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

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## Problems with sign 'K'

Deadline expired on November 11, 2019.

K. 629. Seven ducklings are walking towards the lake in a single file: Loppy, Happy, Tappy, Kippy, Boppy, Poppy, and Sippy. They normally walk in the same order every day, but this time they lined up in reverse order. Regarding the present order, the following information is available:

The ducklings preceding Loppy could line up in six different ways in a single file.

The number of ducklings preceding Boppy is the half of those following him.

The number of ducklings between Poppy and Tappy is one fewer than twice the number of those between Sippy and Happy.

Happy and Tappy both walk behind Kippy.

What is the normal order of the ducklings walking to the lake?

(6 pont)

solution (in Hungarian), statistics

K. 630. A party is breaking up, and everyone is going home. To say goodbye, every female participant shakes hands with every other female participant, and every male participant shakes hands with every other male participant. During this process, a friend of the host turns up, who shakes hands with everyone he knows, males and females alike. Given that 5 of the participating men also brought their wives along, and 83 handshakes took place altogether, what may be the number of persons known by the friend of the host?

(6 pont)

solution (in Hungarian), statistics

K. 631. Explain in detail why the following statement is true: if the product of ten positive integers ends in three zeros, then there are six numbers among them such that their product has the same property.

(6 pont)

solution (in Hungarian), statistics

K. 632. A father had a basket of plums. He gave the plums to his sons in the following way: the first son received 2 pieces plus one $\displaystyle n$th of the remaining plums, then the second son received 4 pieces plus one $\displaystyle n$th of the remaining plums, then the third one received 6 pieces plus one $\displaystyle n$th of the remaining plums, and so on. The rest of the plums the father kept for himself. At the end of this process, it turned out that everyone got the same number of plums. Given that the father had at least 2 sons, what may be the value of $\displaystyle n$?

(6 pont)

solution (in Hungarian), statistics

K. 633. Doris thought of an integer, from 3 to 25, inclusive. She told Anna whether the number is a perfect square, whether it is a prime, and whether it is a multiple of 5. From this information, Anna was able to tell which number it was. What may the number be?

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'C'

Deadline expired on November 11, 2019.

C. 1560. Six classes of a school are planning to take trips to the towns of Pécs, Szeged, Debrecen or Miskolc. (Each class is to visit a single town.) Each town must be visited by at least one class. In how many different ways may they select the trip destinations?

(5 pont)

solution (in Hungarian), statistics

C. 1561. What may be the angles of a triangle, if the triangle formed by the points of tangency of the incircle on the sides is similar to the original triangle?

(5 pont)

solution (in Hungarian), statistics

C. 1562. Prove that if $\displaystyle n^2+1$ is divisible by 5 for some integer $\displaystyle n$, then one of the numbers $\displaystyle {(n-1)}^2+1$ and $\displaystyle {(n+1)}^2+1$ is also divisible by 5.

(5 pont)

solution (in Hungarian), statistics

C. 1563. Let us consider one half of an equilateral triangle (that is, the triangle having angles $\displaystyle 30^\circ$, $\displaystyle 60^\circ$ and $\displaystyle 90^\circ$). We create two more triangles by rotating the original one by $\displaystyle 30^\circ$, and the original one by $\displaystyle 60^\circ$ about its right angle in both cases. Determine the area of the intersection of these 3 triangles.

(5 pont)

solution (in Hungarian), statistics

C. 1564. A $\displaystyle 6 \times 6$ square grid is divided into $\displaystyle n$ rectangles of different areas by cutting along grid lines. Give an example of such a dissection for every possible $\displaystyle n>1$.

(5 pont)

solution (in Hungarian), statistics

C. 1565. The sides of a trapezium are $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 5$, and $\displaystyle 6$ units long, in some order. What is the largest possible area of such a trapezium?

(5 pont)

solution (in Hungarian), statistics

C. 1566. Assume that the probability of a newborn baby being a boy is always $\displaystyle p$. In families with two children, is it more common to have one boy and one girl than to have two children of the same sex?

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on November 11, 2019.

B. 5046. Let $\displaystyle n\ge3$, and consider the graph in which the vertices are the grid points $\displaystyle (i,j)$, where $\displaystyle 1\le i,j\le n$, and the distinct points $\displaystyle (i,j)$ and $\displaystyle (k,l)$ are connected by an edge if and only if $\displaystyle i^2+j^2+k^2+l^2$ is divisible by $\displaystyle 3$. For what values of $\displaystyle n$ is it possible to walk the edges of the graph by traversing each edge exactly once?

Proposed by M. Pálfy, Budapest

(4 pont)

solution (in Hungarian), statistics

B. 5047. In a right-angled triangle $\displaystyle ABC$, point $\displaystyle D$ lies in the interior of leg $\displaystyle AC$, and point $\displaystyle E$ lies on the extension of hypotenuse $\displaystyle AB$ beyond $\displaystyle B$. The second intersection of circles $\displaystyle ADE$ and $\displaystyle BCE$ (different from $\displaystyle E$) is $\displaystyle F$. Show that $\displaystyle \angle CFD=90^\circ$.

