KöMaL Problems in Mathematics, April 2025
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Problems with sign 'K'Deadline expired on May 12, 2025. |
K. 854. Uncle János makes the following statements about himself: ``When I was born, my mother was more than 20 years old. If we add 1 to both digits of her age at that time, and swap the two digits of the resulting number, we get the current age of my mother. My mother is in good health, and she is not yet 90 years old. I'm 56 years old.'' Find the age of the mother of Uncle János when he was born.
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
K. 855. Three ships left the dock at the same time. One ship was travelling north, and the other two were travelling southeast. All three ships arrived at their first stop at the same exact time. The pair travelling in the same direction split up, one of them kept travelling southeast, but the other one started in the direction of the third ship that was stationary at its first stop. Once again, they've reached their destination at the same exact time. In what direction should the two ships being at the same place travel to reach the second destination of the ship that was travelling southeast all along?
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
K. 856. We have a bag filled with red, blue, and green marbles. The total number of the marbles is 25. There is a paper attached to the bag containing the following statements:
\(\displaystyle \bullet\) There are more blue marbles than red in the bag.
\(\displaystyle \bullet\) The total number of the red and blue marbles is 22.
\(\displaystyle \bullet\) The difference between the number of the red and the number of the blue marbles is less than the number of green marbles.
\(\displaystyle \bullet\) There are more blue marbles, than the red and the green marbles combined.
\(\displaystyle \bullet\) There are two colors with the same number of marbles in the bag.
We know that exactly one of these five statements is false. What can be the possible number of marbles of each color?
Proposed by: Mátyás Czett, Budapest
(5 pont)
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Problems with sign 'K/C'Deadline expired on May 12, 2025. |
K/C. 857. For which real numbers \(\displaystyle x\), \(\displaystyle y\) will inequality \(\displaystyle x^2+y^2\geq(x+1)(y-1)\) hold? For which real numbers will equality hold?
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
K/C. 858. We drew four small blue squares in a \(\displaystyle 4\times 1\) rectangle according to the diagram
Find the ratio of the area of a single blue square and the \(\displaystyle 4\times 1\) rectangle.
Proposed by: Bálint Bíró, Eger
(5 pont)
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Problems with sign 'C'Deadline expired on May 12, 2025. |
C. 1853. Some researchers are observing a team of fifty bumblebees buzzing on a flowery sandbank. They excitedly note that each bumblebee collected pollen from exactly four types of flowers before flying off. In fact, they also recorded that each bumblebee chose a different set of four flowers, but all \(\displaystyle 50\) bumblebees visited a common baby's-breath. Prove that the bumblebees collected pollen from at least 9 different types of flowers.
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1854. Circles \(\displaystyle k_1\) and \(\displaystyle k_2\) intersect each other in points \(\displaystyle A\) and \(\displaystyle B\) such that circle \(\displaystyle k_2\) passes through center \(\displaystyle K\) of circle \(\displaystyle k_1\). Let's choose point \(\displaystyle S\) on circle \(\displaystyle k_2\) such that \(\displaystyle S\) is closer to point \(\displaystyle A\) than to point \(\displaystyle B\), and it's not in the interior of circle \(\displaystyle k_1\). Let line \(\displaystyle BS\) intersect circle \(\displaystyle k_1\) at point \(\displaystyle D\) for the second time, and let line \(\displaystyle AD\) intersect circle \(\displaystyle k_2\) at point \(\displaystyle T\) for the second time. Prove that \(\displaystyle KT\) is perpendicular to \(\displaystyle BS\).
Greek competition problem
(5 pont)
solution (in Hungarian), statistics
C. 1855. 19 drones flew over enemy territory following the security protocol requiring the distances of all pairs of drones being different. Since the drones were hacked, they start to fire at each other: each one eliminates the drone that is the nearest to them. (We suppose that each drone managed to fire, the drones can be considered points, and the drones only explode after all the shots reached their goals.) Will there be a survivor among the drones? What is the maximum number of bullets a drone can be hit by from the same plane (that also contains the given drone)?
