KöMaL Problems in Mathematics, October 2025
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Problems with sign 'K'Deadline expired on November 10, 2025. |
K. 869. The average, median, mode and range of five not necessarily distinct integer numbers are 69, 83, 85 and 70, respectively. Find the second smallest of the five numbers.
(5 pont)
K. 870. How can 5 identical squares be dissected into a single big square with at most 5 straight cuts? (During cutting, the individual pieces cannot be stacked, and each cut can be applied to a single piece.) Find a solution.
(5 pont)
K. 871. How many positive integers are there that consists of 4 digits in their base 3 representation, and 3 digits in their base 4 representation?
(5 pont)
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Problems with sign 'K/C'Deadline expired on November 10, 2025. |
K/C. 872. Fill in the parts denoted by \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle \ldots\), \(\displaystyle g\) of the diagram with numbers 1, 2, 3, 4, 5, 6 and 7 such that the numbers in the three circles have the same sum, and \(\displaystyle a\) is perfect square. Find all solutions.
(5 pont)
K/C. 873. Rabbit has written letters to six of his friends and relations and filled out six envelopes with their addresses. Later he gave the whole stack to Tigger and asked him to put the letters into the envelopes and post them, since Tigger boasted several times that Tiggers are the best at posting letters. However, this was not the case, since Tigger actually couldn't read at all. So, not surprisingly, Tigger will put the six letters into the six envelopes randomly (at least he will make sure that each envelope will contain a single letter). In how many cases will exactly two letters be placed in their respective envelopes?
(5 pont)
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Problems with sign 'C'Deadline expired on November 10, 2025. |
C. 1868. Claire observed that the distance (in kilometers) between her home and her favorite lake is equal to the square of the time (in hours) it takes to travel between them on a bicycle. On a given day, it took \(\displaystyle 4\) hours longer to get to the lake, as her speed was \(\displaystyle 3\) km/h slower than usual. Find the distance between the two places.
Based on a Canadian competition problem
(5 pont)
C. 1869. Square \(\displaystyle ABCD\) has a side length of 20 meters. A ray of light is emitted from the vertex \(\displaystyle A\) and reflected back from side \(\displaystyle BC\), \(\displaystyle CD\) and \(\displaystyle DA\) (in this order) and finally arrives at the midpoint \(\displaystyle N\) of the side \(\displaystyle AB\). Find the distance traveled by the ray of light.
Iranian competition problem
(5 pont)
C. 1870. A table tennis competition is held between 10 contestants. Each contestant plays exactly one match against every other contestant. At one of the breaks the organizers observe that if two contestants played the same number of matches so far, then there is no contestant that played with both of them. Supposing that more than 5 matches had already been played until the break, is there a contestant who played exactly two matches so far?
Based on a problem from Polygon, Szeged
(5 pont)
C. 1871. The line through focus \(\displaystyle F\) of the parabola \(\displaystyle x^2-6x-4y+9=0\) and an arbitrary point \(\displaystyle P\) of the parabola intersects the parabola at point \(\displaystyle Q\) for the second time. Find the equation corresponding to the locus of the midpoints of line segment \(\displaystyle PQ\).
Proposed by Bálint Bíró, Eger
(5 pont)
C. 1872. Peti has written the integers from one to fifty on the blackboard. In each step, he chooses two numbers (\(\displaystyle a\) and \(\displaystyle b\)) on the board, erases them and replaces them with the number defined by the formula
\(\displaystyle a^2b-6a^2-7ab+42a+6b-30.\)
He repeats this until a single number remains on the board. What can this number be?
Proposed by Mátyás Czett, Zalaegerszeg
(5 pont)
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Problems with sign 'B'Deadline expired on November 10, 2025. |
B. 5478. Let \(\displaystyle S\) denote the centroid of triangle \(\displaystyle ABC\). Let \(\displaystyle T\) denote the foot of the height from vertex \(\displaystyle A\) in triangle \(\displaystyle ABC\). Ray \(\displaystyle TS\) (starting from point \(\displaystyle T\)) intersect the circumcircle of the triangle in \(\displaystyle V\). Prove that \(\displaystyle S\) trisects line segment \(\displaystyle TV\).
Proposed by Viktor Vígh, Sándorfalva
(3 pont)
B. 5479. Let \(\displaystyle d\) be a given positive integer. Prove that there exist infinitely many pairs \(\displaystyle (x,y)\) of positive integers such that the difference of the arithmetic and geometric mean of \(\displaystyle x\) and \(\displaystyle y\) is \(\displaystyle d\).
Proposed by László Németh, Fonyód
(3 pont)
B. 5480. In triangle \(\displaystyle ABC\) angle \(\displaystyle \angle ACB\) is \(\displaystyle 90^{\circ}\). Let \(\displaystyle F\) denote the midpoint of leg \(\displaystyle AC\), and let \(\displaystyle G\) denote the midpoint of hypotenuse \(\displaystyle BC\). Prove that the circle with diameter \(\displaystyle FG\) is tangent to the incircle of triangle \(\displaystyle ABC\) if and only if \(\displaystyle AC=2BC\).
