KöMaL Problems in Mathematics, April 2021

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

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C. 1665. Each letter of the word \(\displaystyle \textit{KÖMAL}\) denotes a digit in decimal notation. Given the equalities below, determine the value of the five-digit number \(\displaystyle \overline{\textit{KÖMAL}}\).

$$\begin{align*} M+\textit{Ö}+L & =\overline{KA}, \tag{1}\\ \textit{Ö}+L & =\overline{KK}, \tag{2}\\ K+\textit{Ö}+M & =10, \tag{3}\\ A\cdot{L} & =42. \tag{4} \end{align*}$$

(5 pont)

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C. 1666. In an acute-angled triangle \(\displaystyle ABC\), let \(\displaystyle K\) and \(\displaystyle D\), respectively, be the intersections of the interior angle bisector drawn from point \(\displaystyle A\) with the interior angle bisector drawn from \(\displaystyle B\), and with side \(\displaystyle BC\). The perpendicular drawn to angle bisector \(\displaystyle AD\) at point \(\displaystyle K\) intersects side \(\displaystyle AB\) at point \(\displaystyle E\). \(\displaystyle F\) is the foot of the perpendicular drawn from point \(\displaystyle E\) to \(\displaystyle BC\). \(\displaystyle T\) is the foot of the perpendicular drawn from point \(\displaystyle D\) to line \(\displaystyle AB\). Prove that \(\displaystyle T\) lies on the circumscribed circle of triangle \(\displaystyle KEF\).

(5 pont)

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C. 1667. Let

$$\begin{align*} A & ={(-1)}^1+{(-1)}^2+{(-1)}^3+\dots+{(-1)}^{2021},\\ B & ={(-2)}^1+{(-2)}^2+{(-2)}^3+\dots+{(-2)}^{2021} \end{align*}$$

and

\(\displaystyle C={(-3)}^1+{(-3)}^2+{(-3)}^3+\dots+{(-3)}^{2021}. \)

Determine the last digit of the number \(\displaystyle B+C-A\).

(5 pont)

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C. 1668. The midpoints of sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\) of a parallelogram \(\displaystyle ABCD\) are \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle G\), \(\displaystyle H\), respectively. The lines \(\displaystyle AF\) and \(\displaystyle AG\) intersect diagonal \(\displaystyle BD\) at points \(\displaystyle K\) and \(\displaystyle L\), respectively. Show that the sum of the areas of triangles \(\displaystyle EFK\) and \(\displaystyle GHL\) equals the area of triangle \(\displaystyle EKL\).

(5 pont)

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C. 1669. Let \(\displaystyle N\) be \(\displaystyle \overline{abc}\) a three-digit number in decimal notation. The value of a number \(\displaystyle M=\overline{abc}\) represented in some non-decimal notation is \(\displaystyle 2N\). Determine the number \(\displaystyle N\).

(5 pont)

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C. 1670. Given that \(\displaystyle a\) and \(\displaystyle b\) are integers such that \(\displaystyle 3a-2b\) is divisible by \(\displaystyle 13\), prove that \(\displaystyle 4a+19b\) and \(\displaystyle 38a+57b\) are also divisible by \(\displaystyle 13\).

(5 pont)

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C. 1671. Line segments \(\displaystyle AE\), \(\displaystyle BF\), \(\displaystyle CG\), \(\displaystyle DH\) are perpendicular to the plane \(\displaystyle S\) of parallelogram \(\displaystyle ABCD\), in the same half space formed by plane \(\displaystyle S\). \(\displaystyle T\) and \(\displaystyle t\) denote the areas of quadrilaterals \(\displaystyle CGEA\) and \(\displaystyle DHFB\), respectively. Prove that if

\(\displaystyle \frac{T}{t}=\frac{AC}{BD}, \)

then the points \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle G\), \(\displaystyle H\) are coplanar.

(5 pont)

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

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B. 5166. Are there prime numbers \(\displaystyle p\), \(\displaystyle r\) greater than \(\displaystyle 3\) such that the sum of the digits of \(\displaystyle 2p^2+7r^2+2021\) should be a perfect square?

(3 pont)

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B. 5167. Consider two circles in the plane that have common interior tangents. Show that the circle passing through the points of contact of the internal tangents bisects the line segment connecting the centres of the two original circles.

Proposed by the class 8C of Fazekas Mihály Primary and Secondary Grammar School of Budapest

(3 pont)

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B. 5168. Each of the integers 1 to 100 is written on a piece of paper. 16 pieces of paper are selected out of the 100 pieces. Is it certain that there will always be four pieces of paper among the selected ones such that the sum of the numbers on two of them equals the sum of the numbers on the other two?

(6 pont)

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B. 5169. Find the real solutions of the equation

\(\displaystyle \sqrt[3]{2x+11}+\sqrt[3]{3x+4}=\sqrt[3]{x+9}+\sqrt[3]{4x+6}. \)

Proposed by M. Szalai, Szeged

(5 pont)

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B. 5170. Let \(\displaystyle \alpha\) and \(\displaystyle \beta\) be acute angles such that \(\displaystyle \sin^2\alpha+\sin^2\beta=\sin{(\alpha+\beta)}\). Prove that \(\displaystyle \alpha+\beta=90^{\circ}\).

