KöMaL Problems in Mathematics, December 2024
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Problems with sign 'K'Deadline expired on January 10, 2025. |
K. 834. In the addition \(\displaystyle \textrm{I}\textrm{T}+\textrm{I}\textrm{T}+\textrm{I}\textrm{T}=\textrm{D}\textrm{I}\textrm{D}\) different letters stand for different numbers, the same letters stand for the same numbers. Find the numerical value of the addends and the result.
(5 pont)
solution (in Hungarian), statistics
K. 835. Determine the number of four digit numbers, in which one of the digits equals the triple of the sum of the other three digits.
(5 pont)
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K. 836. For right triangle \(\displaystyle ABC\) point \(\displaystyle D\) is chosen on the extension of the unit length leg \(\displaystyle AB\) beyond \(\displaystyle B\) with the property \(\displaystyle BD = AC/2\). Let \(\displaystyle E\) denote the midpoint of leg \(\displaystyle AC\). The ratio of the areas of triangles \(\displaystyle AED\) and \(\displaystyle ABC\) is \(\displaystyle 2:3\). Determine the length of line segment \(\displaystyle BD\).
(5 pont)
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Problems with sign 'K/C'Deadline expired on January 10, 2025. |
K/C. 837. We have a \(\displaystyle 4\times 4\) table with a chessboard pattern. In a step we change the color of each square in a \(\displaystyle 2\times 2\) part of the table: black squares become white and white squares become black.
a) Is it possible to turn the whole table into black?
b) And if we start with a \(\displaystyle 5\times 5\) board, is it possible to turn the whole table into black?
(5 pont)
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K/C. 838. Can the difference of the square of two prime numbers be 2024?
(5 pont)
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Problems with sign 'C'Deadline expired on January 10, 2025. |
C. 1833. Solve the system of equations
\(\displaystyle a+c=b,\)
\(\displaystyle a^3-c=b^2,\)
\(\displaystyle a+b=c^3\)
for natural numbers \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\).
Proposed by: Bálint Bíró, Eger
(5 pont)
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C. 1834. Three handsome princes competed for the hand of the beautiful daughter of the Blue King: the Red Prince, the White Prince, and the Green Prince. The first trial was the test of good taste. The princes were each given a regular blue 20-sided polygon and were allowed to color any part of it with their own color, as tastefully as possible. What the princess did not tell them was that she had already decided in advance: anyone who painted more than one-fifth of the 20-sided polygon with their own color would be considered too egotistical and therefore would not be allowed to continue competing for her hand. The princes created the following patterns.
Which of them advanced to the next round of the trials? (Where the task was actually to defeat and eat a seven-headed dragon, but that's another math problem altogether.)
Proposed by: Zoltán Bertalan, Békéscsaba
(5 pont)
solution (in Hungarian), statistics
C. 1835. \(\displaystyle \lfloor x \rfloor\) denotes the floor of number \(\displaystyle x\), and \(\displaystyle \{x\}\) denotes its fractional part. Prove that equation \(\displaystyle {\left[x\right]\cdot\{x\}=\frac{2024}{2025}}\) has an infinite number of solutions on the set of rational numbers.
Proposed by: Mihály Bence, Brassó
(5 pont)
solution (in Hungarian), statistics
C. 1836. Circle \(\displaystyle k\) containing the endpoints of the base \(\displaystyle AB\) of isosceles triangle \(\displaystyle ABC\) touches lines \(\displaystyle AC\) and \(\displaystyle BC\) at \(\displaystyle A\) and \(\displaystyle B\). Let \(\displaystyle P\) be a point of circle \(\displaystyle k\) that is different from points \(\displaystyle A\) and \(\displaystyle B\). Prove that the distance of \(\displaystyle P\) from side \(\displaystyle AB\) is at most the average of its distances from sides \(\displaystyle AC\) and \(\displaystyle BC\).
Proposed by: Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1837. The ancient Greeks were already familiar with the fact that \(\displaystyle \pi\approx 3.1416\) can be approximated with the fraction \(\displaystyle 22/7 \approx 3.1429\). How many pairs of positive integers \(\displaystyle (a;b)\) satisfy the following properties: \(\displaystyle 1<b<100\) and the decimal form of \(\displaystyle \frac{a}{b}\) starts as \(\displaystyle 3.14\)?
Proposed by: Katalin Abigél Kozma, Győr
(5 pont)
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Problems with sign 'B'Deadline expired on January 10, 2025. |
B. 5422. Two natural numbers are relatives if they differ in at most one digit. For example, 135 and 175 are relatives, and so are 101 and 1 (that is, 001), but 135 and 513 are not. Does there exist a number whose relatives are all composite?
Márton Lovas, Budakalász
(3 pont)
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B. 5423. \(\displaystyle \{x\}\) denotes the fractional part of \(\displaystyle x\). Does there exist a positive integer \(\displaystyle n\) for which \(\displaystyle \bigl\{\sqrt{2} n\bigr\}\cdot\bigl\{\frac{n}{\sqrt{2}}\bigr\}\) is rational?
Proposed by: Bálint Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5424. For an arbitrary positive integer \(\displaystyle n\) let \(\displaystyle K_n\) denote the shape obtained by cutting out an \(\displaystyle {(n-1)\times (n-1)}\) square from all four corners of a \(\displaystyle (2n)\times (2n)\) `chessboard' according to the diagrams.
