KöMaL Problems in Mathematics, April 2026
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Problems with sign 'K'Deadline expired on May 11, 2026. |
K. 899. Anna has written all the positive two-digit numbers on the blackboard once. Boglárka has erased Anna's numbers one by one and replaced them with the numbers obtained by subtracting their first digits from the second. For example, \(\displaystyle 26\) is replaced with \(\displaystyle 6-2=4\) and \(\displaystyle 37\) is replaced with \(\displaystyle 3-7=-4\). Find the sum of Boglárka's numbers.
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
K. 900. We obtain a regular octagon by cutting out isosceles right triangles from the four corners of a square. Prove that the length of the side of the octagon is equal to the difference of the lengths of the diagonal and the side of the square.
Proposed by Márton Ujházy, Budapest
(5 pont)
solution (in Hungarian), statistics
K. 901. Aliz Zilah and her mother, Anna celebrate their birthdays on the same day, 03.30. They celebrate this year the palindrome birthday: we can obtain Anna's age by replacing the digits of Aliz' age. How old was Anna when Aliz was born? (Find all possible solutions.)
Proposed by Márton Ujházy, Budapest
(5 pont)
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Problems with sign 'K/C'Deadline expired on May 11, 2026. |
K/C. 902. We have \(\displaystyle 2026\) unit cubes. From these we create the biggest number of compositions that can be obtained by removing the corner cubes of a \(\displaystyle 3\times 3\times 3\) cube. Find the ratio of surface areas of these cornerless compositions and the remaining unit cubes.
Proposed by Bálint Bíró, Eger
(5 pont)
K/C. 903. In a group of twenty-five friend \(\displaystyle 20\) people can play bridge, \(\displaystyle 19\) people can play chess and \(\displaystyle 18\) people van play. How many people can play all three games
a) at least?
b) at most?
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
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Problems with sign 'C'Deadline expired on May 11, 2026. |
C. 1898. We would like to select some of the nine lattice points of a \(\displaystyle 2\times 2\) square lattice such that no three of them form a right triangle.
a) Prove that it's not possible to select five points with the property above.
b) How many ways are there to select four points satisfying the above property? (Two selections are considered different if there is a point that is selected in one of them and not selected in the other.)
Proposed by Márton Ujházy, Budapest
(5 pont)
C. 1899. 2227 is a special number for two different reasons. The first is that this is the next year when Pluto will be the closer to the Sun than Neptun, and the second is that three of its digits can be chosen such that their product is bigger by one than its remaining digit. How many four-digit numbers are there having the second property?
Proposed by Mátyás Czett, Zalaegerszeg
(5 pont)
C. 1900. In a mathematics class of 30 students they play the following game. The teacher rolls seven octahedral dice of different colors, and the students write down the seven resulting numbers in an order of their choice, thus creating a seven-digit number. This time the resulting numbers were 1, 2, 3, 4, 5, 6, 7. Is it possible that one of the numbers created by the students is a multiple of another number? (Remark. The winner of the game is the student the minimum difference of whose number from the other numbers is the largest.)
Proposed by Zoltán Paulovics, Budapest
(5 pont)
C. 1901. Real numbers \(\displaystyle a\) and \(\displaystyle b\) satisfy \(\displaystyle |a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|\). Find the smallest possible value of \(\displaystyle |a-b|\).
Proposed by Sándor Róka, Nyíregyháza
(5 pont)
C. 1902. On the smaller arc \(\displaystyle AB\) of the circumcircle of square \(\displaystyle ABCD\) we pick internal point \(\displaystyle E\). Let \(\displaystyle E'\) be the reflection of \(\displaystyle E\) across the center of the square. Let \(\displaystyle F\) and \(\displaystyle G\) be the feet of the perpendiculars from \(\displaystyle E\) to diagonals \(\displaystyle AC\) and \(\displaystyle BD\), respectively. We get similarly points \(\displaystyle H\) and \(\displaystyle I\) from point \(\displaystyle E'\).
a) Prove that quadrilateral \(\displaystyle FGHI\) is a diamond.
b) Find the maximum of the area of quadrilateral \(\displaystyle FGHI\).
Proposed by Bálint Bíró, Eger
(5 pont)
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Problems with sign 'B'Deadline expired on May 11, 2026. |
B. 5526. Let \(\displaystyle P\) be an interior point of circle \(\displaystyle k\) with center \(\displaystyle O\), different from \(\displaystyle O\). Let \(\displaystyle O'\) be the reflection of point \(\displaystyle O\) across point \(\displaystyle P\). Let \(\displaystyle X\) be one of the intersection points of the circle with center \(\displaystyle O'\) and radius \(\displaystyle O'P\) and circle \(\displaystyle k\). Line \(\displaystyle XP\) intersects circle \(\displaystyle k\) at point \(\displaystyle Y\) for the second time. Prove that \(\displaystyle P\) trisects line segment \(\displaystyle XY\).
Proposed by Mathematics sophomores of University of Szeged
(3 pont)
B. 5527. The sides of acute triangle \(\displaystyle ABC\) are \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), and its inradius is \(\displaystyle r\). Prove that equality \(\displaystyle 2r^2=(c-a)(c-b)\) (also featured in problem B.5495.) holds if and only if \(\displaystyle a+b=3c\).
