KöMaL Problems in Mathematics, May 2026
Please read the rules of the competition.
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Problems with sign 'K'Deadline expired on June 10, 2026. |
K. 904. There are three streets in a village: street \(\displaystyle T\), \(\displaystyle V\) and \(\displaystyle P\). People in \(\displaystyle T\) street always tell the truth, people in \(\displaystyle V\) street always lie, and people \(\displaystyle P\) street tell the truth and lie alternately. One day the watchman on duty in the fire tower of the village saw a column of smoke rising from one of the streets. A few seconds later the telephone rang. The caller said only this: `There is fire in the street!' The watchman on duty asked: `Which street?' The caller answered: `On \(\displaystyle P\) street.' Which street did the firefighters have to go to?
Estonian Competition Problem
(5 pont)
solution (in Hungarian), statistics
K. 905. Anna is really looking forward to the summer vacation, so she decided to write down in her notebook all the letter sequences that can be formed from the letters of the word VAKÁCIÓ. At what position does the word VAKÁCIÓ appear on this list, if Anna proceeds in alphabetical order?
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
K. 906. One of the diagonals of a rectangle is three times as long as one of the sides, and it is one unit longer than the other side. How long are the sides of the rectangle measured in the given unit?
Proposed by Mátyás Czett, Zalaegerszeg
(5 pont)
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Problems with sign 'K/C'Deadline expired on June 10, 2026. |
K/C. 907. During a house renovation, our next task is to tile a rectangular pantry. We have \(\displaystyle 1\times 1\) square tiles. The side lengths of the pantry are integers (measured in these units). How many tiles are needed if the number of edge tiles (those along the walls or door) is half of the total number of tiles?
Scottish Competition Problem
(5 pont)
solution (in Hungarian), statistics
K/C. 908. Anna has placed a unit disk on the table, and Boglárka has arranged three identical disks around Anna's disk. Every disk on the outside is tangent to Anna's disk and exactly two other disks. Find the radius of Boglárka's disks.
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
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Problems with sign 'C'Deadline expired on June 10, 2026. |
C. 1903. The 17 enthusiastic members of a math club would like to get together for a group math session during the summer break, so they collect everyone’s availability: from each person, they receive a time interval consisting of at least two days. (For example, from July 3 to July 6.)
After collecting the data, it turned out that among any three students there are two who have marked a common day.
Is it guaranteed that there will be a day such that every student will be available?
Is it guaranteed that there will be two days such that every student will be available on at least one of the two days?
Proposed by Zoltán Paulovics, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1904. \(\displaystyle AB\) is the hypotenuse of right triangle \(\displaystyle ABC\). Let \(\displaystyle D\) be the foot of the altitude from \(\displaystyle C\). Let \(\displaystyle E\) be the foot of the perpendicular from \(\displaystyle D\) to \(\displaystyle AC\), and let \(\displaystyle F\) be the foot of the perpendicular from \(\displaystyle E\) to \(\displaystyle AB\). Given that \(\displaystyle F\) is the midpoint of the hypotenuse, find the lengths of the legs of the triangle.
Proposed by Márton Ujházy, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1905. Let \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) be distinct positive integers. Among the four statement below two are true and two are fale:
(i) \(\displaystyle a<b<c<d\),
(ii) \(\displaystyle a+b=c+d\),
(iii) \(\displaystyle a^2+b^2=c^2+d^2\),
(iv) \(\displaystyle a^3-b^3=c^3-d^3\).
Find the smallest possible value of \(\displaystyle d\).
Scottish Competition Problem
(5 pont)
solution (in Hungarian), statistics
C. 1906. We draw all the excircles tangent to the sides of a circumscribed trapezoid. Prove that the sum of the areas of the circles tangent to the bases is at least as big as the sum of the areas of the circles tangent to the legs.
Proposed by Márton Ujházy, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1907. We wrote down all the fractions between \(\displaystyle 0\) and \(\displaystyle 1\) that have a denonimator of at most \(\displaystyle 99\) in increasing order. Find the fraction right after \(\displaystyle \displaystyle{\frac{11}{21}}\).
Proposed by György Birkás, Siófok
(5 pont)
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Problems with sign 'B'Deadline expired on June 10, 2026. |
B. 5534. Solve the equation \(\displaystyle \lbrace x \rbrace + \left\lbrace \dfrac{1}{x}\right\rbrace=1\) on the set of real numbers, where \(\displaystyle \lbrace x \rbrace\) denotes the fractional part of \(\displaystyle x\).
Proposed by Gábor Holló, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5535. \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) are the sides and \(\displaystyle r\) is the inradius of a scalene triangle. Let \(\displaystyle s_a\) be the length of the median corresponding to side \(\displaystyle a\), and let \(\displaystyle t_a\) denote the distance of the incenter from the line of median \(\displaystyle s_a\). Prove that
\(\displaystyle 2t_as_a=|b-c|\cdot r.\)
Proposed by Géza Kiss, Csömör
(3 pont)
solution (in Hungarian), statistics
B. 5536. We have written six non-negative integers with a sum of 2027 at the vertices of a regular hexagon. In a step we replace one of the six numbers with the absolute value of the difference of the numbers at the adjacent vertices.
Is it possible that after a sequence of such steps there will be 0 at every vertex?