Let \(\displaystyle a_n\) denote the number of ways \(\displaystyle K_n\) can be partitioned into \(\displaystyle 1\times 2\) `dominoes' (for example, \(\displaystyle a_1=2\) and \(\displaystyle a_2=8\)). Prove that \(\displaystyle 2a_n\) is a perfect square for every \(\displaystyle n\).
Proposed by: Attila Sztranyák, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5425. Let \(\displaystyle ABCD\) be a cyclic quadrilateral, let \(\displaystyle E\) be the midpoint of \(\displaystyle AC\) and \(\displaystyle O\) be the centre of the circumcircle such that \(\displaystyle O\) and \(\displaystyle E\) are distinct. Prove that if \(\displaystyle OEBD\) is cyclic then \(\displaystyle EC\) bisects angle \(\displaystyle DEB\).
Proposed by: Ákos Somogyi, London
(4 pont)
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B. 5426. Jumpy, the grasshopper is jumping around on the positive integers of the numberline, visiting each exactly once. Is it possible that the lengths of his jumps produce every positive integer exactly once?
Proposed by: Márton Lovas, Budakalász
(5 pont)
solution (in Hungarian), statistics
B. 5427. Let \(\displaystyle P\) be an internal point in triangle \(\displaystyle ABC\). Let lines \(\displaystyle AP\), \(\displaystyle BP\) and \(\displaystyle CP\) intersect sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) in points \(\displaystyle L\), \(\displaystyle M\) and \(\displaystyle N\), respectively. Prove that \(\displaystyle P\) is the centroid of triangle \(\displaystyle ABC\) if and only if \(\displaystyle P\) is the centroid of triangle \(\displaystyle LMN\).
Crux Mathematicorum
(5 pont)
solution (in Hungarian), statistics
B. 5428. Solve the equation \(\displaystyle 5^{a}+12^{b}=13^{c}\) on non-negative integers.
Proposed by: Ákos Somogyi, London
(6 pont)
solution (in Hungarian), statistics
B. 5429. For every points \(\displaystyle X\), \(\displaystyle Y\), \(\displaystyle Z\), denote by \(\displaystyle [XYZ]\) the area of triangle \(\displaystyle XYZ\). Show that if points \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle E\), \(\displaystyle F\) lie on a non-degenerate conic section then \(\displaystyle {[ABC]\cdot[CDE]\cdot[EFA]\cdot[BDF]=[BCD]\cdot[DEF]\cdot[FAB]\cdot[ACE]}\).
Proposed by: Géza Kós, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on January 10, 2025. |
A. 893. In a text editor program, initially there is a footprint symbol (L) that we want to multiply. Unfortunately, our computer has been the victim of a hacker attack, and only two functions are working: Copy and Paste, each costing \(\displaystyle 1\) Dürer dollar to use. Using the Copy function, we can select one or more consecutive symbols from the existing ones, and the computer memorizes their number. When using the Paste function, the computer adds as many new footprint symbols to the sequence as were selected in the last Copy. If no Copy has been done yet, Paste cannot be used. Let \(\displaystyle D(n)\) denote the minimum number of Dürer dollars required to obtain exactly \(\displaystyle n\) footprint symbols. Prove that for any positive integer \(\displaystyle k\), there exists a positive integer \(\displaystyle N\) such that \(\displaystyle D(N)={D(N+1)}+1={D(N+2)}={D(N+3)}+1={D(N+4)}=\ldots ={D(N+2k-1)}+1={D(N+2k)}\).
Based on a problem of the Dürer Competition
(7 pont)
A. 894. In convex polyhedron \(\displaystyle ABCDE\) line segment \(\displaystyle DE\) intersects the plane of triangle \(\displaystyle ABC\) inside the triangle. Rotate the point \(\displaystyle D\) outward into the plane of triangle \(\displaystyle ABC\) around the lines \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CA\); let the resulting points be \(\displaystyle D_1\), \(\displaystyle D_2\), and \(\displaystyle D_3\). Similarly, rotate the point \(\displaystyle E\) outward into the plane of triangle \(\displaystyle ABC\) around the lines \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CA\); let the resulting points be \(\displaystyle E_1\), \(\displaystyle E_2\), and \(\displaystyle E_3\). Show that if the polyhedron has an inscribed sphere, then the circumcircles of \(\displaystyle D_1D_2D_3\) and \(\displaystyle E_1E_2E_3\) are concentric.
Proposed by: Géza Kós, Budapest
(7 pont)
A. 895. Let's call a function \(\displaystyle f\colon\mathbb{R}\to\mathbb{R}\) weakly periodic if it is continuous and satisfies \(\displaystyle {f(x+1)}=f(f(x))+1\) for all \(\displaystyle x\in \mathbb{R}\). a) Does there exist a weakly periodic function such that \(\displaystyle f(x)>x\) for all \(\displaystyle x\in\mathbb{R}\)? b) Does there exist a weakly periodic function such that \(\displaystyle f(x)<x\) for all \(\displaystyle x\in\mathbb{R}\)?
Proposed by: András Imolay, Budapest
(7 pont)
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