KöMaL Problems in Mathematics, May 2025
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Problems with sign 'K'Deadline expired on June 10, 2025. |
K. 859. I have already paid \(\displaystyle 25,000\) forints in advance for a language course, but I still have to pay as much as I would have to pay if the advance had been half of what I still have to pay now. How much does the language course cost in total?
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
K. 860. Anna and Boglárka are playing on a circular table of diameter \(\displaystyle 1.5\) meters. Anna places her rectangular piece of paper of dimensions \(\displaystyle 42\times 30\) cm, and then Boglárka places her identical piece of paper such that two papers are not completely covering each other, but share a diagonal with each other, and neither hangs off the edge of the table. Find the percentage of the table covered by the two pieces of paper.
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
K. 861. A group of eight friends wants to organize a tarot championship. In each game, four players take part, and any pair from the eight friends plays the same number of games. Prove that there will be more than 13 games in the championship.
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
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Problems with sign 'K/C'Deadline expired on June 10, 2025. |
K/C. 862. a) Prove that the fraction below can be simplified (i.e., its numerator and denominator are not relatively prime) for every positive integer \(\displaystyle n\): \(\displaystyle {\frac{6n^2+13n+6}{15n^2+22n+8}}\).
b) Find those positive integers for which the fraction \(\displaystyle {\frac{2n+3}{5n+4}}\) cannot be simplified, i.e., its numerator and denominator are relatively prime.
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
K/C. 863. Anna from problem K. 860. is creating a tulip bed in the middle of her square-shaped garden. In 2D shape \(\displaystyle ABCD\) she is planting tulip bulbs, and she is planting lilies in the remaining part of the garden. The centers of the circular arcs on the perimeter of the bed are the vertices of the square, and their radii equal the side of the square. Find the area of Anna's tulip bed knowing that the area of the whole garden is \(\displaystyle 100~\mathrm{m}^2\).
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
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Problems with sign 'C'Deadline expired on June 10, 2025. |
C. 1858. Tomi numbered 30 cards from 1 to 30 and distributed them into 10 envelopes. In each envelope there are at least 2 and at most 4 cards. On each envelop he wrote the sum of the numbers on the cards contained in the envelop, and arranged them in decreasing order. Find the smallest and the largest possible number, respectively, on the third envelop of the arrangement.
Proposed by: Mátyás Czett, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1859. The side lengths \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) of a non-equilateral triangle satisfy \(\displaystyle a+b=2c\). Prove that the line segment connecting the incenter and the centroid of the triangle is parallel to one of the sides of the triangle.
German competition problem
(5 pont)
solution (in Hungarian), statistics
C. 1860. Let us consider a geometric progression consisting of positive integers. Prove that the contraharmonic mean of any three consecutive terms is an integer. How can the parity of this integer number be determined?
The contraharmonic mean \(\displaystyle C(a_{1},a_{2},\ldots,a_{n})\) of positive real numbers \(\displaystyle a_{1}\), \(\displaystyle a_{2}\), \(\displaystyle \ldots\), \(\displaystyle a_{n}\) is defined as \(\displaystyle C(a_{1},a_{2},\ldots,a_{n})=\frac{a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}}{a_{1}+a_{2}+\ldots+a_{n}}\).
Proposed by: Tamás Unyi, Szada
(5 pont)
solution (in Hungarian), statistics
C. 1861. The base of the pyramid \(\displaystyle ABCDE\) is the square \(\displaystyle ABCD\) of side length 12. We know that the orthogonal projection of the vertex \(\displaystyle E\) on the plane of the base is on the circumcircle of the square \(\displaystyle ABCD\), and also \(\displaystyle BE=CE\) and \(\displaystyle AE=18\). Find the volume of the pyramid.
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1862. \(\displaystyle n\) (\(\displaystyle {>2}\)) children want to play on the two undistinguishable swings of the playground. They have decided that everybody will play together with two other children for two minutes. After each round, the two children have to hand over the swings to another pair who have not yet played on the swings together. (A child handing over the swing to the next pair cannot be in the next pair.) Is the plan of the children feasible?
Based on the idea of László Németh, Fonyód
(5 pont)
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Problems with sign 'B'Deadline expired on June 10, 2025. |
B. 5462. Abel has written down the positive integers from 1 to \(\displaystyle n\) in a certain order. Any positive integer from 1 to \(\displaystyle \frac{n(n+1)}{2}\) can be obtained as the sum of some consecutive terms from Ábel's sequence. For which values of \(\displaystyle n\) is this possible?
Proposed by: Erik Füredi, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5463. The square \(\displaystyle ABCD\) of positive orientation is given. Construct an equilateral triangle \(\displaystyle BEF\) of positive orientation such that its vertex \(\displaystyle F\) is on ray \(\displaystyle BC\) and \(\displaystyle \angle EAF=45^\circ\).
Proposed by: Viktor Vígh, Sándorfalva
(3 pont)
solution (in Hungarian), statistics
B. 5464. Let \(\displaystyle k>1\) be a given integer. Prove that the sequence \(\displaystyle a_n=n^2+n+1\) has infinitely many terms that equal the product of \(\displaystyle k\) distinct terms of the same sequence.