Proposed by Sándor Róka, Nyíregyháza
(5 pont)
solution (in Hungarian), statistics
B. 5537. In circle \(\displaystyle k\) chord \(\displaystyle CD\) bisects chord \(\displaystyle AB\). The tangents of \(\displaystyle k\) at \(\displaystyle C\) and \(\displaystyle D\), respectively intersect line \(\displaystyle AB\) at \(\displaystyle X\) and \(\displaystyle Y\), respectively. Prove that \(\displaystyle XA=YB\).
Crux Mathematicorum
(4 pont)
solution (in Hungarian), statistics
B. 5538. Solve the following equation on the set of real numbers: \(\displaystyle \frac{1}{x^2-5x+9}+\frac{1}{y^2-5y+9}+\frac{1}{z^2-5z+9}=\frac{1}{\sqrt{6x-9}}+\frac{1}{\sqrt{6y-9}}+\frac{1}{\sqrt{6z-9}}\).
Proposed by Mihály Bence, Brassó
(5 pont)
solution (in Hungarian), statistics
B. 5539. For any positive integer \(\displaystyle n\) prove that
\(\displaystyle \dfrac{n!\cdot\big[\tfrac{n}{30}\big]!}{\big[\tfrac{n}{2}\big]!\cdot\big[\tfrac{n}{3}\big]!\cdot\big[\tfrac{n}{5}\big]!}\)
is an integer, and divides the least common multiple of integers \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle \ldots\), \(\displaystyle n\).
Pafnuty Lvovich Chebyshev (1821–1894), Saint Petersburg
(5 pont)
solution (in Hungarian), statistics
B. 5540. Find the smallest and the largest possible area of the convex polygon \(\displaystyle K\) if its orthogonal projections on the \(\displaystyle x\)-axis, the \(\displaystyle y\)-axis and the line \(\displaystyle x=y\) is a line segment of unit length.
Based on a Russian problem
(6 pont)
solution (in Hungarian), statistics
B. 5541. Let \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) be three points of a parabola with focus \(\displaystyle F\), in this order. Let the tangent of the parabola at \(\displaystyle B\) intersect the tangent at \(\displaystyle P\) in point \(\displaystyle A\), and the tangent at \(\displaystyle C\) at point \(\displaystyle Q\). Prove that the point of intersection of circles \(\displaystyle FAP\) and \(\displaystyle FCQ\) different from \(\displaystyle F\) is on line \(\displaystyle AC\).
Proposed by Gábor Holló, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on June 10, 2026. |
A. 935. A criminal starts at the origin and escapes by moving on the integer lattice points of the plane. At each step, they move to a neighboring lattice point, with the restriction that they never move in the same direction twice consecutively. It is known that their first move is upward, and that their path visits every lattice point exactly once. An agent also starts at the origin but cannot see the criminal. Fortunately, the headquarters can track the criminal's movement, but they can communicate with the agent only in the following way: a predetermined infinite set \(\displaystyle S \subseteq \mathbb{N}\) is given, and for each \(\displaystyle s \in S\), after the \(\displaystyle s\)-th step of the criminal, a message consisting of either \(\displaystyle 1\) or \(\displaystyle 2\) is sent to the agent. The agent moves only when a message is received, and always makes exactly as many steps as the criminal has taken since the last message.
What is the value of \(\displaystyle I=\inf_{S} \left(\lim_{n\rightarrow \infty} \frac{|S \cap \{1,2,3,\ldots,n\}|}{n} \right)\) taken over all sets \(\displaystyle S\) for which the agent can uniquely reconstruct the movement of the criminal?
Both the agent and the headquarters are aware of the rules restricting the criminal's movement.
Proposed by Márton Németh, Budapest)
(7 pont)
A. 936. Given an acute, scalene triangle \(\displaystyle ABC\) in the plane. Denote its symmedian point by \(\displaystyle K\) and the center of its Feuerbach circle by \(\displaystyle N\). Construct, using a compass and straightedge, distinct points \(\displaystyle X\), \(\displaystyle X^*\), \(\displaystyle Y\), \(\displaystyle Y^*\) such that
\(\displaystyle \bullet\) the lines \(\displaystyle XY\) and \(\displaystyle X^*Y^*\) intersect at \(\displaystyle K\);
\(\displaystyle \bullet\) the lines \(\displaystyle XX^*\) and \(\displaystyle YY^*\) intersect at \(\displaystyle N\);
\(\displaystyle \bullet\) the points \(\displaystyle X\) and \(\displaystyle X^*\), as well as \(\displaystyle Y\) and \(\displaystyle Y^*\), are isogonal conjugates with respect to the triangle.
Proposed by Áron Bán-Szabó, Palaiseau
(7 pont)
A. 937. Let \(\displaystyle P \in \mathbb{C}[x_1,\ldots,x_n]\) be an irreducible complex polynomial of degree at least 2. Assume that there exists an integer \(\displaystyle M>1\) such that \(\displaystyle P(x_1,\dots,x_n)\mid P(x_1^M,\dots,x_n^M)\). Prove that there exist a nonzero complex number \(\displaystyle c\), a complex root of unity \(\displaystyle \zeta\) and non-negative integers \(\displaystyle a_1\), \(\displaystyle \ldots\), \(\displaystyle a_n\), \(\displaystyle b_1\), \(\displaystyle \dots\), \(\displaystyle b_n\) such that \(\displaystyle P(x_1,\dots,x_n)=c\left(x_1^{a_1}\cdots x_n^{a_n}-\zeta\,x_1^{b_1}\cdots x_n^{b_n}\right)\).
Proposed by Navid Safaei, Tehran
(7 pont)
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