Mathematical and Physical Journal
for High Schools
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KöMaL Problems in Mathematics, May 2022

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Problems with sign 'C'

Deadline expired on June 10, 2022.


C. 1721. Bonnie listed \(\displaystyle 2022\) numbers such that the ratio of the second number divided by the first number equals the third number on the list, and so on, for example, the seventh number equals the ratio of the sixth number divided by the fifth. What is the last number on Bonnie's list if the first number is \(\displaystyle 20\), and the second number is \(\displaystyle 22\)?

(5 pont)

solution (in Hungarian), statistics


C. 1722. In a quadrilateral \(\displaystyle ABCD\), sides \(\displaystyle AD\) and \(\displaystyle DC\) are equal in length. If \(\displaystyle \alpha\) denotes the angle \(\displaystyle DAB\) then \(\displaystyle \angle ABC=2\alpha\), \(\displaystyle \angle BCD=3\alpha\) and \(\displaystyle \angle CDA=4\alpha\). Prove that side \(\displaystyle AB\) is twice as long as side \(\displaystyle AD\).

(German competition problem)

(5 pont)

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C. 1723. Determine all at most four-digit numbers \(\displaystyle \overline{abcd}\) of distinct digits (allowing \(\displaystyle a=0\), too) for which \(\displaystyle 9\cdot\overline{abcd}=\overline{acbcd}\).

Proposed by A. Siposs, Budapest

(5 pont)

solution (in Hungarian), statistics


C. 1724. In a triangle \(\displaystyle ABC\), \(\displaystyle \angle CAB=30^{\circ}\). Find the measures of the other angles of the triangle, given that the median drawn from vertex \(\displaystyle C\) encloses an angle of \(\displaystyle 45^{\circ}\) with line \(\displaystyle AB\).

(5 pont)

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C. 1725. Let \(\displaystyle p\) denote a positive prime number. Given that the roots of the equation \(\displaystyle x^2-px-580p=0\) are integers, find the value of \(\displaystyle p\).

Proposed by M. Szalai, Szeged

(5 pont)

solution (in Hungarian), statistics


C. 1726. Prove that if \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\) are real numbers such that

\(\displaystyle \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=1, \quad\text{then}\quad \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=0. \)

Find all real numbers satisfying this condition.

(5 pont)

solution (in Hungarian), statistics


C. 1727. In a solid sphere of radius \(\displaystyle R\), a cylindrical bore of radius \(\displaystyle r<R\) is made along a line passing through its centre. Express the volume of the remaining solid in terms of the height \(\displaystyle m\) of the remaining solid.

Proposed by B. Szabó, Miskolc, 1986

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on June 10, 2022.


B. 5246. There are 14 people sitting around a table. Each of them is wearing either a blue shirt or a yellow shirt. What is the maximum possible number of people who have adjacent neighbors with shirts of different color?

(3 pont)

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B. 5247. The ends of a rope are fixed to the ground at two points separated by a distance shorter than the length of the rope. The rope will become taut if its midpoint is raised to a height of 150 cm. The rope will also become taut if a point of the rope 90 cm from one end is raised to a height of 90 cm. How long is the rope?

(3 pont)

solution (in Hungarian), statistics


B. 5248. Solve the following simultaneous equations over the set of real numbers:

$$\begin{align*} \frac{x^2}{y}+\frac{y^2}{x}+x+y & =\frac{8}{xy},\\ x(x+1)+y(y+1) & =6. \end{align*}$$

(4 pont)

solution (in Hungarian), statistics


B. 5249. Let \(\displaystyle T_0\) denote the area of the triangle formed by the points of tangency of the inscribed circle of triangle \(\displaystyle ABC\) on the sides, and let \(\displaystyle T_1\) denote the area of the triangle formed by the centres of the escribed circles. Show that the geometric mean of \(\displaystyle T_0\) and \(\displaystyle T_1\) equals the area of triangle \(\displaystyle ABC\).

Proposed by P. Bártfai

(5 pont)

solution (in Hungarian), statistics


B. 5250. Prove that for all non-negative integers \(\displaystyle n\),

\(\displaystyle 2^{2^{n}(n-2)+n+2}\le (2^n)!\le 2^{2^{n}(n-1)+1}. \)

Proposed by I. Blahota, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics


B. 5251. The vertices of a rectangle \(\displaystyle ABCD\) in the coordinate plane are \(\displaystyle A(0,0)\), \(\displaystyle B(2022,0)\), \(\displaystyle C(2022,2)\), \(\displaystyle D(0,2)\). Consider those triangles of unit area that have all three vertices at lattice points lying on the longer sides of the rectangle. These triangles are to be coloured so that no triangles of the same colour have an interior point in common. What is the minimum number of colours needed?

