KöMaL Problems in Mathematics, February 2026
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Problems with sign 'K'Deadline expired on March 10, 2026. |
K. 889. In chess, the queen can move any number of squares in a straight line or diagonally. Find the smallest number of queens that can be placed on a \(\displaystyle 6\times 6\) chessboard such that any square not occupied by a queen can be reached by one of the queens.
(5 pont)
solution (in Hungarian), statistics
K. 890. A square and its center are given by its vertices.
a) How many straight lines can be drawn that passes through at least two of the five given points?
b) How many circles can be drawn that passes through at least three of the five given points?
Based on a problem by György Birkás
(5 pont)
solution (in Hungarian), statistics
K. 891. On a cube with an edge length of 5 cm we draw all the diagonals of the faces using red color.
a) From one of the vertices of the cube an Ulrich's ground beetle starts travelling following the red lines, passing through all the vertices of the cube, and returning to the original vertex. Find the least distance travelled by the bug.
b) We cut up the large cube into smaller cubes with an edge length of 1 cm. How many of the small cubes will not contain a red line on their surface?
(5 pont)
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Problems with sign 'K/C'Deadline expired on March 10, 2026. |
K/C. 892. We call a number zig-zag number, if (reading it from left to right) its second digit is smaller than its first digit, its third digit is bigger than its second digit, its fourth digit is smaller than its third digit, and so on. How many five-digit zig-zag numbers can be created from digits 1, 2, 3, 4, 5, 6, 7?
(5 pont)
solution (in Hungarian), statistics
K/C. 893. In triangle \(\displaystyle ABC\), \(\displaystyle \angle ABC=15^{\circ}\), \(\displaystyle \angle BCA=30^{\circ}\), and point \(\displaystyle D\) on side \(\displaystyle BC\) satisfies \(\displaystyle \angle ADC=45^{\circ}\). Prove that \(\displaystyle D\) is the midpoint of side \(\displaystyle BC\).
(5 pont)
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Problems with sign 'C'Deadline expired on March 10, 2026. |
C. 1888. Anna has written all the three-digit numbers on the blackboard, each appearing exactly once. Boglárka has erased Anna's numbers one by one, replacing them with the number obtained by subtracting the last digit from the sum of the first and the second digit. For example, 225 is replaced by \(\displaystyle 2+2-5=-1\) and 973 is replaced by \(\displaystyle 9+7-3=13\). Find the sum of numbers written on the blackboard by Boglárka.
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1889. Numbers 1 and \(\displaystyle \sqrt{5}\) are marked on the number line, and nothing else. Mark 0 on the line by construction. (The basic constructions, e.g., bisecting an angle or reflecting across a line don't have to be detailed.)
Proposed by Ferenc Veszprémi, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1890. Prove that the irrational number \(\displaystyle \sqrt[3]{a}\) cannot be written as \(\displaystyle b+c\sqrt{d}\), where \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) are non-negative integer numbers.
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1891. Let \(\displaystyle I\) denote the incenter of triangle \(\displaystyle ABC\), and let line \(\displaystyle AI\) intersect the circumcircle of the triangle at point \(\displaystyle D\). Let the circumcircle of triangle \(\displaystyle CDI\) intersect line segment \(\displaystyle BC\) at point \(\displaystyle E\). Prove that \(\displaystyle BE=IE\).
Proposed by David Nguyen, Sydney, Australia
(5 pont)
solution (in Hungarian), statistics
C. 1892. Prove that \(\displaystyle \binom{\binom{n}{2}}{3}=15\binom{n}{6}+30\binom{n}{5}+16\binom{n}{4}+\binom{n}{3}\) is true for every positive integer \(\displaystyle n\).
Proposed by Zoltán Paulovics, Budapest
(5 pont)
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Problems with sign 'B'Deadline expired on March 10, 2026. |
B. 5510. We write real numbers on the faces of a cube such that the sum of the numbers on opposite faces is always 7. Subsequently at each vertex we take the product of the numbers written on the faces containing the given vertex. What is the largest possible value of the sum of these eight numbers?
Proposed by László Németh, Fonyód
(3 pont)
solution (in Hungarian), statistics
B. 5511. Prove that if a chord intersects the diameter of the unit circle in an angle of 45 degrees, then the sum of the squares of the lengths of the two line segments created on the chord by the diameter equals 2.
