Mathematical and Physical Journal
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KöMaL Problems in Mathematics, November 2021

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Problems with sign 'K'

Deadline expired on December 10, 2021.


K. 704. There were 5 participants in a chess tournament. Each player played every other player once. 1 point was awarded for winning the game, 0.5 point for a draw and 0 for losing. At the end, it turned out that:

– the player finishing in the first place had no draws;

– the player in the second place lost no game;

– each player had a different number of points. Find the score of each player.

(5 pont)

solution (in Hungarian), statistics


K. 705. Three different numbers are chosen from 1, 2, 3, 4, 5, 6, 7, 8 and added. This is performed for every possible selection of three numbers. Some of the sums obtained will be even and some will be odd. Which kind of result will occur more frequently: even or odd?

(5 pont)

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K. 706. Three numbers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) are entered (left to right) in the first row of a three column table. The numbers in the second row are \(\displaystyle a-b\), \(\displaystyle b-c\), \(\displaystyle c-a\). In the third row, the numbers are obtained by the same rule from the second row (the same operations carried out with the numbers of the first, second and third fields), and so on. Show that from the fourth row onwards 2021 cannot occur in the table.

(5 pont)

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Problems with sign 'K/C'

Deadline expired on December 10, 2021.


K/C. 707. A few children (at least two) are standing around a circle. They are playing an ``elimination game'' as follows: counting from the starting player, every second child is eliminated from the circle. The player remaining in the circle alone will win the game. For example, if there are six players A, B, C, D, E, F and A starts then the players eliminated (in this order) are B, D, F, C, A. Thus the winner is E. With how many players can the starting player win the game?

(5 pont)

solution (in Hungarian), statistics


K/C. 708. Jean the butler is instructed by his lord to place candles in the 10 three-prong candle holders in the living room. Jean either needs to put three candles of different colours in each holder, or put 30 candles of the same colour in all of them. Jean goes to the shop on the corner to buy the candles, and finds that the shop only has 70 candles altogether. Show that Jean can buy an appropriate selection of 30 candles.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on December 10, 2021.


C. 1689. Solve the following simultaneous equations for integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\):

$$\begin{align*} a + d & = 9,\\ ad + b & = 8,\\ bd + c & = 74,\\ cd & = 18. \end{align*}$$

Proposed by E. Berkó, Szolnok

(5 pont)

solution (in Hungarian), statistics


C. 1690. The centre of a semicircle of unit radius and diameter \(\displaystyle AB\) is \(\displaystyle O\). The semicircle of diameter \(\displaystyle OB\), centred at \(\displaystyle K\) is drawn inside the larger semicircle. A ray drawn from point \(\displaystyle A\) touches the small semicircle at point \(\displaystyle C\). The perpendicular dropped from point \(\displaystyle O\) to \(\displaystyle AC\) intersects the arc of diameter \(\displaystyle AB\) at a point \(\displaystyle D\). Prove that the midpoint of line segment \(\displaystyle BD\) is \(\displaystyle C\).

(5 pont)

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C. 1691. What are the positive prime numbers \(\displaystyle p\), \(\displaystyle q\) for which \(\displaystyle p^{5}-q^{3}+{(p+q)}^{4}=9900\)?

(5 pont)

solution (in Hungarian), statistics


C. 1692. \(\displaystyle P\) is an interior point of side \(\displaystyle DA\) of a square \(\displaystyle ABCD\). The angle bisector of \(\displaystyle \angle PBC\) intersects side \(\displaystyle CD\) at point \(\displaystyle Q\), and the foot of the perpendicular dropped from point \(\displaystyle Q\) to line \(\displaystyle BP\) is \(\displaystyle R\). Find the angle of the lines \(\displaystyle AR\) and \(\displaystyle BQ\).

(5 pont)

solution (in Hungarian), statistics


C. 1693. Four vertices of a cube are selected at random. Every selection of four vertices is equally probable. What is the probability that the four vertices form a tetrahedron? What is the probability that the four vertices form a regular tetrahedron?

Proposed by N. Zagyva, Baja

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on December 10, 2021.


B. 5198. I have a tortoise, a cat and a dog. If the cat is standing on the floor and I place the tortoise on the table, the head of the tortoise will be 70 cm above the head of the cat. If the dog is standing on the floor and I place the cat on the table, the head of the cat will be 80 cm above the head of the dog. Finally, if the tortoise is standing on the floor and I place the dog on the table, the head of the dog will be 120 cm above the head of the tortoise. How tall is the table?

Based on the idea of Sz. Kocsis, Budapest

(3 pont)

solution (in Hungarian), statistics


B. 5199. A coin is placed on each field of a chessboard, heads facing up. In each move, we can (simultaneously) turn over three adjacent coins in any row or column. Is it possible to achieve an arrangement where all coins show tails on top?

Proposed by M. E. Gáspár, Budapest

(4 pont)

solution (in Hungarian), statistics


B. 5200. The diameter of a semicircular arc is \(\displaystyle A_0A_1=1\). A point \(\displaystyle A_2\) is selected on the arc, such that \(\displaystyle \angle A_0A_1A_2 =1^\circ\). Then a point \(\displaystyle A_3\) is selected on the arc \(\displaystyle A_1A_2\), such that \(\displaystyle \angle A_1A_2A_3 =2^\circ\). The procedure is continued: point \(\displaystyle A_{k+1}\) is selected on the arc \(\displaystyle A_{k-1}A_k\), so that the measure of angle \(\displaystyle A_{k-1}A_kA_{k+1}\) is \(\displaystyle k\) degrees (\(\displaystyle k=3,4,\ldots,9\)). What will be the length of the line segment \(\displaystyle A_9A_{10}\)? (The figure is not to scale.)

