KöMaL Problems in Mathematics, September 2025
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Problems with sign 'K'Deadline expired on October 10, 2025. |
K. 864. The arrangement in the figure was created using matches. Find the smallest number of matches which can be moved in a way that instead of the original five congruent squares there are only four squares visible, all of which are congruent to the original ones.
(5 pont)
solution (in Hungarian), statistics
K. 865. A merchant bought a product from the wholesaler for HUF 3000. He wants to set the price of the product such that even after a 10% discount he earns an extra 20% of the original price. Find the new price of the product.
(5 pont)
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K. 866. Find the values of distinct positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\) if we know that
\(\displaystyle a(b+c+d)=16,\)
\(\displaystyle b(a+c+d)=30,\)
\(\displaystyle c(a+b+d)=42.\)
(5 pont)
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Problems with sign 'K/C'Deadline expired on October 10, 2025. |
K/C. 867. We have two identical rectangular pieces of paper. We cut the first piece of paper into two rectangles with a straight cut. The sum of the perimeters of the two resulting rectangles is 84 cm. We also cut the second piece of paper into two rectangles with a straight cut. The sum of the perimeters of the two resulting rectangles is 96 cm. Find the lengths of sides of the original rectangles.
(5 pont)
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K/C. 868. We have written a different digit on each of five number cards. First we've arranged the five cards such that we have obtained the largest possible five digit number, and then we have arranged them such that we have obtained the smallest possible five digit number. The sum of the two numbers we have obtained this way is \(\displaystyle 96\,478\). Find the five digits we have written on the five cards. (The number cards must not be rotated – so, for example, you may not turn a 9 into a 6.)
(5 pont)
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Problems with sign 'C'Deadline expired on October 10, 2025. |
C. 1863. There are four children in a family. The ratio of their ages in 2004 was \(\displaystyle 2:3:4:5\). The sum of their years of birth is 7960.
a) What were the sum of the ages of the children in 2000?
b) Find the age of the oldest child in 2004.
Canadian Competition Problem
(5 pont)
solution (in Hungarian), statistics
C. 1864. The angles of a hexagon are equal, and the lengths of four consecutive sides are 5, 3, 6 and 7 (in this order). Find the lengths of the next two sides.
Indian Competition Problem
(5 pont)
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C. 1865. In the school arm wrestling championship 17 students took part. Every student played with every other student exactly once, and there were no draws. We call a group of participants strong, if every participant not in the group was beaten by at least one participant in the group. Prove that it is always possible to find a strong group containing at most 9 participants.
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1866. Three segments are given with lengths \(\displaystyle 1\), \(\displaystyle a\) and \(\displaystyle b\) (where \(\displaystyle a\neq 1\) and \(\displaystyle b\neq 1\)). Describe a way to construct segments of lengths \(\displaystyle {\frac{1}{a}}\), \(\displaystyle a\cdot b\) and \(\displaystyle a^2\cdot b^3\). The elementary constructions, e.g. bisecting an angle, reflecting across a line, etc. do not have to be described in detail.
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1867. Find the coefficient of term \(\displaystyle a^nb^nc^n\) after the expansion and collection of like terms of algebraic expression \(\displaystyle (a+b)^n(b+c)^n(c+a)^n\), where \(\displaystyle n\) is a positive integer. (The answer can be given as a sum of at most \(\displaystyle n\) terms.)
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
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Problems with sign 'B'Deadline expired on October 10, 2025. |
B. 5470. 1001, 2002 and 3003 are three consecutive elements of the \(\displaystyle 14^{\text{th}}\) row of Pascal's triangle. Does there exist another row where three consecutive elements are of the form \(\displaystyle n\), \(\displaystyle 2n\) and \(\displaystyle 3n\) for some positive integer \(\displaystyle n\)?
Proposed by: Bálint Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5471. At most how many positive integers can be chosen from the first 50 positive integers such that no two have a product which is a perfect \(\displaystyle 4^\text{th}\) power?
Proposed by: Péter Pál Pach, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5472. Convex quadrilateral \(\displaystyle ABCD\) satisfies property \(\displaystyle AB=BC=CD\). Prove that if \(\displaystyle \angle BCD=2\angle DAB\), then \(\displaystyle \angle ABC=2\angle CDA\).
