KöMaL Problems in Mathematics, March 2025
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Problems with sign 'K'Deadline expired on April 10, 2025. |
K. 849. Find the smallest number of matches that can be removed in the figure such that no square remains.
(5 pont)
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K. 850. The teacher has written a two-digit positive prime number and a non-zero digit on the blackboard. Each one of Sanyi, Kati, and Joli has created a three-digit number from the numbers on the blackboard. Sanyi has written the digit behind the prime number written on the blackboard, Kati has written the digit between the digits of the given prime number, and Joli has written the digit in front of the given prime number. Sanyi's number was 45 more than Kati's number and 225 more than Joli's number. Find the prime number and the digit chosen by the teacher.
(5 pont)
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K. 851. Divide the shape on the figure (consisting of five congruent squares) into three parts with two cuts such that the pieces can be combined together to form a rectangle in which one side is twice as long as the other.
(5 pont)
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Problems with sign 'K/C'Deadline expired on April 10, 2025. |
K/C. 852. The sum of the digits of the sum of the squares of three consecutive positive prime numbers is 11. What can be the three prime numbers?
(5 pont)
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K/C. 853. The length of the sides of the bases of a regular hexagon based right prism is \(\displaystyle a\), and the length of its height is \(\displaystyle b\). Let's add together the lengths of all the diagonals of all the bases and all the joining faces of the prism. For what value of \(\displaystyle {\frac{a}{b}}\) will the sum equal \(\displaystyle 12(a\sqrt{3}+3b)\)?
Proposed by: Bálint Róka, Budapest
(5 pont)
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Problems with sign 'C'Deadline expired on April 10, 2025. |
C. 1848. The sequence \(\displaystyle a_n\) is defined in the following way: \(\displaystyle a_1=1\), \(\displaystyle a_2=2\), and for all positive integers \(\displaystyle n\), \(\displaystyle a_{n+2}=a_n^2+a_{n+1}^2\). Find the last digit of the \(\displaystyle 2025^{\text{th}}\) member of the sequence.
Australian competition problem
(5 pont)
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C. 1849. Let \(\displaystyle N_1\), \(\displaystyle N_2\) and \(\displaystyle N_3\) be the quadrisection points on side \(\displaystyle AB\) of convex quadrilateral \(\displaystyle ABCD\) going from \(\displaystyle A\) to \(\displaystyle B\). Let \(\displaystyle M_1\), \(\displaystyle M_2\) and \(\displaystyle M_3\) be the quadrisection points on side \(\displaystyle DC\) going from \(\displaystyle D\) to \(\displaystyle C\). Point \(\displaystyle N_4\) is obtained by extending side \(\displaystyle AB\) beyond point \(\displaystyle B\) with the quarter of the length of side \(\displaystyle AB\). Similarly, point \(\displaystyle M_4\) is obtained by extending side \(\displaystyle DC\) beyond point \(\displaystyle C\) with the quarter of the length of side \(\displaystyle DC\). Determine the area of quadrilateral \(\displaystyle ABCD\) if it's given that the area of quadrilateral \(\displaystyle AN_1M_1D\) is \(\displaystyle 8\) units and the area of quadrilateral \(\displaystyle BN_4M_4C\) is \(\displaystyle 10\) units.
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1850. Prove that if positive numbers \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) satisfy \(\displaystyle abc^6=\frac{b^2}{c^2}=16\), then \(\displaystyle a+4b>16\).
Proposed by: Mátyás Czett, Budapest
(5 pont)
solution (in Hungarian), statistics
C. 1851. The interior angle bisector at vertex \(\displaystyle C\) of triangle \(\displaystyle ABC\) intersects side \(\displaystyle AB\) at \(\displaystyle D\). Find the lengths of the sides of the triangle if \(\displaystyle AD=15\), \(\displaystyle DB=20\), and \(\displaystyle CD=f_{c}=12\sqrt{2}\).
Proposed by: László Németh, Fonyód
(5 pont)
solution (in Hungarian), statistics
C. 1852. Solve the follwing systen of equations on the set of triples of real numbers: \(\displaystyle \log_5(22+x)=\log_3(12-y)\), \(\displaystyle \log_5(22+y)=\log_3(12-z)\), \(\displaystyle \log_5(22+z)={\log_3(12-x)}\).
Proposed by: Mihály Bencze, Brașov
(5 pont)
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Problems with sign 'B'Deadline expired on April 10, 2025. |
B. 5446. What can be the value of \(\displaystyle \ell\), if up to congruence there are exactly three triangles with a side of unit length, another side of length \(\displaystyle \ell\) and an angle of \(\displaystyle 60\) degrees?
Proposed by: Bálint Hujter, Budapest
(3 pont)
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B. 5447. The sum of positive real numbers \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\) is \(\displaystyle 2025\). Find the smallest possible value of \(\displaystyle x^2+y^2+z^2+20x+2y+5z\).
Proposed by: Márton Lovas, Budakalász
(3 pont)
solution (in Hungarian), statistics
B. 5448. We have two congruent regular \(\displaystyle n\)-gon based pyramids. On the lateral faces of both pyramids, the numbers \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle \ldots\), \(\displaystyle n\) are written in a random order. For which values of \(\displaystyle n\) is it guaranteed that the bases of the pyramids can always be glued together in such a way that at least two edges of the resulting bipyramid satisfy the property that the two numbers on either side of the edges are equal to each other?