(3 pont)

solution (in Hungarian), statistics


B. 5201. Let \(\displaystyle 1=d_1<d_2<\dots<d_k=n\) be the divisors of a positive integer \(\displaystyle n\). Determine those composite numbers \(\displaystyle n\) for which the numbers \(\displaystyle d_1\), \(\displaystyle d_1+d_2\), \(\displaystyle d_1+d_2+d_3\), \(\displaystyle \dots,\) \(\displaystyle d_1+d_2+\dots+d_{k-1}\) all divide \(\displaystyle n\).

Proposed by Cs. Sándor, Budapest

(4 pont)

solution (in Hungarian), statistics


B. 5202. Two rational numbers are said to be acquainted to each other if they can be represented in the forms \(\displaystyle p/q\) and \(\displaystyle r/s\), respectively (\(\displaystyle p\), \(\displaystyle q\), \(\displaystyle r\), \(\displaystyle s\) are integers), such that \(\displaystyle |ps-qr|=1\). If two rational numbers are acquainted to each other, how many common acquaintances may they have?

Proposed by Sz. Kocsis, Budapest

(5 pont)

solution (in Hungarian), statistics


B. 5203. In a triangle \(\displaystyle ABC\), \(\displaystyle AB>BC\), the inscribed circle touches sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) at the points \(\displaystyle A_0\), \(\displaystyle B_0\) és \(\displaystyle C_0\), respectively, and the escribed circle drawn to side \(\displaystyle AC\) touches \(\displaystyle AC\) at point \(\displaystyle B_1\). Show that the intersection of the line segments \(\displaystyle A_0B_1\) and \(\displaystyle B_0C_0\) lies on the interior angle bisector drawn from vertex \(\displaystyle B\) if and only if the angle at vertex \(\displaystyle C\) measures \(\displaystyle 90^{\circ}\).

Proposed by G. Holló, Budapest

(5 pont)

solution (in Hungarian), statistics


B. 5204. Let \(\displaystyle 1\le a, b, c, d\le 4\) denote real numbers. Prove that

\(\displaystyle 16\le (a+b+c+d)\left(\frac{1}a+\frac{1}b+\frac{1}c+\frac{1}d\right)\le 25. \)

Proposed by J. Szoldatics, Budapest

(6 pont)

solution (in Hungarian), statistics


B. 5205. There are four circles given in the plane: circle \(\displaystyle k_2\) lies in the interior of circle \(\displaystyle k_1\), circle \(\displaystyle k_3\) lies in the interior of \(\displaystyle k_2\), and circle \(\displaystyle k_4\) lies in the interior of \(\displaystyle k_3\). Also given are three lines \(\displaystyle e_1\), \(\displaystyle e_2\) and \(\displaystyle e_3\) which are pairwise non-parallel and each line intersects each circle. For all \(\displaystyle i=1,2,3\) let the intersections of line \(\displaystyle e_i\) with the circles be \(\displaystyle A_i\), \(\displaystyle B_i\), \(\displaystyle C_i\), \(\displaystyle D_i\), \(\displaystyle E_i\), \(\displaystyle F_i\), \(\displaystyle G_i\) and \(\displaystyle H_i\), in this order. Prove that if \(\displaystyle A_1B_1+E_1F_1=C_1D_1+G_1H_1\) and \(\displaystyle A_2B_2+E_2F_2=C_2D_2+G_2H_2\) then \(\displaystyle A_3B_3+E_3F_3=C_3D_3+G_3H_3\).

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on December 10, 2021.


A. 809. Let the lengths of the sides of triangle \(\displaystyle ABC\) be denoted by \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) using the standard notations. Let \(\displaystyle S\) denote the centroid of triangle \(\displaystyle ABC\). Prove that for an arbitrary point \(\displaystyle P\) in the plane of the triangle the following inequality is true:

\(\displaystyle a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\ge 3abc\cdot PS. \)

Proposed by János Schultz, Szeged

(7 pont)

solution, statistics


A. 810. For all positive integers \(\displaystyle n\) let \(\displaystyle r_n\) be defined as

\(\displaystyle r_n=\sum_{t=0}^{n} {(-1)}^t \binom{n}{t} \frac{1}{(t+1)!}. \)

Prove that \(\displaystyle \sum_{n=1}^{\infty} r_n=0\).

(7 pont)

solution (in Hungarian), statistics


A. 811. Let \(\displaystyle A\) be a given set with \(\displaystyle n\) elements. Let \(\displaystyle k<n\) be a given positive integer. Find the maximum value of \(\displaystyle m\) for which it is possible to choose sets \(\displaystyle B_i\) and \(\displaystyle C_i\) for \(\displaystyle i=1,2,\dots,m\) satisfying the following conditions:

\(\displaystyle (i)\) \(\displaystyle B_i \subset A\), \(\displaystyle |B_i|=k\),

\(\displaystyle (ii)\) \(\displaystyle C_i \subset B_i\) (there is no additional condition for the number of elements in \(\displaystyle C_i\)),

\(\displaystyle (iii)\) \(\displaystyle B_i \cap C_j \ne B_j \cap C_i\) for all \(\displaystyle i \ne j\).

(7 pont)

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