KöMaL Problems in Mathematics, February 2025
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Problems with sign 'K'Deadline expired on March 10, 2025. |
K. 844. In a football championship, 5 teams play a round-robin tournament, where everyone plays against everyone once. A win earns 3 points, a draw earns 1 point, and a loss earns 0 points. At the end of the tournament, the points of four teams are 1, 2, 5, and 7. How many points did the \(\displaystyle 5^{\text{th}}\) team have?
(5 pont)
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K. 845. Fill in a \(\displaystyle 3 \times 3\) table with the numbers 1, 2, 3, 4, 5, 6, 7, 9, and 10 such that the sum of any two adjacent cells (horizontally or vertically) is a prime number. How many different solutions exist for this task? (Two solutions are considered different if there is a number that has different neighbors in one arrangement compared to the other one.)
(5 pont)
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K. 846. The LIGO company’s new building toy consists of identical building blocks. Each building block consists of four glued-together small cubes with 2 cm edges. What is the maximum number of building blocks that can fit into a box with dimensions \(\displaystyle 6~\mathrm{cm}\times 6~\mathrm{cm}\times 8~\mathrm{cm}\)?
(5 pont)
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Problems with sign 'K/C'Deadline expired on March 10, 2025. |
K/C. 847. How many non-empty subsets of the set \(\displaystyle \{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) have the property that the product of its elements is even and also the sum of its elements is even?
(5 pont)
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K/C. 848. Trapezoid \(\displaystyle ABCD\) is inscribed in a circle with radius 10 such that \(\displaystyle AB\) is the diameter of the circle and \(\displaystyle \angle ABC=75^{\circ}\). Calculate the area of trapezoid \(\displaystyle ABCD\).
(5 pont)
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Problems with sign 'C'Deadline expired on March 10, 2025. |
C. 1843. Boglárka has drawn a rectangle with sides of lengths \(\displaystyle 75\) cm and \(\displaystyle 105\) cm. She divided the rectangle into \(\displaystyle 75 \cdot 105= 7875\) squares of area \(\displaystyle 1~\mathrm{cm}^{2}\), and she also drew one of the diagonals of the rectangle. How many small squares does the diagonal cross?
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
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C. 1844. Ági uses red and Laci uses blue to color the squares in an \(\displaystyle n\times n\) white board. Let \(\displaystyle (i,j)\) denote the square in the \(\displaystyle i^{\text{th}}\) row and \(\displaystyle j^{\text{th}}\) coloumn. In the first round Ági colors the squares in the main diagonal (from the top left to the bottom right) red. Now they take turns: if Laci colors square \(\displaystyle (i,j)\) blue, then Ági colors \(\displaystyle (j,i)\) red. They color every square exactly once. The \(\displaystyle k^{\text{th}}\) row is special if for every blue \(\displaystyle (k,j)\) there exists \(\displaystyle l\) such that both \(\displaystyle (k,l)\) and \(\displaystyle (l,j)\) is red. Prove that after finishing the coloring Ági is guaranteed to find a special row.
Proposed by: Zoltán Paulovics, Budapest
(5 pont)
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C. 1845. Ezekiel multiplied two integers together. One of the factors was 74 greater than the other. He made a mistake during the multiplication, as he accidentally wrote a digit that was 3 less than it should have been in the tens place of the product. When checking the multiplication by dividing by the smaller factor, he got exactly 61 as the quotient. What could the two numbers have been?
Proposed by: Gergely Sánta, Budapest
(5 pont)
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C. 1846. What ratio of the five-digit palindromes that are divisible by \(\displaystyle 11\) are not divisible by \(\displaystyle 121\)? (Palindromes are positive integers which have the same value when read backwards.)
Proposed by: László Németh, Fonyód
(5 pont)
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C. 1847. Let point \(\displaystyle P\) be chosen on side \(\displaystyle AD\) of square \(\displaystyle ABCD\) such that \(\displaystyle \angle CPA=105^{\circ}\). Let \(\displaystyle Q\) be the foot of the perpendicular from \(\displaystyle A\) to \(\displaystyle CP\). Find the exact value of the ratio of the areas of triangles \(\displaystyle ABQ\) and \(\displaystyle ACP\).
Proposed by: Bálint Bíró, Eger
(5 pont)
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Problems with sign 'B'Deadline expired on March 10, 2025. |
B. 5438. How many different results can we get by adding together two different \(\displaystyle n\)-digit numbers, if each digit in both numbers can only be \(\displaystyle 4\) or \(\displaystyle 7\)?
Proposed by: Attila Sztranyák, Budapest
(3 pont)
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B. 5439. Rectangle \(\displaystyle ABCD\) satisfies \(\displaystyle AD<AB<2AD\). Let point \(\displaystyle O\) be chosen on side \(\displaystyle AB\) satisfying \(\displaystyle OB=AD\). The circle with center \(\displaystyle O\) and radius \(\displaystyle OB\) intersects side \(\displaystyle AD\) at point \(\displaystyle E\). Prove that the area of rectangle \(\displaystyle ABCD\) equals \(\displaystyle BE^2/2\).
Proposed by: Viktor Vígh, Sándorfalva
(3 pont)
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B. 5440. We extended the sides of a triangle in both directions along the lines of the sides, and at each vertex the length of the two extensions equals the length of the side opposite the vertex. Prove the resulting six points are concyclic.