Proposed by: Márton Lovas, Budakalász
(4 pont)
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B. 5449. Find all pairs of positive integers \(\displaystyle (a,b)\) satisfying \(\displaystyle a^6=2b^2-1\).
Proposed by: Erik Füredi, Budapest
(4 pont)
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B. 5450. Let positive integer \(\displaystyle d\) be called a block-divisor of positive integer \(\displaystyle n\) if \(\displaystyle d\mid n\) and moreover, \(\displaystyle d\) and \(\displaystyle \frac{n}{d}\) are coprimes. Let \(\displaystyle B(n)\) denote the sum of the block-divisors of \(\displaystyle n\). Find all positive integers \(\displaystyle n\) such that \(\displaystyle n\) has at most three distinct prime factors and \(\displaystyle B(n)=2n\).
Proposed by: Norbert Csizmazia, Harkány
(5 pont)
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B. 5451. Convex quadrilateral \(\displaystyle ABCD\) satisfies properties \(\displaystyle \angle BAC=2\angle CAD\), \(\displaystyle \angle ADB=2\angle DBA\), and \(\displaystyle \angle CBD=30^{\circ}\). What can be the measure of \(\displaystyle \angle DCA\)?
Based on the idea of Béla Kovács, Szatmárnémeti
(5 pont)
solution (in Hungarian), statistics
B. 5452. For positive integer \(\displaystyle n\geq 3\) let points \(\displaystyle P_1\), \(\displaystyle P_2\), \(\displaystyle \ldots\), \(\displaystyle P_n\) be chosen on a parabola with focus \(\displaystyle F\) such that \(\displaystyle \angle P_1FP_2=\angle P_2FP_3=\ldots=\angle P_nFP_1=\frac{360^\circ}{n}\). Prove that the harmonic mean of the distances \(\displaystyle FP_1\), \(\displaystyle \ldots\), \(\displaystyle FP_n\) eaquals the parameter of the parabola.
Proposed by: Gábor Holló, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5453. The faces of a convex polyhedron are quadrilaterals \(\displaystyle ABCD\), \(\displaystyle ABFE\), \(\displaystyle CDHG\), \(\displaystyle ADHE\) and \(\displaystyle EFGH\) according to the diagram. The edges from points \(\displaystyle A\) and \(\displaystyle G\), respectively are pairwise perpendicular. Prove that \(\displaystyle [ABCD]^2+[ABFE]^2+[ADHE]^2 = [BCGF]^2+[CDHG]^2+[EFGH]^2,\) where \(\displaystyle [XYZW]\) denotes the area of quadrilateral \(\displaystyle XYZW\).
Proposed by: Géza Kós, Budapest)
(6 pont)
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Problems with sign 'A'Deadline expired on April 10, 2025. |
A. 902. In triangle \(\displaystyle ABC\), interior point \(\displaystyle D\) is chosen such that triangle \(\displaystyle BCD\) is equilateral. Let \(\displaystyle E\) be the isogonal conjugate of point \(\displaystyle D\) with respect to triangle \(\displaystyle ABC\). Define point \(\displaystyle P\) on the ray \(\displaystyle AB\) such that \(\displaystyle AP = BE\). Similarly, define point \(\displaystyle Q\) on the ray \(\displaystyle AC\) such that \(\displaystyle AQ = CE\). Prove that line \(\displaystyle AD\) bisects segment \(\displaystyle PQ\).
Proposed by: Áron Bán-Szabó, Budapest
(7 pont)
A. 903. Let the irrational number \(\displaystyle \alpha={1-\frac1{2a_1-\frac1{2a_2-\frac1{2a_3-\frac1{\dots}}}}}\), where coefficients \(\displaystyle a_1\), \(\displaystyle a_2\), \(\displaystyle \ldots\) are positive integers, infinitely many of which are greater than \(\displaystyle 1\). Prove that for every positive integer \(\displaystyle N\) at least half of the numbers \(\displaystyle \lfloor \alpha \rfloor, \lfloor 2\alpha \rfloor,\ldots , \lfloor N\alpha \rfloor\) are even. \(\displaystyle \lfloor x \rfloor\) denotes the floor function of \(\displaystyle x \), which is the greatest integer less than or equal to \(\displaystyle x\).
Proposed by: Géza Kós, Budapest
(7 pont)
A. 904. Let \(\displaystyle n\) be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular \(\displaystyle 2n\)-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let \(\displaystyle P(n)\) denote the number of different ways Luca can make \(\displaystyle 2n\) jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.) a) Find the value of \(\displaystyle P(n)\) if \(\displaystyle n\) is odd. b) Prove that if \(\displaystyle n\) is even, then \(\displaystyle P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n,d\in \mathbb{N}}\left(\varphi\left (\dfrac{n}{d}\right) \cdot \binom{2d}{d} \right)\), where \(\displaystyle \varphi(k)\) denotes the number of positive integers not greater than \(\displaystyle k\) that are coprime to \(\displaystyle k\).
Proposed by: Péter Csikvári and Kartal Nagy, Budapest
(7 pont)
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