KöMaL Problems in Mathematics, November 2025

Problems with sign 'K' The deadline is: December 10, 2025 24:00 (UTC+01:00).

K. 874. András, Bori, Cili, Dezső, Elemér, Feri, Gabi and Hugó are standing in a circle in this order, each holding some beans, with a total of 240 beans altogether. If András gives 1 bean each to Bori, Cili, Dezső, Elemér, Feri, Gabi, and Hugó, then Bori gives 2 beans each to Cili, Dezső, Elemér, Feri, Gabi and Hugó, then Cili gives 3 beans each to Dezső, Elemér, Feri, Gabi and Hugó, then Dezső gives 4 beans each to Elemér, Feri, Gabi and Hugó, then Elemér gives 5 beans each to Feri, Gabi, and Hugó, then Feri gives 6 beans each to Gabi and Hugó, then Gabi gives Hugó 7 beans, each of them will have the same number of beans in their hands. How many beans did each of them have initially?

(5 pont)

This problem is for grade 9 students only.

K. 875. Divide an equilateral triangle into four congruent 2D-shapes. Divide an equilateral triangle into six congruent 2D-shapes. Create a total of at least four different divisions.

(5 pont)

This problem is for grade 9 students only.

K. 876. Think of a positive integer. If the number is even, divide it by 2, if odd, add 1 to it. Continue in the same manner with the result: if it's even, divide it by 2, if it's odd, add 1 to it. Is it true that if the initial number is less than 2025, we will get 1 in less than 25 steps?

(5 pont)

This problem is for grade 9 students only.

Problems with sign 'K/C' The deadline is: December 10, 2025 24:00 (UTC+01:00).

K/C. 877. Fill in the diagram with numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 in a way that in the three grey parts containing four small triangles the product of the numbers is a perfect square. How many ways are there to achieve this?

(5 pont)

This problem is for grades 1–10 students only.

K/C. 878. How many four-digit positive integers divisible by 91 are there that contain exactly two different kinds of digits, with each digit appearing exactly twice?

(5 pont)

This problem is for grades 1–10 students only.

Problems with sign 'C' The deadline is: December 10, 2025 24:00 (UTC+01:00).

C. 1873. Solve the following system of equations on the set of triples of real numbers: \(\displaystyle x^2+13x+144=5y+32z\), \(\displaystyle y^2+13y+144=5z+32x\), \(\displaystyle z^2+13z+144=5x+32y\).

Proposed by Mátyás Czett, Zalaegerszeg

(5 pont)

C. 1874. Two points, \(\displaystyle G\) and \(\displaystyle H\) are given in the plane. Construct acute isosceles triangles using a straightedge and compass such that \(\displaystyle H\) is orthocenter and \(\displaystyle G\) is the centroid of the triangle. (The elementary steps of construction such as bisecting an angle, reflecting across a line etc. do not need to be detailed.)

(5 pont)

C. 1875. Thanks to a generous donation, during the lunch break of the KöMaL Ankét on Saturday, \(\displaystyle m\) types of books were distributed among the \(\displaystyle n\) students present. There were multiple copies available of each book type, and everyone could choose freely: they could take books of several types, but at most one copy of each type. We know that for any two books there were a difference of at least three students between the set of students choosing each book. Prove that \(\displaystyle m \leq \frac{2^{n}}{n+1}\).

Proposed by Zoltán Paulovics, Budapest

(5 pont)

C. 1876. Winnie the Pooh tries to decide whether the following statement is true: 'If 30 days of a given year is chosen, there will not necessarily be at least five of them falling on the same day of the week.' He proposes the following argument: 'The statement is false. The day's of the week are the pigeon holes, and let's fill them with pigeons corresponding to the chosen days. We can distribute 28 (identical) pigeons such that every pigeon hole contains exactly 4 pigeons, however, the \(\displaystyle 29^{\text{th}}\) pigeon will surely be placed in a pigeon hole containing four pigeons, therefore there must be a day of the week containing at least five of the chosen days.' Pooh's friend, Christopher Robin points out the missing part of the argument: 'Pooh, you've only considered seven cases: when each pigeon hole contains four pigeons, except for one containing five'. How many cases were omitted by Pooh?

Based on the idea of László Németh, Fonyód

(5 pont)

This problem is for grades 11–12 students only.

C. 1877. Let \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\) be the midpoints of the arcs \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) not containing points \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\), respectively, on the circumcircle of acute triangle \(\displaystyle ABC\). Line segment \(\displaystyle DE\) intersects sides \(\displaystyle BC\) and \(\displaystyle CA\) in points \(\displaystyle P\) and \(\displaystyle Q\), respectively. Line segment \(\displaystyle EF\) intersects sides \(\displaystyle CA\) and \(\displaystyle AB\) in points \(\displaystyle R\) and \(\displaystyle S\), respectively. Line segment \(\displaystyle FD\) intersects sides \(\displaystyle AB\) and \(\displaystyle BC\) in points \(\displaystyle T\) and \(\displaystyle U\), respectively. Prove that triangles \(\displaystyle PRT\) and \(\displaystyle QSU\) have equal areas.

Proposed by Bálint Bíró, Eger

(5 pont)

This problem is for grades 11–12 students only.

Problems with sign 'B' The deadline is: December 10, 2025 24:00 (UTC+01:00).

B. 5486. Let \(\displaystyle E\) be the intersection of the diagonals of convex quadrilateral \(\displaystyle ABCD\). Suppose that \(\displaystyle AB=CD\), and the areas of triangles \(\displaystyle ABE\) and \(\displaystyle CDE\) are equal. Prove that the quadrilateral is a parallelogram or an isosceles trapezoid.

