KöMaL Problems in Mathematics, January 2025
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Problems with sign 'K'Deadline expired on February 10, 2025. |
K. 839. How many squares can be drawn in the diagram following the grid lines that contain at least one of the small black squares?
(5 pont)
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K. 840. The digital digits are made of small hexagons. For 0 we need six hexagons, for 1 we need two, for 2 we need five etc. How many pairs of consecutive positive integers are there the representation of which using digital digits contain the same number of hexagons?
(5 pont)
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K. 841. We celebrate the \(\displaystyle 125^{\text{th}}\) anniversary of writer István Fekete's birth on 2025.01.25 (YYYY.MM.DD). Number 20250125 is not divisible by 11, and gives a remainder of 2 when divided by 3. How many numbers can be obtained by rearranging the digits of this number that are divisible by 11 and give a remainder of 2 when divided by 3 just as well?
(5 pont)
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Problems with sign 'K/C'Deadline expired on February 10, 2025. |
K/C. 842. How many non-empty subsets of set \(\displaystyle H=\{1; 2; 3; 4; 5, 6; 7; 8; 9\}\) are there where the sum of the elements is even?
(5 pont)
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K/C. 843. Let \(\displaystyle ABC\) be an isosceles right triangle. Let \(\displaystyle D\) be a point on the extension of leg \(\displaystyle AB\) beyond \(\displaystyle B\) satisfying \(\displaystyle BD=AB\). Let \(\displaystyle E\) denote the midpoint of leg \(\displaystyle AC\), and let line segment \(\displaystyle ED\) intersect hypotenuse \(\displaystyle BC\) in \(\displaystyle F\). Find the ratio of the areas of triangles \(\displaystyle AED\) and \(\displaystyle FEC\).
(5 pont)
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Problems with sign 'C'Deadline expired on February 10, 2025. |
C. 1838. We have written numbers 1, 2, 3, \(\displaystyle \ldots\), 2024, 2025 on the blackboard. Farkas and Piroska take turns deleting the numbers from the blackboard until only two numbers remain on the blackboard. Piroska wins if the sum of the last two numbers is divisible by 11. How can Piroska win this game, if she starts the game? (Based on a German competition problem)
(5 pont)
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C. 1839. Two parallel lines and two points, \(\displaystyle A\) and \(\displaystyle B\) on one of them are given. Construct the midpoint of line segment \(\displaystyle AB\) with only a straightedge.
(5 pont)
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C. 1840. Let \(\displaystyle -20.25<K<0\). Solve the following system of equations for real numbers:
$$\begin{gather*} x+y+z=K,\\ x^3+y^3+z^3=1,\\ xyz=-2024. \end{gather*}$$Proposed by: Erzsébet Berkó, Szolnok
(5 pont)
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C. 1841. In the convex quadrilateral \(\displaystyle PQRS\), \(\displaystyle QR=4\), \(\displaystyle RS=6\), \(\displaystyle SP=5\), and the internal angles at vertices \(\displaystyle P\) and \(\displaystyle Q\) are \(\displaystyle 60^{\circ}\). Knowing that \(\displaystyle 2PQ=a+\sqrt{b}\), where \(\displaystyle a\) and \(\displaystyle b\) are positive integers, find the value of sum \(\displaystyle a+b\).
(Australian competition problem)
(5 pont)
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C. 1842. Solve equation \(\displaystyle 9^x+(6x-23)\cdot 3^x+5x^2-39x+76=0\) for real numbers.
Proposed by: Mihály Bencze, Brasov
(5 pont)
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Problems with sign 'B'Deadline expired on February 10, 2025. |
B. 5430. At most how many positive integers can be written on the circumference of a circle such that the product of any two neighbouring numbers is less than \(\displaystyle 2025\)?
Proposed by: Attila Sztranyák, Budapest
(3 pont)
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B. 5431. Let \(\displaystyle k\) denote the incircle of triangle \(\displaystyle ABC\), and let \(\displaystyle I\) denote its center. Line \(\displaystyle AI\) intersects circle \(\displaystyle k\) in points \(\displaystyle D\) and \(\displaystyle E\) such that \(\displaystyle D\) is closer to \(\displaystyle A\) than \(\displaystyle E\). Let \(\displaystyle k\) be tangent to \(\displaystyle AC\) at \(\displaystyle F\). Let the tangent of \(\displaystyle K\) at point \(\displaystyle E\) intersect \(\displaystyle AC\) at \(\displaystyle G\). Prove that \(\displaystyle GI\) is parallel to \(\displaystyle FD\).
Proposed by: Márton Lovas, Budakalász
(3 pont)
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B. 5432. Is it possible to find real numbers \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) such that none of the functions below has a root?
$$\begin{gather*} p_1(x)=ax^2-(b^2+1)x+c,\\ p_2(x)=bx^2-(c^2+1)x+a,\\ p_3(x)=cx^2-(a^2+1)x+b. \end{gather*}$$Based on an idea of Ildikó Kámán, Budapest
(4 pont)
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B. 5433. Prove that in any pyramid with a quadrilateral base the line segments connecting the centroids of the triangular faces to the midpoints of the opposite edges of the base are concurrent.