(4 pont)

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B. 5171. Let \(\displaystyle OLMN\) be a tetrahedron, and the vertices \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) of another tetrahedron \(\displaystyle OABC\) lie on the rays \(\displaystyle OL\), \(\displaystyle OM\) and \(\displaystyle ON\), respectively. The centre of the inscribed circle of triangle \(\displaystyle LMN\) coincides with the centroid of triangle \(\displaystyle ABC\). Show that the volume of tetrahedron \(\displaystyle OLMN\) is greater than or equal to the volume of tetrahedron \(\displaystyle OABC\). On what condition will the volumes of the two tetrahedra be equal?

(From the British qualifying competition for the olympiad, 1980)

(5 pont)

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B. 5172. Six regular dice are placed in a cup, and rolled simultaneously. Those dice that do not show a 6 are returned to the cup, and rolled again. If there are dice that still not show a 6, those dice are rolled a third time. The procedure is repeated until every dice shows a 6. What is the probability that exactly six rolls are needed?

(6 pont)

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B. 5173. The orthocentre of an acute-angled triangle \(\displaystyle ABC\) is \(\displaystyle H\), and the centre of the circumscribed circle is \(\displaystyle O\). Let \(\displaystyle D\) and \(\displaystyle E\) denote interior points on the line segments \(\displaystyle AB\) and \(\displaystyle AC\), respectively. The orthocentre and circumcentre of triangle \(\displaystyle ADE\) are \(\displaystyle H'\) and \(\displaystyle O'\), respectively. Show that lines \(\displaystyle HH'\) and \(\displaystyle OO'\) are parallel if and only if \(\displaystyle BD = CE\).

Proposed by Á. Bán-Szabó, Budapest

(6 pont)

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

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A. 797. We call a system of non-empty sets \(\displaystyle H\) entwined, if for every disjoint pair of sets \(\displaystyle A\) and \(\displaystyle B\) in \(\displaystyle H\) there exists \(\displaystyle b\in B\) such that \(\displaystyle A\cup\{b\}\) is in \(\displaystyle H\) or there exists \(\displaystyle a\in A\) such that \(\displaystyle B\cup\{a\}\) is in \(\displaystyle H\).

Let \(\displaystyle H\) be an entwined system of sets containing the following \(\displaystyle n\) one-element sets: \(\displaystyle \{1\}, \{2\},\dots,\{n\}\). Prove that if \(\displaystyle n>k(k+1)/2\), then \(\displaystyle H\) contains a set with at least \(\displaystyle k+1\) elements, and this is sharp for every \(\displaystyle k\), i.e. if \(\displaystyle n=k(k+1)\), it is possible that every set in \(\displaystyle H\) have at most \(\displaystyle k\) elements.

(7 pont)

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A. 798. Let \(\displaystyle 0<p<1\) be given. Initially we have \(\displaystyle n\) coins, all of which has probability \(\displaystyle p\) of landing on heads, and probability \(\displaystyle 1-p\) landing on tails (the results of the tosses are independent from each other). In each round we toss our coins and remove those that result in heads. We keep repeating this until all our coins are removed. Let \(\displaystyle k_n\) denote the expected number of rounds that was needed to get rid of all the coins. Prove that there exists \(\displaystyle c>0\) for which the following inequality holds for all positive integers \(\displaystyle n\):

\(\displaystyle c\left(1+\frac12+\cdots+\frac1{n}\right)<k_n<1+c\left(1+\frac12+\cdots+\frac1{n}\right). \)

(7 pont)

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A. 799. For a given quadrilateral \(\displaystyle A_1A_2B_1B_2\) point \(\displaystyle P\) is called phenomenal, if line segments \(\displaystyle A_1A_2\) and \(\displaystyle B_1B_2\) subtend the same angle at point \(\displaystyle P\) (i.e. triangles \(\displaystyle PA_1A_2\) and \(\displaystyle PB_1B_2\) which can be also also degenerate have equal inner angles at point \(\displaystyle P\) disregarding orientation).

Three non-collinear points, \(\displaystyle A_1\), \(\displaystyle A_2\) and \(\displaystyle B_1\) are given on the plane. Prove that it is possible to find a disc on the plane such that for every point \(\displaystyle B_2\) on the disc quadrilateral \(\displaystyle A_1A_2B_1B_2\) is convex for which it is possible to construct seven distinct phenomenal points only using a right ruler.

With a right ruler the following two steps are allowed:

\(\displaystyle i)\) given two points it is possible to draw the straight line connecting them;

\(\displaystyle ii)\) given a point and a straight line, it is possible to draw the straight line passing through the given point which is perpendicular to the given line.

Proposed by Á. Bán-Szabó, Budapest

(7 pont)

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