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

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Problems with sign 'K'

Deadline expired on March 10, 2022.


K. 719. Each integer on the number line is coloured either red or blue. Is it certain for all possible colourings that

\(\displaystyle a)\) there will be two numbers of the same colour separated by a distance of 3;

\(\displaystyle b)\) there will be two numbers of the same colour separated by a distance of 3 or 4?

(5 pont)

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K. 720. Divide the area of a regular hexagon into three equal parts with two lines passing through the same vertex.

(5 pont)

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K. 721. Alex made some wooden sticks of integer lengths such that no three of them could be used to form a triangle. Given that there were sticks of lengths 1 and 10 and that the longest stick was 100 units long, what is the maximum possible number of sticks that Alex may have made?

(5 pont)

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Problems with sign 'K/C'

Deadline expired on March 10, 2022.


K/C. 722. The arithmetic mean of two three-digit numbers equals the number obtained by writing them next to each other, separated by a decimal point. What may be the two numbers?

(5 pont)

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K/C. 723. The Hungarian Handball Federation nominated 17 players for women's handball in the Tokyo Olympic Games: 3 goalkeepers, 1 right winger, 4 right backs, 2 playmakers, 3 pivots, 2 left backs and 2 left wingers. In how many different ways may the players line up for the anthem if players of the same position must stand together? (During the anthem, players line up next to each other in a single line.)

Proposed by B. Róka Budapest

(5 pont)

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Problems with sign 'C'

Deadline expired on March 10, 2022.


C. 1704. For which real numbers \(\displaystyle a\) will the minimum of the function

\(\displaystyle f(x)=4x^2-4ax+a^2-2a+2 \)

defined on the segment \(\displaystyle [0;2]\) be equal to 3?

(MC&IC)

(5 pont)

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C. 1705. Given that a certain quadrilateral is a kite, it is cyclic and its sides are 42 and 56 units long, what is the distance between the centres of the inscribed and circumscribed circles?

Proposed by A. Siposs, Budapest

(5 pont)

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C. 1706. Prove that in every set of 2022 positive integers there exist two numbers such that their difference or sum is divisible by 4040.

Proposed by L. Sáfár, Ráckeve

(5 pont)

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C. 1707. In a triangle \(\displaystyle ABC\) (with the usual notations) \(\displaystyle b=6\), \(\displaystyle a=2\) and they enclose an angle of \(\displaystyle \gamma=120^{\circ}\). Find the exact length of the interior angle bisector of angle \(\displaystyle \gamma\).

(MC&IC)

(5 pont)

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C. 1708. Solve the following equation over the set of real number pairs:

\(\displaystyle \log_2^2(x+y)+\log_2^2(xy)+1=2\log_2(x+y). \)

(MC&IC)

(5 pont)

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Problems with sign 'B'

Deadline expired on March 10, 2022.


B. 5222. Let \(\displaystyle A\) denote the set of even positive integers for which the sum of the digits decreases by 2 if the number is halved. Let \(\displaystyle B\) denote the set of positive integers for which the sum of the digits increases by 5 if the number is multiplied by 5. What is the number of elements in the set \(\displaystyle A\cap B\) and in the set \(\displaystyle B\setminus A\)?

Proposed by T. Káspári, Paks

(3 pont)

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B. 5223. Define the sequence \(\displaystyle \{a_n\}\) as follows:

\(\displaystyle a_1=-3,\qquad a_{n+1}=4+a_n+4\sqrt{a_n+4}\,. \)

Determine the value of \(\displaystyle a_{2022}\).

Proposed by T. Káspári, Paks

(3 pont)

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B. 5224. \(\displaystyle P\) is a point on side \(\displaystyle BC\) of a unit square \(\displaystyle ABCD\), and \(\displaystyle Q\) is a point on side \(\displaystyle CD\), such that \(\displaystyle \angle PAQ =45^{\circ}\). For which positions of points \(\displaystyle P\) and \(\displaystyle Q\) will the sum \(\displaystyle BP+PQ+QD\) be minimal?

Proposed by J. Szoldatics, Budapest

(4 pont)

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B. 5225. The inscribed circle of triangle \(\displaystyle ABC\) is centred at \(\displaystyle I\) and has a radius \(\displaystyle \varrho\), the radius of the circumscribed circle is \(\displaystyle R\). Prove that if \(\displaystyle \overline{AI}=R\), then the area of the triangle \(\displaystyle ABC\) is \(\displaystyle \frac{a\cdot R}{4}+\varrho \cdot a\), where \(\displaystyle a\) denotes the length of the side opposite to vertex \(\displaystyle A\).

Proposed by Sz. Kocsis, Budapest

(4 pont)

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B. 5226. The length of each side of a triangle is at most 2 units. Each pair of vertices is joined with an arc of a unit circle, not longer than a semicircle. Prove that

\(\displaystyle a'+b'>2c'/3, \)

where \(\displaystyle a'\), \(\displaystyle b'\), \(\displaystyle c'\) denote the lengths of the arcs.