Proposed by Viktor Vígh, Sándorfalva
(3 pont)
solution (in Hungarian), statistics
B. 5512. Prove that
\(\displaystyle \frac{1}{\sqrt[n]{k}}+\frac{1}{\sqrt[k]{n}}\geq 1+\frac1{kn}\)
is true for every positive integer \(\displaystyle k\) and \(\displaystyle n\).
Proposed by Gábor Holló, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5513. Inside regular hexagon \(\displaystyle ABCDEF\) point \(\displaystyle P\) is given such that the areas of triangles \(\displaystyle ABP\), \(\displaystyle BCP\) and \(\displaystyle CDP\) are \(\displaystyle 25\), \(\displaystyle 31\) and \(\displaystyle 32\) units, respectively. Find the areas of triangles \(\displaystyle DEP\), \(\displaystyle EFP\) and \(\displaystyle FAP\).
Proposed by Géza Kós, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5514. The sides of triangle \(\displaystyle ABC\) satisfy \(\displaystyle a \geq b \geq c\) (with the usual notations). Let \(\displaystyle P\) be the endpoint of the angle bisector from point \(\displaystyle C\) on side \(\displaystyle AB\). Prove that the diagonals of the cyclic trapezoid with side lengths \(\displaystyle CP\), \(\displaystyle BP\), \(\displaystyle CP\), \(\displaystyle PA\) equals \(\displaystyle \sqrt{ab}\).
Proposed by Mihály Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5515. Is it possible to write a digit on each of the vertices of a regular 1000-gon such that numbers 000, 001, 002, \(\displaystyle \ldots\), 998, 999 can all be read from some three consecutive vertices, moving in the clockwise direction?
Based on an idea of Csongor Beke, Cambridge
(5 pont)
solution (in Hungarian), statistics
B. 5516. A right triangle is inscribed in a parabola such that the foot of the altitude corresponding to the hypotenuse is the focus of the parabola. What can be the size of the angles of the triangle?
Proposed by Gábor Holló, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5517. For which prime numbers \(\displaystyle p\) does there exist a permutation \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \dots\), \(\displaystyle a_{p-1}\) of numbers 1, 2, \(\displaystyle \ldots\), \(\displaystyle p-1\) such that numbers \(\displaystyle a_1^1\), \(\displaystyle a_2^2\), \(\displaystyle \ldots\), \(\displaystyle a_{p-1}^{p-1}\) have pairwise different remainders modulo \(\displaystyle p\)?
(Proposed by Péter Pál Pach, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on March 10, 2026. |
A. 926. Let
\(\displaystyle A=\{[\frac{3+\sqrt{5}}{2}\cdot n]\colon n=1,2,3,\ldots\}=\{2,5,7,10,13,15,\ldots\}.\)
Two players, First and Second, play a game with a pile of \(\displaystyle N\) coins according to the following rules. The players move alternately, with First moving first. On each move, the player on turn chooses a number \(\displaystyle k\in A\) and removes \(\displaystyle k\) coins from the pile. A player who is unable to make a legal move loses the game. Determine, in terms of \(\displaystyle N\), which player has a winning strategy, and describe such a winning strategy.
Proposed by Attila Sztranyák, Budapest
(7 pont)
A. 927. Let \(\displaystyle ABCDEF\) be a bicentric hexagon, that is a hexagon which is both cyclic and tangential. Assume that there exists a point \(\displaystyle P\) such that line \(\displaystyle AP\) is perpendicular to line \(\displaystyle BF\), \(\displaystyle CP\) is perpendicular to \(\displaystyle BD\), and \(\displaystyle EP\) is perpendicular to \(\displaystyle DF\). Prove that there are two opposite sides of the hexagon whose sum is equal to the circumdiameter.
Proposed by Andrei Chirita, Cambridge
(7 pont)
A. 928. Let \(\displaystyle a_0=0<a_1<a_2<\ldots<a_n\) be integers such that the sequence \(\displaystyle b_k=\frac{a_{k+1}-a_k}{2k+1}\) (\(\displaystyle k=0\), 1, \(\displaystyle \ldots\), \(\displaystyle n-1\)) is non-decreasing. Suppose that \(\displaystyle c_1\), \(\displaystyle c_2\), \(\displaystyle \ldots\), \(\displaystyle c_n\) are real numbers such that the polynomial \(\displaystyle 1+\sum_{k=1}^n c_kx^{a_k}\) is divisible by the polynomial \(\displaystyle (x+1)^n\). Show that \(\displaystyle 2>|c_1|>|c_2|>\ldots>|c_n|\).
Proposed by Géza Kós, Budapest
(7 pont)
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