungarian), statistics
B. 5501. The incircle of triangle \(\displaystyle ABC\) is tangent to sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) at points \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\), respectively. Let \(\displaystyle G\) be the antipode of point \(\displaystyle F\) on the incircle. Lines \(\displaystyle GD\) and \(\displaystyle GE\) intersect line \(\displaystyle AB\) at points \(\displaystyle D'\) and \(\displaystyle E'\), respectively. Prove that circles \(\displaystyle AED'\) and \(\displaystyle BDE'\) intersect each other on the incircle.
Proposed by Roland Jármai, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on January 12, 2026. |
A. 920. Given a non-isosceles triangle \(\displaystyle ABC\), let its circumcircle be denoted by \(\displaystyle \Omega\). Let \(\displaystyle \omega_A\) be the mixtilinear incircle corresponding to vertex \(\displaystyle A\) (that is, the circle tangent to sides \(\displaystyle AB\), \(\displaystyle AC\), and internally tangent to \(\displaystyle \Omega\)). Define \(\displaystyle \omega_B\) and \(\displaystyle \omega_C\) analogously. Let the points where \(\displaystyle \omega_A,\omega_B,\omega_C\) touch \(\displaystyle \Omega\) be denoted by \(\displaystyle T_A, T_B, T_C\), respectively.
Show that the Monge line of the circles \(\displaystyle \omega_A,\omega_B,\omega_C\) (that is, the line passing through the pairwise external homothety centers of the three circles) coincides with the Pascal line of the hexagon \(\displaystyle AT_CBT_ACT_B\) (that is, the line passing through the points \(\displaystyle AT_C \cap T_AC\), \(\displaystyle T_CB \cap CT_B\), \(\displaystyle BT_A \cap T_BA\)).
Proposed by Sha Jingyuan, Budapest
(7 pont)
A. 921. Let \(\displaystyle n \ge 2\) be an integer. Let \(\displaystyle x_1,\ldots,x_n\) be positive real numbers whose product is \(\displaystyle n-1\). Prove that
\(\displaystyle x_1+\ldots+x_n-\left(\frac{1}{x_1^{n-1}}+\ldots+\frac{1}{x_n^{n-1}}\right)<(n-1)^{1+\frac{1}{n}}.\)
Proposed by Áron Bán-Szabó, Palaiseau
(7 pont)
A. 922. Fix two positive integers \(\displaystyle a\) and \(\displaystyle b\). Let \(\displaystyle n\) be a positive integer, and let \(\displaystyle T\) be a black–white coloring of the cells of an \(\displaystyle an\times bn\) grid, containing at least two white cells. Csigusz the snail starts on a white cell, visits all white cells exactly once, and then returns to the starting cell, always moving between side-adjacent white cells. He then notices that he was able to do this in exactly one way (that is, once he made his first move, there was a unique way to complete the cycle). Let \(\displaystyle \mathcal{T}_n\) denote the set of all colorings \(\displaystyle T\) satisfying this property, and let \(\displaystyle \phi(T)\) denote the number of white lattice points in \(\displaystyle T\). Show that
\(\displaystyle L=\lim_{n\to \infty}\frac{\max_{T\in \mathcal{T}_n} \phi(T)}{abn^2}\)
exists for every choice of positive integers \(\displaystyle a\) and \(\displaystyle b\), and determine the value of \(\displaystyle L\).
Proposed by Márton Lovas and Csongor Beke, Cambridge
(7 pont)
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