Proposed by Géza Kiss, Csömör
(4 pont)
B. 5528. Prove that no matter how we color the natural numbers using 100 colors, it is always possible to find numbers \(\displaystyle a<b<c<d\) with the same color satisfying \(\displaystyle a+d=b+c\).
Proposed by Dömötör Pálvölgyi, Budapest
(4 pont)
B. 5529. How many of the perfect powers \(\displaystyle 6^2\), \(\displaystyle 6^3\), \(\displaystyle 6^4\), \(\displaystyle \ldots\), \(\displaystyle 6^{2026}\) have a first digit that is smaller than \(\displaystyle 6\)?
Proposed by Attila Sztranyák, Budapest
(3 pont)
B. 5530. Five different positive integers satisfy the following property: no matter how we choose some of them, their geometric mean is always an integer. Find the smallest possible value of the biggest of these numbers.
Proposed by Péter Pál Pach, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5531. The incenter of triangle \(\displaystyle ABC\) is \(\displaystyle I\), the circumcenters of triangles \(\displaystyle ABI\), \(\displaystyle BCI\) and \(\displaystyle CAI\) are \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\), respectively. Prove that the area of triangle \(\displaystyle CDE\) is at least as large as the area of triangle \(\displaystyle ABC\).
Proposed by Gábor Holló, Budapest
(5 pont)
B. 5532. An opera house has 8 boxes. With a box subscription, one can reserve the same box for at most 7 different evenings of the 2027/28 season, chosen by the buyer. There is one performance each evening. A box can be rented by only one person for a given evening, but multiple subscriptions may refer to the same box as long as the sets of selected evenings are disjoint. What is the minimum number of different performances in the repertoire such that the opera house can always arrange the program (knowing the subscribers' chosen dates) so that every box subscriber sees only different performances?
Proposed by András Imolay, Budapest
(6 pont)
B. 5533. In acute triangle \(\displaystyle ABC\) let \(\displaystyle A_0\) be the foot of the altitude from \(\displaystyle A\), and let \(\displaystyle B_0\) be the foot of the altitude from \(\displaystyle B\). On circle \(\displaystyle A_0B_0C\) let \(\displaystyle X\) and \(\displaystyle Y\) be the two points for which circles \(\displaystyle ABX\) and \(\displaystyle ABY\) are tangent to circle \(\displaystyle A_0B_0C\). Prove that line \(\displaystyle XY\) bisects side \(\displaystyle AB\).
Proposed by Géza Kós, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on May 11, 2026. |
A. 932. Let \(\displaystyle ABC\) be an acute triangle with orthocenter \(\displaystyle H\). Points \(\displaystyle D\) and \(\displaystyle E\) lie on the lines \(\displaystyle AC\) and \(\displaystyle AB\), respectively, such that the points \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle E\) are concyclic, and the line \(\displaystyle DE\) bisects the side \(\displaystyle BC\). Suppose that the lines \(\displaystyle BD\) and \(\displaystyle CE\) meet at \(\displaystyle M\). Prove that the line \(\displaystyle HM\) is perpendicular to the symmedian through \(\displaystyle A\).
Proposed by Áron Bán-Szabó, Palaiseau
(7 pont)
A. 933. Let \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) be real numbers in the interval \(\displaystyle (1,4)\). Consider the following three inequalities: \(\displaystyle {bx(x+y-z)}+{cx(x+z-y)}\geq {a(2yz+x)}\), \(\displaystyle {cy(y+z-x)}+{ay(y+x-z)} \geq {b(2zx+y)}\), \(\displaystyle {az(z+x-y)}+{bz(z+y-x)} \geq {c(2xy+z)}\).
a) Determine, in terms of \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), the maximum number of these inequalities that can hold simultaneously for positive real numbers \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\) satisfying \(\displaystyle x+y+z=1\).
b) Find an explicit formula for \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\) in terms of \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), without using case distinctions, such that \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z>0\), \(\displaystyle x+y+z=1\), and none of the three inequalities holds.
Proposed by Áron Bán-Szabó, Palaiseau
(7 pont)
A. 934. Let \(\displaystyle \mathcal{T}\) denote the set of countable, undirected trees. A subset \(\displaystyle X\) of the real numbers is called bariton if every nonempty subset of \(\displaystyle X\) has a least element.
a) Prove that for every bariton \(\displaystyle X\) there exists a function \(\displaystyle f\colon X\to \mathcal{T}\) such that for all \(\displaystyle y\), \(\displaystyle z\in X\) we have \(\displaystyle y\leq z\) if and only if \(\displaystyle f(y)\) is isomorphic to a subgraph of \(\displaystyle f(z)\).
b) Does the statement remain true if we additionally require that for every \(\displaystyle x\in X\), all vertices of \(\displaystyle f(x)\) have finite degree? (A graph with vertex set \(\displaystyle V\) is called countable if there exists an injective map \(\displaystyle V\to\mathbb{N}\).)
Proposed by Márton Németh, Budapest
(7 pont)
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