(4 pont)

solution (in Hungarian), statistics

B. 5048. The base of a pyramid is a convex polygon, and the areas of the lateral faces are equal. Select an arbitrary point on the base, and consider the sum of the distances of this point from the lateral faces. Prove that the sum is independent of the choice of the point on the base.

(Croatian problem)

(3 pont)

solution (in Hungarian), statistics

B. 5049. Prove that there are infinitely many pairs of positive integers $\displaystyle (a,b)$ for which

$\displaystyle 2019 < \frac{2^a}{3^b} < 2020.$

Proposed by S. Róka, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics

B. 5050. Solve the equation

$\displaystyle \cos(3x)+\cos^2x=0.$

(3 pont)

solution (in Hungarian), statistics

B. 5051. The sides of quadrilateral $\displaystyle ABCD$ are $\displaystyle AB=8$, $\displaystyle BC=5$, $\displaystyle CD=17$ and $\displaystyle DA=10$. The intersection of diagonals $\displaystyle AC$ and $\displaystyle BD$ is $\displaystyle E$, and $\displaystyle BE:ED=1:2$. What is the area of the quadrilateral?

Proposed by S. Róka, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics

B. 5052. Two players, First and Second take turns in writing a number 0 or 1 in the fields of a $\displaystyle 19\times 19$ table, initially all blank. When all fields are filled in, they calculate the sum of each row, and the sum of each column. Let the largest row sum be $\displaystyle A$, and let the largest column sum be $\displaystyle B$. If $\displaystyle A>B$ then First wins the game. Second wins if $\displaystyle A<B$, and it is a draw if $\displaystyle A=B$. Does either of the players have a winning strategy?

(6 pont)

solution (in Hungarian), statistics

B. 5053. Let $\displaystyle G$ denote the inscribed sphere of a tetrahedron $\displaystyle ABCD$, and let $\displaystyle G_A$ be the escribed sphere touching the face $\displaystyle BCD$. Let $\displaystyle T$ be the tetrahedron formed by the points of tangency of $\displaystyle G$ on the planes of the faces, and let $\displaystyle T_A$ be the tetrahedron formed by the points of tangency of $\displaystyle G_A$ on the planes of the faces. Show that

$\displaystyle \frac{V^3(T)}{V^3(T_A)} =\frac{V^2(G)}{V^2(G_A)}$

holds for the volumes of the tetrahedra and the spheres.

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on November 11, 2019.

A. 758. In quadrilateral $\displaystyle ABCD$, $\displaystyle AB=BC=\frac1{\sqrt2}DA$, and $\displaystyle \angle ABC$ is a right angle. The midpoint of side $\displaystyle BC$ is $\displaystyle E$, the orthogonal projection of $\displaystyle C$ on $\displaystyle AD$ is $\displaystyle F$, and the orthogonal projection of $\displaystyle B$ on $\displaystyle CD$ is $\displaystyle G$. The second intersection point of circle $\displaystyle BCF$ (with center $\displaystyle H$) and line $\displaystyle BG$ is $\displaystyle K$, and the second intersection point of circle $\displaystyle BHC$ and line $\displaystyle HK$ is $\displaystyle L$. The intersection of lines $\displaystyle BL$ and $\displaystyle CF$ is $\displaystyle M$. The center of the Feurbach circle of triangle $\displaystyle BFM$ is $\displaystyle N$. Prove that $\displaystyle \angle BNE$ is a right angle.

Proposed by Zsombor Fehér, Cambridge

(7 pont)

statistics

A. 759. We choose a random permutation of numbers $\displaystyle 1, 2,\dots, n$ with uniform distribution. Prove that the expected value of the length of the longest increasing subsequence in the permutation is at least $\displaystyle \sqrt{n}\,$.

Proposed by László Surányi, Budapest

(7 pont)

statistics

A. 760. An illusionist and his assistant are about to perform the following magic trick.

Let $\displaystyle k$ be a positive integer. A spectator is given $\displaystyle n = k! + k - 1$ balls numbered $\displaystyle 1, 2,\dots, n$. Unseen by the illusionist, the spectator arranges the balls into a sequence as he sees fit. The assistant studies the sequence, chooses some block of $\displaystyle k$ consecutive balls, and covers them under her scarf. Then the illusionist looks at the newly obscured sequence and guesses the precise order of the $\displaystyle k$ balls he does not see.

Devise a strategy for the illusionist and the assistant to follow so that the trick always works.

(The strategy needs to be constructed explicitly. For instance, it should be possible to implement the strategy, as described by the solver, in the form of a computer program that takes $\displaystyle k$ and the obscured sequence as input and then runs in time polynomial in $\displaystyle n$. A mere proof that an appropriate strategy exists does not qualify as a complete solution.)

Proposed by Nikolai Beluhov, Bulgaria, and Palmer Mebane, USA

(7 pont)

statistics