Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, November 2019

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## Problems with sign 'K'

Deadline expired on December 10, 2019.

K. 634. A sheet of graph paper has a grid of unit squares on it. A rectangle is drawn with sides lying along grid lines. Is it possible to draw a closed broken line in the rectangle along grid lines such that it should never leave the rectangle but it should pass through all the grid points in the interior and on the boundary of the rectangle, if the dimensions of the rectangle are

$\displaystyle a)$ $\displaystyle 2019\times2020$ units;

$\displaystyle b)$ $\displaystyle 2018\times2020$ units?

If so, determine the length of the possible broken lines, too.

(6 pont)

solution (in Hungarian), statistics

K. 635. Consider a concave quadrilateral, and draw the diagonal that lies in its interior. The diagonal divides the quadrilateral into two triangles. Prove that the areas of the two triangles are equal if and only if the line of this diagonal bisects the other diagonal.

(6 pont)

solution (in Hungarian), statistics

K. 636. Find all possible values of the digits $\displaystyle x$ and $\displaystyle y$ for which every nonzero digit occurs the same number of times in the prime factorization of the eight-digit number $\displaystyle \overline{xyxyxyxy}$ in decimal notation. (In making the prime factorization, identical prime factors are not written as a power but written down as separate factors.)

(6 pont)

solution (in Hungarian), statistics

K. 637. Let us consider the integer $\displaystyle 12345678901234567890\ldots 1234567890$ consisting of 2020 digits. First we remove the digits at every odd position. Then, from the remaining 1010 digits, we remove the digits at every even position. Then, repeating in the same way, from the remaining 505 digits, we remove the digits at every odd position. This alternating process is continued until a single digit remains. Determine this digit.

(6 pont)

solution (in Hungarian), statistics

K. 638. Fibonacci-type sequences are defined as sequences in which, from the third term onwards, each term is the sum of the preceding two terms. For example, the sequence $\displaystyle 1, 1, 2, 3, 5, 8, \dots$ starting with 1, 1 (the Fibonacci sequence itself), and the sequence $\displaystyle 1, 3, 4, 7, 11, 18, 29, 47, \dots$ starting with 1, 3 are both Fibonacci-type sequences. Find the Fibonacci-type sequence that contains only positive integers, contains 2010 as a term, and has the largest possible number of terms before 2010.

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'C'

Deadline expired on December 10, 2019.

C. 1567. Find the real solutions of the equation

$\displaystyle 2x^2-4xy+4y^2-8x+16=0.$

Proposed by M. Szalai, Szeged

(5 pont)

solution (in Hungarian), statistics

C. 1568. Let $\displaystyle D$ be the midpoint of side $\displaystyle AB$ of a triangle $\displaystyle ABC$, $\displaystyle E$ the midpoint of side $\displaystyle AC$, and $\displaystyle P$, $\displaystyle Q$ the centres of the circumscribed circles of triangles $\displaystyle DEB$ and $\displaystyle DEC$, respectively (assume that $\displaystyle P\ne Q$). Prove that line $\displaystyle PQ$ is perpendicular to line $\displaystyle BC$.

Proposed by D. Hegedűs, Gyöngyös

(5 pont)

solution (in Hungarian), statistics

C. 1569. In a class of 24, there are an odd number of students whose first name is Sophia. When the class is listed in alphabetical order (of family names) and students are numbered in this order, the number of the first Sophia on the list is equal to the number of Sophias in the class, and the number of the third Sophia on the list is three times the number of Sophias in the class. Given that each Sophia on the list is immediately preceded or followed by another Sophia, determine the numbers assigned to all the Sophias on the list.

Based on a problem by L. Hommer, Kemence

(5 pont)

solution (in Hungarian), statistics

C. 1570. In a hexagon, the measure of each angle is $\displaystyle 120^{\circ}$, and the diagonals connecting opposite vertices are equal in length. Prove that the hexagon has rotational symmetry.

Proposed by K. Fried, Budapest

(5 pont)

solution (in Hungarian), statistics

C. 1571. The positive integers from 1 to $\displaystyle n^2$ are written in increasing order in an $\displaystyle n\times n$ table: the numbers 1 to $\displaystyle n$ are entered in the first row, ($\displaystyle n+1$) to $\displaystyle 2n$ in the second row; and so on. Prove that the sum of the numbers in one diagonal equals the sum of the numbers in the other diagonal.

(5 pont)

solution (in Hungarian), statistics

C. 1572. In a trapezium $\displaystyle ABCD$, let $\displaystyle M$ denote the intersection of diagonals $\displaystyle AC$ and $\displaystyle BD$, and let $\displaystyle N$ and $\displaystyle P$ denote the centres of the circumscribed circles of triangles $\displaystyle ABC$ and $\displaystyle ACD$, respectively. Prove that $\displaystyle M$, $\displaystyle N$ and $\displaystyle P$ are collinear if and only if $\displaystyle ABCD$ is a parallelogram or a cyclic trapezium.

(5 pont)

solution (in Hungarian), statistics

C. 1573. Show that the sum

$\displaystyle 12^{2n}+7^{2n-1}+3^{3n}+4^{4n-2}-2^{2n}-11^{2n}$

is divisible by $\displaystyle 23$ for all positive integers $\displaystyle n$.

Proposed by T. Imre, Marosvásárhely

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on December 10, 2019.

B. 5054. Are there positive integers $\displaystyle n$ and $\displaystyle k$ such that

$\displaystyle 20^k+19^k=2019^n-10^n?$

Proposed by T. Imre, Marosvásárhely

(4 pont)

solution (in Hungarian), statistics

B. 5055. Given a circle $\displaystyle k$ in the plane, determine the locus of the orthocentres of all triangles inscribed in $\displaystyle k$.