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1856. Prove the following inequality for every triangle (using the usual notations for the sides and the angles): \(\displaystyle a\sin\alpha+b\sin\beta+c\sin\gamma\ge\frac{a+b+c}{3}\left(\sin\alpha+\sin\beta+\sin\gamma\right)\). When will the equality hold?
Proposed by: Gábor Holló, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1857. Find all triples of positive prime numbers \(\displaystyle p\), \(\displaystyle q\) and \(\displaystyle r\) that satisfy the system of equations \(\displaystyle p+q=r+1\), \(\displaystyle p\cdot r=q^2+6\).
German competition problem
(5 pont)
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Problems with sign 'B'Deadline expired on May 12, 2025. |
B. 5454. Positive integers \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle a_3\), \(\displaystyle \ldots\), \(\displaystyle a_{2024}\), \(\displaystyle a_{2025}\) are chosen such that fractions \(\displaystyle \frac{a_1}{a_2}\), \(\displaystyle \frac{a_2}{a_3}\), \(\displaystyle \frac{a_3}{a_4}\), \(\displaystyle \ldots\), \(\displaystyle \frac{a_{2024}}{a_{2025}}\) have pairwise different values. At least how many of the numbers must be different?
Proposed by: Attila Sztranyák, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5455. Let \(\displaystyle P\) be the intersection of lines \(\displaystyle AB\) and \(\displaystyle CD\), where \(\displaystyle ABCD\) is a convex quadrilateral. Let points \(\displaystyle M\) and \(\displaystyle N\) be the midpoints of the diagonals. Prove that the area of triangle \(\displaystyle PMN\) equals one quarter of the area of quadrilateral \(\displaystyle ABCD\).
Proposed by: Sándor Róka, Nyíregyháza
(4 pont)
solution (in Hungarian), statistics
B. 5456. Which can be the smallest and the largest of numbers \(\displaystyle a^{\log_b c}\), \(\displaystyle b^{\log_c a}\), \(\displaystyle c^{\log_a b}\), provided that \(\displaystyle 1<a<b<c\)?
Proposed by: Bálint Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5457. The angles of a triangle are \(\displaystyle \alpha \leq \beta \leq \gamma\), and its longest side is of unit length. Prove that the triangle is obtuse if and only if the reciprocal of its area is more than \(\displaystyle 2\left(\tan\alpha +\tan\beta\right)\).
Proposed by: Mihály Hujter, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5458. Let \(\displaystyle I\) be the incenter of triangle \(\displaystyle ABC\), and let \(\displaystyle AB<AC\). The perpendicular to \(\displaystyle AI\) passing through \(\displaystyle I\) intersects line \(\displaystyle BC\) at \(\displaystyle P\), and line segment \(\displaystyle AP\) intersects the circumcircle of triangle \(\displaystyle ABC\) at \(\displaystyle Q\) for the second time. Prove that \(\displaystyle IQ\) is perpendicular to \(\displaystyle AP\).
Proposed by: Géza Kós, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5459. Find all functions \(\displaystyle f\colon\mathbb{Q}\rightarrow\mathbb{Q}\) satisfying \(\displaystyle f(x)+f(y)=\frac{f(x+2y)+f(2x-y)}{5}\) for all rational numbers \(\displaystyle x\), \(\displaystyle y\).
Proposed by: Erik Füredi, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5460. In acute triangle \(\displaystyle ABC\) let \(\displaystyle D\), \(\displaystyle E\) and \(\displaystyle F\) denote the feet of the altitudes from \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\), respectively. Let line \(\displaystyle DE\) intersect circle \(\displaystyle ABC\) at points \(\displaystyle G\), \(\displaystyle H\). Similarly, let line \(\displaystyle DF\) intersect circle \(\displaystyle ABC\) at points \(\displaystyle I\), \(\displaystyle J\). Prove that the radical axis of circles \(\displaystyle EIJ\) and \(\displaystyle FGH\) passes through the orthocenter of triangle \(\displaystyle ABC\).
