KöMaL Problems in Mathematics, May 2021

Problems with sign 'C' Deadline expired on June 10, 2021.

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C. 1672. Find all number pairs \(\displaystyle p\), \(\displaystyle r\) such that \(\displaystyle p\), \(\displaystyle r\) and \(\displaystyle \frac{p+r}{p-r}\) are all positive and prime.

(5 pont)

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C. 1673. A trapezium is divided into four triangles by its diagonals. The sum of the areas of the triangles lying on the bases of the trapezium make up \(\displaystyle \frac{13}{18}\) of the area of the trapezium. Given that the length of one base is 5 cm, what may be the length of the other base?

(5 pont)

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C. 1674. Prove that there are infinitely many right-angled triangles in which the measures of the sides are positive integers, and the hypotenuse is one unit longer than one of the legs.

Proposed by L. Németh, Fonyód

(5 pont)

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C. 1675. Let \(\displaystyle D\) be an interior point of side \(\displaystyle AB\) in triangle \(\displaystyle ABC\), and

\(\displaystyle \frac{AD}{DB}=\frac{m}{n}<\frac{1}{2}, \)

where \(\displaystyle m\), \(\displaystyle n\) are positive integers. A point \(\displaystyle E\), different from \(\displaystyle D\), is marked on the circumference of the triangle such that line \(\displaystyle DE\) divides the area of the triangle in a \(\displaystyle 1:2\) ratio. Depending on the numbers \(\displaystyle m\) and \(\displaystyle n\), on which side of the triangle will point \(\displaystyle E\) lie, and in what ratio will it divide that side?

(5 pont)

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C. 1676. Show that \(\displaystyle 2019^{2021}+2021^{2019}\) is divisible by \(\displaystyle 4040\). Determine whether the following generalization of the problem is also true: if \(\displaystyle a\) and \(\displaystyle b\) are consecutive odd positive integers then \(\displaystyle a^{b}+b^{a}\) is divisible by \(\displaystyle a+b\).

(5 pont)

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C. 1677. Solve the equation

\(\displaystyle \left|2\cdot\log_2\sqrt{x^2-x}+3+\frac{1}{\log_4\sqrt{x^2-x}}\right|=2 \)

over the set of real numbers.

(5 pont)

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C. 1678. The length of each edge of a square-based regular pyramid is \(\displaystyle a\). Connect the centres of the faces of the pyramid in every possible way. Prove that one can always construct a triangle using any three such line segments.

(5 pont)

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Problems with sign 'B' Deadline expired on June 10, 2021.

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B. 5174. Prove that

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

for all positive integers \(\displaystyle n\).

Proposed by M. Szalai, Szeged

(3 pont)

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B. 5175. In a triangle \(\displaystyle ABC\), \(\displaystyle AC=BC\), \(\displaystyle D\) is an interior point of side \(\displaystyle AC\), and \(\displaystyle K\) is the centre of the circle \(\displaystyle ABD\). Show that quadrilateral \(\displaystyle BCDK\) is cyclic.

(3 pont)

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B. 5176. The first \(\displaystyle n\) positive integers need to be written on the circumference of a circle (each number exactly once), so that the sums of all sets of three adjacent numbers should form exactly two different values. Find all possible values of \(\displaystyle n\).

(Scottish competition problem)

(4 pont)

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B. 5177. In a right-angled triangle \(\displaystyle ABC\), line segment \(\displaystyle CD\) is the altitude drawn to the hypotenuse. The circle \(\displaystyle k\) of diameter \(\displaystyle CD\) intersects the legs \(\displaystyle AC\) and \(\displaystyle BC\) again at points \(\displaystyle E\) and \(\displaystyle F\), respectively. The tangent drawn to circle \(\displaystyle k\) at point \(\displaystyle E\) intersects the line of leg \(\displaystyle BC\) at point \(\displaystyle P\), and the hypotenuse \(\displaystyle AB\) at point \(\displaystyle M\). The tangent drawn to circle \(\displaystyle k\) at point \(\displaystyle F\) intersects the line of leg \(\displaystyle AC\) at point \(\displaystyle Q\), and the hypotenuse \(\displaystyle AB\) at point \(\displaystyle N\). Prove that

\(\displaystyle 4\cdot MN^2=PE^2+QF^2+2\cdot EF^2. \)

(5 pont)

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B. 5178. Let \(\displaystyle x\) be a positive real number. Show that

\(\displaystyle \sqrt{6x+9}+\sqrt{16x+64}\le \left(\sqrt{x}+\frac{3}{\sqrt{x}}\right) \left(\sqrt{x}+\frac{8}{\sqrt{x}}\right). \)

Proposed by J. Szoldatics, Budapest

(4 pont)

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B. 5179. Is there a set \(\displaystyle H\) of integers with the following property: every nonzero integer can be represented in infinitely many ways as a sum of some distinct elements of \(\displaystyle H\), but \(\displaystyle 0\) cannot be represented at all?

(6 pont)

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B. 5180. The radius of the circumscribed circle of a regular heptagon \(\displaystyle ABCDEFG\) is \(\displaystyle r\). Prove that the circle of radius \(\displaystyle 2r\) centred at \(\displaystyle A\) passes through the orthocentre of triangle \(\displaystyle BCE\).

(5 pont)

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B. 5181. Given eight points on the plane, no three of which are collinear and no five of which are concyclic, what is the maximum possible number of circles that pass through four of the points each?

Proposed by A. Imolay, Budapest

(6 pont)

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Problems with sign 'A' Deadline expired on June 10, 2021.

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A. 800. In a finite, simple, connected graph \(\displaystyle G\) we play the following game: initially we color all the vertices with a different color. In each step we choose a vertex randomly (with uniform distribution), and then choose one of its neighbors randomly (also with uniform distribution), and color it to the the same color as the originally chosen vertex (if the two chosen vertices already have the same color, we do nothing). The game ends when all the vertices have the same color.

Knowing graph \(\displaystyle G\) find the probability for each vertex that the game ends with all vertices having the same color as the chosen vertex.

Submitted by Dávid Matolcsi, Budapest

(7 pont)

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A. 801. For which values of positive integer \(\displaystyle m\) is it possible to find polynomials \(\displaystyle p,q\in \mathbb{C}[x]\) with degrees at least two such that \(\displaystyle x(x+1)\cdots(x+m-1) =p\big(q(x)\big)\)?

Submitted by Navid Safaei, Tehran

(7 pont)

solution (in Hungarian), statistics

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A. 802. Let \(\displaystyle P\) be a given regular 100-gon. Prove that if we take the union of two polygons that are congruent to \(\displaystyle P\), the ratio of the perimeter and area of the resulting shape cannot be more than the ratio of the perimeter and area of \(\displaystyle P\).

(7 pont)

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