Proposed by: Gábor Holló, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5465. Let \(\displaystyle AA'\), \(\displaystyle BB'\), and \(\displaystyle CC'\) be diameters in the circumcircle of the non-equilateral acute triangle \(\displaystyle ABC\). Let \(\displaystyle A_1\) be the reflection of point \(\displaystyle A'\) across line \(\displaystyle BC\), \(\displaystyle B_1\) be the reflection of \(\displaystyle B'\) across line \(\displaystyle AC\), and \(\displaystyle C_1\) be the reflection of \(\displaystyle C'\) across line \(\displaystyle AB\). Prove that triangle \(\displaystyle A_1B_1C_1\) is similar to triangle \(\displaystyle ABC\).
Proposed by: Szilveszter Kocsis, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5466. The side lengths of a triangle are \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\). Prove that \(\displaystyle \frac{(b+c)^2}{a^2+bc}+\frac{(c+a)^2}{b^2+ca}+\frac{(a+b)^2}{c^2+ab}\ge 6\).
Crux Mathematicorum
(5 pont)
solution (in Hungarian), statistics
B. 5467. Prove that for any odd positive integers \(\displaystyle a\) and \(\displaystyle b\) the remainder of \(\displaystyle (a+b+7)^7-a^7-b^7\) modulo 168 is 7.
Proposed by: Mihály Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5468. Let \(\displaystyle r\) be the inradius of the acute triangle \(\displaystyle ABC\), let \(\displaystyle R\) be the circumradius of the same triangle, and let the inradius of the orthic triangle of the triangle \(\displaystyle ABC\) be \(\displaystyle \varrho\). Prove that \(\displaystyle r \ge \sqrt{R\varrho}\).
Proposed by: Viktor Vígh, Sándorfalva
(6 pont)
solution (in Hungarian), statistics
B. 5469. There are three cities in Bergengócia, \(\displaystyle A\), \(\displaystyle B\), and \(\displaystyle C\), inhabited by a total of 2025 people. Starting from the current year, at the end of each year, half of the inhabitants of \(\displaystyle A\) move to \(\displaystyle B\), a third of the inhabitants of \(\displaystyle B\) move to \(\displaystyle C\), and a quarter of the inhabitants of \(\displaystyle C\) move to \(\displaystyle A\) (if any of the values of these fractions is not an integer, we round the value up). Prove that after a few years, the number of inhabitants of all three cities will remain unchanged.
Proposed by: Benedek Váli, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on June 10, 2025. |
A. 908. Let \(\displaystyle n \geq 2\) be an integer. Little Red Riding Hood and the wolf are playing a game in which the numbers \(\displaystyle \{1, 2, \ldots, n\}\) are being colored red and blue. In each round, Little Red Riding Hood selects a pair \(\displaystyle (i, j)\) such that \(\displaystyle 1 \leq i < j \leq n\), and both \(\displaystyle i\) and \(\displaystyle j\) are still uncolored. Then the wolf colors one of them blue and the other red. The game ends when all but at most one of the numbers from \(\displaystyle \{1, 2, \ldots, n\}\) have been colored. Show that there exists a constant \(\displaystyle c>0\) independent of \(\displaystyle n\), and a strategy for Little Red Riding Hood such that she can ensure that at some point during the game, there exist numbers \(\displaystyle 1 \leq a \leq b \leq n\) such that the set \(\displaystyle \{a, a+1, \ldots, b\}\) contains at least \(\displaystyle c \cdot \sqrt[3]{n}\) more red numbers than blue numbers.
Proposed by: Dömötör Pálvölgyi, Budapest
(7 pont)
A. 909. Prove that for any positive integer \(\displaystyle N\), there exists a positive integer \(\displaystyle n>2\) such that the difference between any two distinct prime factors of \(\displaystyle n^3-1\) is at least \(\displaystyle N\).
(7 pont)
A. 910. Points \(\displaystyle A_1\), \(\displaystyle B_1\), \(\displaystyle C_1\), \(\displaystyle A_2\), \(\displaystyle B_2\), \(\displaystyle C_2\) are collinear in this order. The semicircles with diameters \(\displaystyle A_1A_2\) and \(\displaystyle B_1B_2\) intersect at point \(\displaystyle P\), the semicircles with diameters \(\displaystyle B_1B_2\) and \(\displaystyle C_1C_2\) intersect at point \(\displaystyle Q\), and the semicircles with diameters \(\displaystyle C_1C_2\) and \(\displaystyle A_1A_2\) intersect at point \(\displaystyle R\). Circles \(\displaystyle k\) and \(\displaystyle l\) are tangent to all three semicircles as shown in the figure. Let \(\displaystyle K\) denote the point of tangency of the circle \(\displaystyle k\) and the semicircle with diameter \(\displaystyle A_1A_2\), while let \(\displaystyle L\) denote the point of tangency of the circle \(\displaystyle l\) and the semicircle with diameter \(\displaystyle C_1C_2\).
Prove that \(\displaystyle \frac{A_1R\cdot B_1P\cdot B_2Q\cdot C_2R}{A_2R\cdot B_1Q\cdot B_2P\cdot C_1R}=\frac{A_1K\cdot C_2L}{A_2K\cdot C_1L}\).
(Proposed by: Áron Bán-Szabó, Budapest
(7 pont)
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