Proposed by Z. L. Nagy Budapest

(5 pont)

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B. 5252. A polyhedron \(\displaystyle ABCA_1B_1C_1\) has six vertices. Faces \(\displaystyle ABC\) and \(\displaystyle A_1B_1C_1\) are triangles. The edges \(\displaystyle AA_1\), \(\displaystyle BB_1\) and \(\displaystyle CC_1\) are parallel. Faces \(\displaystyle AA_1B_1B\), \(\displaystyle BB_1C_1C\) and \(\displaystyle CC_1A_1A\) are trapeziums in which the diagonals intersect at points \(\displaystyle P\), \(\displaystyle Q\) and \(\displaystyle R\), respectively. Show that the volumes of polyhedra \(\displaystyle ABCPQR\) and \(\displaystyle A_1B_1C_1PQR\) are equal.

Proposed by Sz. Kocsis, Budapest

(6 pont)

solution (in Hungarian), statistics


B. 5253. Is it true that if \(\displaystyle \binom{n}{k}\) is even, then the \(\displaystyle k\)-element subsets of an \(\displaystyle n\)-element set \(\displaystyle S\) can be paired up so that the symmetric difference of every pair should have exactly 2 elements?

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on June 10, 2022.


A. 827. Let \(\displaystyle n>1\) be a given integer. In a deck of cards the cards are of \(\displaystyle n\) different suites and \(\displaystyle n\) different values, and for each pair of a suite and a value there is exactly one such card. We shuffle the deck and distribute the cards among \(\displaystyle n\) players giving each player \(\displaystyle n\) cards. The players' goal is to choose a way to sit down around a round table so that they will be able to do the following: the first player puts down an arbitrary card, and then each consecutive player puts down a card that has a different suite and different value compared to the previous card that was put down on the table. For which \(\displaystyle n\) is it possible that the cards were distributed in such a way that the players cannot achieve their goal? (The players work together, and they can see each other's cards.)

Proposed by Anett Kocsis, Budapest

(7 pont)

solution, statistics


A. 828. Triangle \(\displaystyle ABC\) has incenter \(\displaystyle I\) and excircles \(\displaystyle \Omega_A\), \(\displaystyle \Omega_B\), and \(\displaystyle \Omega_C\). Let \(\displaystyle \ell_A\) be the line through the feet of the tangents from \(\displaystyle I\) to \(\displaystyle \Omega_A\), and define lines \(\displaystyle \ell_B\) and \(\displaystyle \ell_C\) similarly. Prove that the orthocenter of the triangle formed by lines \(\displaystyle \ell_A\), \(\displaystyle \ell_B\), and \(\displaystyle \ell_C\) coincides with the Nagel point of triangle \(\displaystyle ABC\).

(The Nagel point of triangle \(\displaystyle ABC\) is the intersection of segments \(\displaystyle AT_A\), \(\displaystyle BT_B\), and \(\displaystyle CT_C\), where \(\displaystyle T_A\) is the tangency point of \(\displaystyle \Omega_A\) with side \(\displaystyle BC\), and points \(\displaystyle T_B\) and \(\displaystyle T_C\) are defined similarly.)

Proposed by Nikolai Beluhov, Bulgaria

(7 pont)

solution, statistics


A. 829. Let \(\displaystyle G\) be a simple graph on \(\displaystyle n\) vertices with at least one edge, and let us consider those \(\displaystyle S\colon V(G)\to \mathbb{R}^{\ge 0}\) weighings of the vertices of the graph for which \(\displaystyle \sum\limits_{v\in V(G)}S(v)=1\). Furthermore define

\(\displaystyle f(G)=\max_S \min_{(v,w)\in E(G)} S(v)S(w), \)

where \(\displaystyle S\) runs through all possible weighings.

Prove that \(\displaystyle f(G)=\frac{1}{n^2}\) if and only if the vertices of \(\displaystyle G\) can be covered with a disjoint union of edges and odd cycles.

(\(\displaystyle V(G)\) denotes the vertices of graph \(\displaystyle G\), \(\displaystyle E(G)\) denotes the edges of graph \(\displaystyle G\).)

(7 pont)

solution (in Hungarian), statistics


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