Proposed by: Géza Kós, Budapest and Viktor Vígh, Sándorfalva
(4 pont)
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B. 5473. The sum of some distinct positive integers is 1000. What is the largest possible value of their product?
Proposed by: Katalin Abigél Kozma, Győr
(4 pont)
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B. 5474. Prove that squares with side lengths 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19 and 23 cannot be arranged in a \(\displaystyle 45\times 45\) square without overlap.
Proposed by: Sándor Bozóki, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5475. Show an example of a regular polygon, the area of which equals the product of two of its diagonals.
Proposed by: Mihály Hujter, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5476. Each of Ali and Baba has a random number generator (RNG). Ali's RNG chooses one of numbers 1, 2, \(\displaystyle \ldots\), 100 with uniform distribution (each number is chosen with the same probability), and Baba's RNG chooses one of numbers 0, 1, 2, \(\displaystyle \ldots\), 100 with uniform distribution. Ali calculated the expected number of times he has to use his RNG to get a sum which is at least 101, and Baba calculated the expected number of times he has to use his RNG to get a sum which is at least 100. Whose number is bigger?
Proposed by: Attila Sztranyák, Budapest
(6 pont)
B. 5477. \(\displaystyle \Omega\) is the circumcircle of cyclic quadrilateral \(\displaystyle ABCD\). Circles \(\displaystyle \omega_1\) and \(\displaystyle \omega_2\) are internally tangent to \(\displaystyle \Omega\), and both are tangent to line segments \(\displaystyle AB\) and \(\displaystyle CD\). Circles \(\displaystyle \omega_1\) and \(\displaystyle \omega_2\) intersect each other in points \(\displaystyle P\) and \(\displaystyle Q\). Prove that line \(\displaystyle PQ\) bisects arcs \(\displaystyle AB\) and \(\displaystyle CD\) of circle \(\displaystyle \Omega\).
Proposed by: Géza Kós, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on October 10, 2025. |
A. 911. For which integers \(\displaystyle n \ge 4\) is it true that the area of the regular \(\displaystyle n\)-gon is equal to the product of two of its diagonals?
Based on the idea of Mihály Hujter, Budapest
(7 pont)
A. 912. Let \(\displaystyle ABC \) be a triangle, and let the symmedian through \(\displaystyle A\) intersect the circumcircle of the triangle at a second point \(\displaystyle A'\). Prove that the following three points lie on a line:
- the reflection of point \(\displaystyle A\) over the line \(\displaystyle BC\);
- the inverse image of point \(\displaystyle A\) under inversion with respect to the Feuerbach circle of triangle \(\displaystyle ABC\);
- the midpoint of the chord \(\displaystyle AA'\).
Proposed by: Ábel Szakács, Budapest
(7 pont)
A. 913. Let \(\displaystyle n\) be a positive integer. Marci writes down the first \(\displaystyle n\) positive integers in some order in his notebook, which we cannot see. Let this order be \(\displaystyle m(1),m(2),\ldots,m(n)\); we would like to determine this order. In one step we may also list the first \(\displaystyle n\) positive integers for Marci in some order, let this be \(\displaystyle a(1),\ldots,a(n)\). Then Marci, in his notebook (which we cannot see), draws a directed graph on \(\displaystyle n\) vertices, labelled with \(\displaystyle 1,2,\ldots,n\), respectively. After this, for every integer \(\displaystyle 1\leq i\leq n\), he draws a directed edge from the vertex labelled \(\displaystyle i\) to the vertex labelled \(\displaystyle m(a(i))\). Finally, the resulting graph with \(\displaystyle n\) edges decomposes into a disjoint union of directed cycles (each of which may consist of a single loop), and Marci tells us the number of these cycles.
a) Prove that the secret permutation can be determined in at most \(\displaystyle n\log_2(n)\) steps.
b) Does there exist a constant \(\displaystyle c<1\) such that for every positive integer \(\displaystyle n\) the secret permutation can be determined in at most \(\displaystyle cn\log_2(n)\) steps?
Proposed by: Márton Németh, Budapest
(7 pont)
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