Proposed by: Géza Kiss, Csömör
(4 pont)
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B. 5434. For which positive integers \(\displaystyle m>1\) does there exist polynomial \(\displaystyle f(x)\) with integer coefficients such that exactly one of \(\displaystyle k\) and \(\displaystyle f(k)\) is divisible by \(\displaystyle m\)? (5 points)
Proposed by: Bálint Hujter, Budapest and Géza Kós, Budapest
(5 pont)
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B. 5435. The incircle of triangle \(\displaystyle ABC\) is tangent to the sides \(\displaystyle BC\), \(\displaystyle CA\), \(\displaystyle AB\) at \(\displaystyle D\), \(\displaystyle E\), \(\displaystyle F\), respectively. Let \(\displaystyle P\) be the point on the segment \(\displaystyle AD\) for which \(\displaystyle 2\cdot AP=PD\), and define points \(\displaystyle Q\), \(\displaystyle R\) on the segments \(\displaystyle BE\) and \(\displaystyle CF\), respectively in a similar manner. Show that at least one of the points \(\displaystyle P\), \(\displaystyle Q\), \(\displaystyle R\) lies on or inside the incircle.
Proposed by: Márton Lovas, Budakalász
(5 pont)
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B. 5436. Let \(\displaystyle p\) be an odd prime. Let \(\displaystyle d_k(n)\) denote the number of positive divisors of \(\displaystyle n\) with remainder \(\displaystyle k\) modulo \(\displaystyle p\) (\(\displaystyle 0\leq k<p\)). Let \(\displaystyle A\subseteq\{0,1,\dots,p-1\}=S\) satisfy
\(\displaystyle \sum_{k\in A}d_k(n)\geq\sum_{j\in S\setminus A}d_j(n) \)
for every positive integer \(\displaystyle n\). Find the minimum number of elements of \(\displaystyle A\).
Proposed by: Márton Lovas, Budakalász
(6 pont)
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B. 5437. Let \(\displaystyle ABC\) be a triangle satisfying \(\displaystyle BC<AC\), and let \(\displaystyle D\) be a point on side \(\displaystyle AC\) satisfying \(\displaystyle BC=DC\). Let circle \(\displaystyle \Gamma\) be tangent to arc \(\displaystyle AC\) not containing \(\displaystyle B\) of the circumcircle of \(\displaystyle ABC\) at point \(\displaystyle X\), and to line segment \(\displaystyle AC\) at \(\displaystyle D\). Let the tangent line from \(\displaystyle B\) to \(\displaystyle \Gamma\) that is closer to \(\displaystyle C\) intersect \(\displaystyle AC\) at \(\displaystyle Y\), and be tangent to \(\displaystyle \Gamma\) at \(\displaystyle Z\). Prove that if \(\displaystyle A\), \(\displaystyle Y\), \(\displaystyle X\) and the circumcenter of triangle \(\displaystyle ABC\) are concyclic, then \(\displaystyle AZ=CZ\).
Proposed by: Márton Lovas, Budakalász
(6 pont)
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Problems with sign 'A'Deadline expired on February 10, 2025. |
A. 896. Marine biologists are studying a new species of shellfish whose first generation consists of 100 shellfish, and their colony reproduces as follows: if a given generation consists of \(\displaystyle N\) shellfish (where \(\displaystyle 5 \mid N\) always holds), they divide themselves into \(\displaystyle N/5\) groups of 5 shellfish each. Each group collectively produces 15 offspring, who form the next generation. Some of the shellfish contain a pearl, but a shellfish can only contain a pearl if none of its direct ancestors contained a pearl. The value of a pearl is determined by the generation of the shellfish containing it: in the \(\displaystyle n\)-th generation, its value is \(\displaystyle 1 / 3^n\). Find the maximum possible total value of the pearls in the colony.
Proposed by: Csongor Beke, Cambridge
(7 pont)
A. 897. Let \(\displaystyle O\) denote the origin and let \(\displaystyle \gamma\) be the circle with center \(\displaystyle (1,0)\) and radius 1 in the Cartesian system of coordinates. Let \(\displaystyle \lambda\) be a real number from the interval \(\displaystyle (0,2)\), and let the line \(\displaystyle x=\lambda\) intersect the circle \(\displaystyle \gamma\) at points \(\displaystyle P\) and \(\displaystyle Q\). The lines \(\displaystyle OP\) and \(\displaystyle OQ\) intersect the line \(\displaystyle x=2-\lambda\) at the points \(\displaystyle P'\) and \(\displaystyle Q'\), respectively. Let \(\displaystyle \mathcal{G}\) denote the locus of such points \(\displaystyle P'\) and \(\displaystyle Q'\) as \(\displaystyle \lambda\) varies over the interval \(\displaystyle (0,2)\). Prove that there exist points \(\displaystyle R\) and \(\displaystyle S\) different from the origin in the plane such that for every \(\displaystyle A\in \mathcal{G}\) there exists a point \(\displaystyle A'\) on line \(\displaystyle OA\) satisfying
\(\displaystyle A'R^2=(A'S-OS)^2=A'A\cdot A'O. \)
Proposed by: Áron Bán-Szabó, Budapest
Attention! A minor discussion error has slipped into the problem statement. The correct form of the statement to be proven is: \(\displaystyle A'R^2 = (A'S \pm OS)^2 = A'A \cdot A'O.\)
(7 pont)
A. 898. Let \(\displaystyle n\) be a given positive integer. Ana and Bob play the following game: Ana chooses a polynomial \(\displaystyle p\) of degree \(\displaystyle n\) with integer coefficients. In each round, Bob can choose a finite set \(\displaystyle S\) of positive integers, and Ana responds with a list containing the values of the polynomial \(\displaystyle p\) evaluated at the elements of \(\displaystyle S\) with multiplicity (sorted in increasing order). Determine, in terms of \(\displaystyle n\), the smallest positive integer \(\displaystyle k\) such that Bob can always determine the polynomial \(\displaystyle p\) in at most \(\displaystyle k\) rounds.
Proposed by: Andrei Chirita, Cambridge
(7 pont)
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