(3 pont)

solution (in Hungarian), statistics

B. 5056. Consider the quadratic function $\displaystyle f(x)=x^2+bx+c$ defined on the set of real numbers. Given that the zeros of $\displaystyle f$ are some distinct prime numbers $\displaystyle p$ and $\displaystyle q$, and $\displaystyle f(p-q)=6pq$, determine the primes $\displaystyle p$ and $\displaystyle q$, and determine the function $\displaystyle f$.

Proposed by B. Bíró, Eger

(3 pont)

solution (in Hungarian), statistics

B. 5057. Let $\displaystyle D$ and $\displaystyle E$ be points on leg $\displaystyle BC$ of a right-angled triangle with hypotenuse $\displaystyle AB$ such that $\displaystyle \angle DAC = \angle EAD= \angle BAE$. The feet of the perpendiculars dropped from vertex $\displaystyle C$ onto the line segment $\displaystyle AD$ and from point $\displaystyle D$ onto the hypotenuse $\displaystyle AB$ are $\displaystyle F$ and $\displaystyle K$, respectively. Line $\displaystyle CK$ intersects line segment $\displaystyle AE$ at point $\displaystyle H$, and the parallel line drawn from point $\displaystyle H$ to the line $\displaystyle AD$ intersects line segment $\displaystyle BC$ at point $\displaystyle M$. Show that point $\displaystyle F$ is the circumcentre of triangle $\displaystyle CHM$.

Proposed by B. Bíró, Eger

(5 pont)

solution (in Hungarian), statistics

B. 5058. Let $\displaystyle P$ be an arbitrary point in the interior of triangle $\displaystyle ABC$. Lines $\displaystyle AP$, $\displaystyle BP$ and $\displaystyle CP$ intersect sides $\displaystyle BC$, $\displaystyle AC$, and $\displaystyle AB$ at points $\displaystyle A_1$, $\displaystyle B_1$ and $\displaystyle C_1$, respectively. Prove that

$\displaystyle \frac{AP}{A_1P}\cdot\frac{BP}{B_1P}\cdot\frac{CP}{C_1P}\ge 8.$

Proposed by L. Németh, Fonyód

(4 pont)

solution (in Hungarian), statistics

B. 5059. For a positive integer $\displaystyle c$, define the sequence $\displaystyle \{ a_n\}$ by the following recurrence relation: $\displaystyle a_0=c$ and $\displaystyle a_{n+1} = \big[ a_{n} + \sqrt{a_{n}}\,\big]$ for $\displaystyle n \ge 0$. Prove that if $\displaystyle 2019$ occurs as a term of the sequence, then no preceding term is a perfect square, but there are infinitely many perfect squares among the following terms.

(5 pont)

solution (in Hungarian), statistics

B. 5060. In the plane $\displaystyle \Sigma$, given a circle $\displaystyle k$ and a point $\displaystyle P$ in its interior, not coinciding with the center of $\displaystyle k$. Call a point $\displaystyle O$ of space, not lying on $\displaystyle \Sigma$, a proper projection center if there exists a plane $\displaystyle \Sigma'$, not passing through $\displaystyle O$, such that, by projecting the points of $\displaystyle \Sigma$ from $\displaystyle O$ to $\displaystyle \Sigma'$, the projection of $\displaystyle k$ is also a circle, and its center is the projection of $\displaystyle P$. Show that the proper projection centers lie on a circle.

(6 pont)

solution (in Hungarian), statistics

B. 5061. A function $\displaystyle f\colon \mathbb R \to \mathbb R$ is called area preserving if the area of the triangle formed by the points $\displaystyle \big(a,f(a)\big)$, $\displaystyle \big(b,f(b)\big)$ and $\displaystyle \big(c,f(c)\big)$ is equal to the area of the triangle formed by points $\displaystyle \big(a+x,f(a+x)\big)$, $\displaystyle \big(b+x,f(b+x)\big)$ and $\displaystyle \big(c+x,f(c+x)\big)$ for all $\displaystyle a<b<c$ and $\displaystyle x$. Which continuous functions $\displaystyle f$ are area preserving?

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on December 10, 2019.

A. 761. Let $\displaystyle n \ge 3$ be a positive integer. We say that a set $\displaystyle S$ of positive integers is good if $\displaystyle |S| = n$, no element of $\displaystyle S$ is a multiple of $\displaystyle n$, and the sum of all elements of $\displaystyle S$ is not a multiple of $\displaystyle n$ either. Find, in terms of $\displaystyle n$, the least positive integer $\displaystyle d$ for which there exists a good set $\displaystyle S$ such that there are exactly $\displaystyle d$ nonempty subsets of $\displaystyle S$ the sum of whose elements is a multiple of $\displaystyle n$.

Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria

(7 pont)

statistics

A. 762. In a forest there are $\displaystyle n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos?

Proposed by Gábor Mészáros, Sunnyvale, Kalifornia

(7 pont)

statistics

A. 763. Let $\displaystyle k\ge 2$ be an integer. We want to determine the weight of $\displaystyle n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $\displaystyle k$ of the answers can be wrong. Let $\displaystyle f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $\displaystyle f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $\displaystyle a_k$ and $\displaystyle b_k$ for which $\displaystyle \big|f_k(n)-a_kn\big|\le b_k$ holds.

Proposed by Surányi László, Budapest and Bálint Virág, Toronto

(7 pont)

statistics