Proposed by: Viktor Csaplár, Bátorkeszi
(6 pont)
solution (in Hungarian), statistics
B. 5461. Prove that for any positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) there exists infinitely many positive integers \(\displaystyle n\) for which \(\displaystyle a^n+bc\) and \(\displaystyle b^{n+d}-1\) are not relatively primes.
Proposed by: Géza Kós, Budapest, based on IMO2024/2
(6 pont)
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Problems with sign 'A'Deadline expired on May 12, 2025. |
A. 905. We say that a strictly increasing sequence of positive integers \(\displaystyle n_1\), \(\displaystyle n_2\), \(\displaystyle \ldots\) is non-decelerating if \(\displaystyle n_{k+1}-n_k\le n_{k+2}-n_{k+1}\) holds for all positive integers \(\displaystyle k\). We say that a strictly increasing sequence \(\displaystyle n_1\), \(\displaystyle n_2\), \(\displaystyle \ldots\) is convergence-inducing, if the following statement is true for all real sequences \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \ldots\): if subsequence \(\displaystyle a_{m+n_1}\), \(\displaystyle a_{m+n_2}\), \(\displaystyle \ldots\) is convergent and tends to 0 for all positive integers \(\displaystyle m\), then sequence \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \ldots\) is also convergent and tends to 0. Prove that a non-decelerating sequence \(\displaystyle n_1\), \(\displaystyle n_2\), \(\displaystyle \ldots\) is convergence-inducing if and only if sequence \(\displaystyle n_2-n_1\), \(\displaystyle n_3-n_2\), \(\displaystyle \ldots\) is bounded from above.
Proposed by: András Imolay, Budapest
(7 pont)
A. 906. Let \(\displaystyle \mathcal{V}_c\) denote the infinite parallel ruler with the parallel edges being at distance \(\displaystyle c\) from each other. The following construction steps are allowed using ruler \(\displaystyle \mathcal{V}_c\):
\(\displaystyle \bullet\) the line through two given points;
\(\displaystyle \bullet\) line \(\displaystyle l'\) parallel to a given line \(\displaystyle l\) at distance \(\displaystyle c\) (there are two such lines, both of which can be constructed using this step);
\(\displaystyle \bullet\) for given points \(\displaystyle A\) and \(\displaystyle B\) with \(\displaystyle |AB|\ge c\) two parallel lines at distance \(\displaystyle c\) such that one of them passes through \(\displaystyle A\), and the other one passes through \(\displaystyle B\) (if \(\displaystyle |AB|>c\), there exists two such pairs of parallel lines, and both can be constructed using this step). On the perimeter of a circular piece of paper three points are given that form a scalene triangle. Let \(\displaystyle n\) be a given positive integer. Prove that based on the three points and \(\displaystyle n\) there exists \(\displaystyle C>0\) such that for any \(\displaystyle 0<c\le C\) it is possible to construct \(\displaystyle n\) points using only \(\displaystyle \mathcal{V}_c\) on one of the excircles of the triangle. We are not allowed to draw anything outside our circular paper. We can construct on the boundary of the paper; it is allowed to take the intersection point of a line with the boundary of the paper.
Proposed by: Áron Bán-Szabó, Budapest
(7 pont)
A. 907. 2025 light bulbs are operated by some switches. Each switch works on a subset of the light bulbs. When we use a switch, all the light bulbs in the subset change their state: bulbs that were on turn off, and bulbs that were off turn on. We know that every light bulb is operated by at least one of the switches. Initially, all lamps were off. Find the biggest number \(\displaystyle k\) for which we can surely turn on at least \(\displaystyle n\) light bulbs.
Based on an OKTV problem
(7 pont)
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