(5 pont)

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B. 5227. Give an example of a positive integer \(\displaystyle k\), along with a finite tree graph \(\displaystyle F\) of at least \(\displaystyle k\) vertices in which the degree of each vertex is at most 3, and \(\displaystyle F\) will fall apart into at least \(\displaystyle 2022\) components if an arbitrary connected subgraph of \(\displaystyle k\) vertices is deleted from \(\displaystyle F\).

Based on a Monthly problem

(6 pont)

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B. 5228. A parabola intersects side \(\displaystyle AB\) of a triangle \(\displaystyle ABC\) at interior points \(\displaystyle C_1\) and \(\displaystyle C_2\), side \(\displaystyle BC\) at interior points \(\displaystyle A_1\) and \(\displaystyle A_2\), and it intersects side \(\displaystyle CA\) at interior points \(\displaystyle B_1\) and \(\displaystyle B_2\). Prove that if \(\displaystyle AC_1 = C_2B\) and \(\displaystyle BA_1 = A_2C\) then \(\displaystyle CB_1=B_2A\).

Proposed by G. Holló, Budapest

(5 pont)

solution (in Hungarian), statistics


B. 5229. Let \(\displaystyle a\ne 0\) be a real number, and let \(\displaystyle f\colon \mathbb{R} \to \mathbb{R}\) be a function, such that

\(\displaystyle f\big(x+f(y)\big) = f(x) + f(y) + ay \)

for all \(\displaystyle x,y\in \mathbb{R}\). Prove that \(\displaystyle f\) is additive, that is, \(\displaystyle f(x+y) = f(x) + f(y)\) for all \(\displaystyle x,y\in \mathbb{R}\).

Proposed by G. Stoica, Saint John, New Brunswick, Canada

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on March 10, 2022.


A. 818. Find all pairs of positive integers \(\displaystyle m\), \(\displaystyle n\) such that \(\displaystyle 9^{|m-n|}+3^{|m-n|}+1\) is divisible by \(\displaystyle m\) and \(\displaystyle n\) simultaneously.

(7 pont)

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A. 819. Let \(\displaystyle G\) be an arbitrarily chosen finite simple graph. We write non-negative integers on the vertices of the graph such that for each vertex \(\displaystyle v\) in \(\displaystyle G\) the number written on \(\displaystyle v\) is equal to the number of vertices adjacent to \(\displaystyle v\) where an even number is written. Prove that the number of ways to achieve this is a power of 2.

(7 pont)

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A. 820. Let \(\displaystyle ABC\) be an arbitrary triangle. Let the excircle tangent to side \(\displaystyle a\) be tangent to lines \(\displaystyle AB\), \(\displaystyle BC\) and \(\displaystyle CA\) at points \(\displaystyle C_a\), \(\displaystyle A_a\) and \(\displaystyle B_a\), respectively. Similarly, let the excircle tangent to side \(\displaystyle b\) be tangent to lines \(\displaystyle AB\), \(\displaystyle BC\) and \(\displaystyle CA\) at points \(\displaystyle B_c\), \(\displaystyle B_a\) and \(\displaystyle B_b\), respectively. Finally, let the excircle tangent to side \(\displaystyle c\) be tangent to lines \(\displaystyle AB\), \(\displaystyle BC\) and \(\displaystyle CA\) at points \(\displaystyle C_c\), \(\displaystyle A_c\) and \(\displaystyle B_c\), respectively. Let \(\displaystyle A'\) be the intersection of lines \(\displaystyle A_bC_b\) and \(\displaystyle A_cB_c\). Similarly, let \(\displaystyle B'\) be the intersection of lines \(\displaystyle B_aC_a\) and \(\displaystyle A_cB_c\), and let \(\displaystyle C'\) be the intersection of lines \(\displaystyle B_aC_a\) and \(\displaystyle A_bC_b\). Finally, let the incircle be tangent to sides \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) at points \(\displaystyle T_a\), \(\displaystyle T_b\) and \(\displaystyle T_c\), respectively.

\(\displaystyle a)\) Prove that lines \(\displaystyle A'A_a\), \(\displaystyle B'B_b\) and \(\displaystyle C'C_c\) are concurrent.

\(\displaystyle b)\) Prove that lines \(\displaystyle A'T_a\) and \(\displaystyle B'T_b\) and \(\displaystyle C'T_c\) are also concurrent, and their point of intersection is on the line defined by the orthocentre and the incentre of triangle \(\displaystyle ABC\).

Submitted by Viktor Csaplár, Bátorkeszi and Dániel Hegedűs, Gyöngyös

(7 pont)

solution (in Hungarian), statistics


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