Proposed by: Sándor Róka, Nyíregyháza
(4 pont)
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B. 5441. Let \(\displaystyle \alpha\), \(\displaystyle \beta\) and \(\displaystyle \gamma\) denote the angles of a triangle. Prove that \(\displaystyle \sin \frac{\alpha}{2}+\sin \frac{\beta}{2}+\sin \frac{\gamma}{2} \geq \cos \alpha+\cos \beta+\cos \gamma\).
Proposed by: Gábor Holló, Budapest
(4 pont)
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B. 5442. Let \(\displaystyle n\), \(\displaystyle k\) and \(\displaystyle \ell\) denote positive integers, and let \(\displaystyle g(n,k,\ell)\) denote the number of ways that an \(\displaystyle n\times k\) table can be filled with the elements of \(\displaystyle \{1,2,\ldots, {\ell+1}\}\) such that in each row from left to right and in each column from top to bottom the numbers are monotonously increasing. Prove that \(\displaystyle g(n,k,\ell)=g(k,\ell,n)\).
Proposed by: Zoltán Gyenes, Budapest
(5 pont)
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B. 5443. For an arbitrary positive integer \(\displaystyle n\) let \(\displaystyle a_n\) denote the number of positive perfect powers that are at most \(\displaystyle n\) (for example, \(\displaystyle a_9=4\)). We call \(\displaystyle n\) `interesting', if \(\displaystyle a_n \mid n\). Prove that there exists infinitely many interesting positive integers.
Proposed by: Attila Sztranyák, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5444. In cyclic hexagon \(\displaystyle ABCDEF\) let \(\displaystyle P\) denote the intersection of diagonals \(\displaystyle AD\) and \(\displaystyle CF\), and let \(\displaystyle Q\) denote the intersection of diagonals \(\displaystyle AE\) and \(\displaystyle BF\). Prove that if \(\displaystyle BC=CP\) and \(\displaystyle DP=DE\), then \(\displaystyle PQ\) bisects angle \(\displaystyle BQE\).
Proposed by: Géza Kós, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5445. Decide whether the following statement is true: if an infinite arithmetic sequence of positive integers includes both a perfect square and a perfect cube, then it also includes a perfect \(\displaystyle 6^{\text{th}}\) power.
Proposed by: Sándor Róka, Nyíregyháza
(6 pont)
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Problems with sign 'A'Deadline expired on March 10, 2025. |
A. 899. The world famous infinite hotel with infinitely many floors (where the floors and the rooms on each floor are numbered with the positive integers) is full of guests: each room is occupied by exactly one guest. The manager of the hotel wants to carpet the corridor on each floor, and an infinite set of carpets of finite length (numbered with the positive integers) was obtained. Every guest marked an infinite number of carpets that they liked. Luckily, any two guests living on a different floor share only a finite number of carpets that they both like. Prove that the carpets can be distributed among the floors in a way that for every guest there are only finitely many carpets they like that are placed on floors different from the one where the guest is.
Proposed by: András Imolay, Budapest
(7 pont)
A. 900. In a room, there are \(\displaystyle n\) lights numbered with positive integers \(\displaystyle 1\), \(\displaystyle 2\), \(\displaystyle \ldots\), \(\displaystyle n\). At the beginning of the game subsets \(\displaystyle S_1\), \(\displaystyle S_2\), \(\displaystyle \ldots\), \(\displaystyle S_k\) of \(\displaystyle \{1,2,\ldots,n\}\) can be chosen. For every integer \(\displaystyle 1\leq i\leq k\), there is a button that turns on the lights corresponding to the elements of \(\displaystyle S_i\) and also a button that turns off all the lights corresponding to the elements of \(\displaystyle S_i\). For any positive integer \(\displaystyle n\), determine the smallest \(\displaystyle k\) for which it is possible to choose the sets \(\displaystyle S_1\), \(\displaystyle S_2\), \(\displaystyle \ldots\), \(\displaystyle S_n\) in such a way that allows any combination of the \(\displaystyle n\) lights to be turned on, starting from the state where all the lights are off.
Proposed by: Kristóf Zólomy, Budapest
(7 pont)
A. 901. Let \(\displaystyle A'B'C'\) denote the reflection of scalene and acute triangle \(\displaystyle ABC\) across its Euler-line. Let \(\displaystyle P\) be an arbitrary point of the nine-point circle of \(\displaystyle ABC\). For every point \(\displaystyle X\), let \(\displaystyle p(X)\) denote the reflection of \(\displaystyle X\) across \(\displaystyle P\). a) Let \(\displaystyle e_{AB}\) denote the line connecting the orthogonal projection of \(\displaystyle A\) to line \(\displaystyle BB'\) and the orthogonal projection of \(\displaystyle B\) to line \(\displaystyle AA'\). Lines \(\displaystyle e_{BC}\) and \(\displaystyle e_{CA}\) are defined analogously. Prove that these three lines are concurrent (and denote their intersection by \(\displaystyle K\)). b) Prove that there are two choices of \(\displaystyle P\) such that lines \(\displaystyle Ap(A')\), \(\displaystyle Bp(B')\) and \(\displaystyle Cp(C')\) are concurrent, and the four points \(\displaystyle p(A)p(A')\cap BC\), \(\displaystyle p(B)p(B')\cap CA\), \(\displaystyle p(C)p(C')\cap AB\), and \(\displaystyle K\) are collinear.
Proposed by: Áron Bán-Szabó, Budapest
(7 pont)
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