Proposed by Mihály Hujter, Budapest

(3 pont)

B. 5487. Positive numbers \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \dots\), \(\displaystyle a_{2025}\) satisfy \(\displaystyle a_1=1\) and \(\displaystyle \frac{1}{a_1+a_2}+\frac{1}{a_2+a_3}+\ldots+\frac{1}{a_{n-1}+a_n}=a_n-1\) for every \(\displaystyle 2\le n\le 2025\). Find the value of \(\displaystyle a_{2025}\).

Based on the idea of Mihály Bencze, Brasov

(3 pont)

B. 5488. Is it possible to color the points of the plane with three given colors such that each color is used at least once, and if the vertices of a triangle are colored with a color chosen from only two of the three given colors, then every point inside and on the perimeter of this triangle is also colored with one of these two colors?

Proposed by Márton Lovas, Budakalász

(4 pont)

B. 5489. In right triangle \(\displaystyle ABC\) the angles are the following: \(\displaystyle \angle ABC=15^\circ\) and \(\displaystyle \angle CAB=75^{\circ}\). Let \(\displaystyle F\) be the midpoint of hypotenuse \(\displaystyle AB\). Let point \(\displaystyle D\) be chosen on leg \(\displaystyle BC\) such that \(\displaystyle BD=CA\), and let point \(\displaystyle E\) be chosen on ray \(\displaystyle CA\) beyond point \(\displaystyle A\) such that \(\displaystyle CE=BC\). Let \(\displaystyle M\) be the point of intersection of lines \(\displaystyle BE\) and \(\displaystyle CF\). Prove that lines \(\displaystyle DM\) and \(\displaystyle CM\) are tangent to the circumcircle of triangle \(\displaystyle AEF\).

Proposed by Bálint Bíró, Eger

(4 pont)

B. 5490. Prove that there exist infinitely many positive integers \(\displaystyle n\) such that numbers \(\displaystyle {2^n-2025}\), \(\displaystyle {2^n-2024}\), \(\displaystyle \dots\), \(\displaystyle {2^n+2025}\) are all composite.

Proposed by Csaba Sándor, Budapest

(5 pont)

B. 5491. Does there exist a set \(\displaystyle H\) of polynomials with integer coefficients and degree at least two such that every integer value is taken by exactly one polynomial in \(\displaystyle H\) at an integer place?

Proposed by Péter Pál Pach, Budapest

(5 pont)

B. 5492. Kornél thinks about a closed subinterval of \(\displaystyle I=[0, 2^k]\) (where \(\displaystyle k\) is a positive integer) with integer endpoints and length at least \(\displaystyle 1\). Kristóf can ask the following question: he can choose an arbitrary closed subinterval with integer length, but not necessarily integer endpoints, and Kornél tells him the length of the intersection of the interval he picked and the interval chosen by Kristóf. (The answer is \(\displaystyle 0\) if the intersection of the two intervals is empty or consists of a single point.) Find the smallest number of questions with which Kristóf can guess the interval chosen by Kornél in all cases.

Proposed by Dávid Matolcsi, Berkeley/Brussels

(6 pont)

B. 5493. Show a function \(\displaystyle f\) that assigns a non-negative value to every vector in the plane satisfying the following: for an arbitrary triangle \(\displaystyle ABC\) with vertices on hyperbola \(\displaystyle x^2-y^2=1\) the area of the triangle equals \(\displaystyle f{(\overrightarrow{AB})\cdot f(\overrightarrow{BC})\cdot f(\overrightarrow{CA})}\).

Proposed by Géza Kós, Budapest

(6 pont)

Problems with sign 'A' The deadline is: December 10, 2025 24:00 (UTC+01:00).

A. 917. Let a set \(\displaystyle S\) of complex numbers be called symmetric if the complex conjugate of every element of \(\displaystyle S\) also belongs to \(\displaystyle S\). For each positive integer \(\displaystyle n\), determine the largest positive integer \(\displaystyle K_n\) for which there exists an \(\displaystyle n\)-element symmetric set \(\displaystyle S = \{z_1, z_2, \ldots, z_n\}\) not containing \(\displaystyle 0\), such that \(\displaystyle z_1^k + z_2^k + \ldots + z_n^k \leq 0\) holds true for every integer \(\displaystyle 1 \leq k \leq K_n\).

Proposed by Navid Safaei, Tehran and Géza Kós, Budapest

(7 pont)

A. 918. Given a tree with \(\displaystyle n \ge 2\) vertices labelled \(\displaystyle v_1\), \(\displaystyle v_2\), \(\displaystyle \dots\), \(\displaystyle v_n\). Each vertex hosts a dwarf, and every dwarf has some number of coins, at least \(\displaystyle n\). We then go through the vertices in the order of increasing indices: the dwarf on the current vertex takes one coin from its richest neighbour; if there are several richest neighbours, he takes one coin from each of them. Determine, as a function of \(\displaystyle n\), the smallest integer \(\displaystyle k\) such that for every tree with \(\displaystyle n\) vertices there exists an initial coin distribution in which the numbers of coins of any two dwarfs differ by at most \(\displaystyle k\), and after the process every dwarf ends up with exactly the same number of coins as at the beginning.

Proposed by Márton Németh, Budapest

(7 pont)

A. 919. Let \(\displaystyle \mathcal{P}\) be a convex cyclic polygon with at least four vertices such that no two diagonals of \(\displaystyle \mathcal{P}\) are congruent. Prove that there is at most one triangulation of \(\displaystyle \mathcal{P}\) where all triangles have the same perimeter.

Proposed by Andrei Chirita, Cambridge

(7 pont)