Proposed by Attila Sztranyák, Budapest
(4 pont)
B. 5481. Prove the following divisibility for every positive integer \(\displaystyle n\):
\(\displaystyle (n!)^n \mid n^2 \cdot (n^2-1) \cdot (n^2-2) \cdot \ldots \cdot (n+2) \cdot (n+1).\)
Proposed by Bálint Hujter, Budapest
(4 pont)
B. 5482. We have selected an odd number of integers. We have changed one of them such that their average increased by 1, but their standard deviation did not change. Prove that the average of the originally selected numbers is an integer.
Based on a proposal by Erzsébet Berkó, Szolnok
(5 pont)
B. 5483. A deck of \(\displaystyle 2n\) cards consists of \(\displaystyle n\) suits, each suit consisting of two values, 1 and 2. \(\displaystyle n\) players play the following cooperative game: They start by shuffling the \(\displaystyle 2n\) cards, then each player is dealt two cards, which they reveal to the other players. Subsequently, they decide together the order they sit down around a round table, and they also choose the player starting the game. During the game, each consecutive player (in the clockwise direction) tries to put down a card, however, a card with value 2 can only be put down if the card of the same suit with value 1 is already on the table. They win the game if they manage to put all the cards on the table, however, they lose the game if somebody holding at least one card in their hand cannot put down a card on the table. Prove that the players can always win the game, regardless of the hand they were dealt.
Proposed by Anett Kocsis and András Imolay, Budapest, based on a Dürer competition problem
(5 pont)
B. 5484. Let \(\displaystyle M\) denote the orthocenter of acute triangle \(\displaystyle ABC\), and let \(\displaystyle F\) denote the midpoint of side \(\displaystyle BC\). Let \(\displaystyle T\) denote the foot of the perpendicular from \(\displaystyle M\) to the inner angle bisector at vertex \(\displaystyle A\). Line \(\displaystyle FT\) intersects side \(\displaystyle AC\) at point \(\displaystyle D\). Prove that the circumcircle of triangle \(\displaystyle CDF\) contains the foot of the height of triangle \(\displaystyle ABC\) from vertex \(\displaystyle A\).
Proposed by Bálint Bíró, Eger
(6 pont)
B. 5485. Prove that there exists a rational number arbitrarily close to \(\displaystyle \frac1{2025}\) that cannot be obtained as the sum of the reciprocals of 2025 positive integers.
Proposed by Mihály Hujter, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on November 10, 2025. |
A. 914. We write the ordered pair \(\displaystyle (1,0)\) on the board. In one step, if the ordered pair \(\displaystyle (a,b)\) is on the board, we erase it and replace it with either \(\displaystyle (a+b,b)\) or \(\displaystyle (2a+b,a+b)\). (For example, after two steps, the pair on the board may be one of \(\displaystyle (1,0)\), \(\displaystyle (2,1)\), \(\displaystyle (3,1)\), or \(\displaystyle (5,3)\).) Prove that after \(\displaystyle n\) steps, the ordered pair on the board can take exactly \(\displaystyle 2^n\) different values.
Proposed by András Imolay, Budapest
(7 pont)
A. 915. Given a circle with three points \(\displaystyle A\), \(\displaystyle B\), and \(\displaystyle C\) on it, which do not form an isosceles triangle. For every point \(\displaystyle P\notin\{A,B,C\}\) on the circle, let \(\displaystyle A_P\), \(\displaystyle B_P\) and \(\displaystyle C_P\) denote the intersections of the tangent at \(\displaystyle P\) with the tangents at \(\displaystyle A\), \(\displaystyle B\), and \(\displaystyle C\), respectively. Prove that there exist exactly three points \(\displaystyle P\notin\{A,B,C\}\) on the circle for which the points \(\displaystyle A_P\), \(\displaystyle B_P\) and \(\displaystyle C_P\) are well-defined and the perpendiculars from \(\displaystyle A_P\) to \(\displaystyle BC\), from \(\displaystyle B_P\) to \(\displaystyle CA\), and from \(\displaystyle C_P\) to \(\displaystyle AB\) are concurrent. Furthermore, show that these three points \(\displaystyle P\) form an equilateral triangle.
Proposed by Zoltán Gyenes, Budapest
(7 pont)
A. 916. Let \(\displaystyle a \geq 3\) be an integer, and define \(\displaystyle f(n) = a^n - 1\) for every positive integer \(\displaystyle n\). Denote by \(\displaystyle f^{(k)}\) the \(\displaystyle k\)-iterate of \(\displaystyle f\), that is, \(\displaystyle f^{(1)}(n) = f(n)\) and \(\displaystyle f^{(k+1)}(n)=f(f^{(k)}(n))\) for \(\displaystyle k \geq 1\).
a) Prove that for any positive integer \(\displaystyle K\) there exists a positive integer \(\displaystyle M\) such that for every integer \(\displaystyle 1 \leq k \leq K\), the number \(\displaystyle f^{(k)}(M)\) is divisible by \(\displaystyle M\) if and only if \(\displaystyle k\) is divisible by \(\displaystyle 2025\).
b) Does there exist a positive integer \(\displaystyle N\) such that for every positive integer \(\displaystyle k\), the number \(\displaystyle f^{(k)}(N)\) is divisible by \(\displaystyle N\) if and only if \(\displaystyle k\) is divisible by \(\displaystyle 2025\)?
Proposed by Boldizsár Varga, Budapest
(7 pont)
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