Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, April 2020

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## Problems with sign 'C'

Deadline expired on May 11, 2020.

C. 1602. Two tenth-grade students and two eleventh-grade students sat down to solve the exercises of type C in the April issue of KöMaL. (There are seven exercises each month. Exercises 1–5 are for students in grade 10 at most, while exercises 3–7 may be solved by 11th and 12th grade students.) After an hour, they observed that each exercise was solved by exactly one of them, and that everyone solved at least one exercise. In how many different arrangements may they have solved the exercises? (Two arrangements are considered different if there is at least one exercise that is solved by a different student.)

(5 pont)

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C. 1603. The altitude drawn from vertex $\displaystyle A$ of an isosceles triangle $\displaystyle ABC$ intersects the leg $\displaystyle BC$ at $\displaystyle T$. Let $\displaystyle M$ denote the orthocentre, and let $\displaystyle O$ be the centre of the inscribed circle. Prove that if line $\displaystyle OT$ is parallel to the base $\displaystyle AB$, then $\displaystyle MC=2AM$.

(5 pont)

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C. 1604. A farmer brought 1225 packets of seeds to an agricultural fair: 1 packet of 1 gram of seeds, 2 packets of 2 grams, 3 packets of 3 grams, ..., $\displaystyle k$ packets of $\displaystyle k$ grams of seeds in each – every positive integer $\displaystyle 1$ to $\displaystyle k$ occurred. What was the average mass of seeds in a packet?

(5 pont)

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C. 1605. The diagonals of a convex quadrilateral $\displaystyle ABCD$ intersect at $\displaystyle M$. The area of triangle $\displaystyle ABM$ is greater than the area of triangle $\displaystyle CDM$. The midpoint of side $\displaystyle BC$ of the quadrilateral is $\displaystyle P$, and the midpoint of side $\displaystyle CD$ is $\displaystyle Q$, $\displaystyle AP+AQ=\sqrt2\,$. Prove that the area of quadrilateral $\displaystyle ABCD$ is less than 1.

(5 pont)

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C. 1606. The areas of two faces of a cuboid are 40 and 56 units. The length of the diagonal of the cuboid is $\displaystyle \sqrt{138}$ units of length. Calculate the possible surface area and the volume of the cuboid.

S. Kiss, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics

C. 1607. Between 4 and 9, some digits of 4 are inserted, followed by the same number of digits of 8 (e.g. 4489). Prove that the resulting number is a perfect square.

(5 pont)

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C. 1608. We are making a Vietnamese hat for a costume party. The hat is a right circular cone of apex angle $\displaystyle 97.18^\circ$. The slant height is 28 cm. Is it possible to make a hat like this out of a $\displaystyle 50\times 70$ cardboard sheet available at the stationery store?

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on May 11, 2020.

B. 5094. Prove that if two right-angled triangles have the same perimeter and the same area, then they are congruent.

S. Kiss, Nyíregyháza

(3 pont)

solution (in Hungarian), statistics

B. 5095. Let $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ denote distinct nonzero integers. Prove that if the sum of the three numbers $\displaystyle \frac{ab}{c}$, $\displaystyle \frac{bc}{a}$ and $\displaystyle \frac{ca}{b}$ is an integer, then each of the three numbers is an integer.

(3 pont)

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B. 5096. In a regular triangle $\displaystyle ABC$ of unit sides, let $\displaystyle P$ be an arbitrary point on the circumference of the incircle. Let $\displaystyle D$, $\displaystyle E$, and $\displaystyle F$ denote the orthogonal projections of point $\displaystyle P$ on the sides $\displaystyle BC$, $\displaystyle AC$ and $\displaystyle AB$, respectively. Prove that the area of triangle $\displaystyle DEF$ is a constant, independent of the choice of $\displaystyle P$.

(4 pont)

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B. 5097. The product of the positive numbers $\displaystyle x_1,x_2,\dots,x_n$ is $\displaystyle 1$. Prove that

$\displaystyle x_1^4+x_2^4+\dots+x_n^4 \ge x_1^3+x_2^3+\dots+x_n^3.$

Dinu Ovidiu-Gabriel, Bălcești, Romania

(4 pont)

solution (in Hungarian), statistics

B. 5098. Two players, First and Second, are playing the following game:

First selects a positive integer not greater than 2020, which Second is trying to find out by guessing (by naming a number as a guess).

The possible answers of First are as follows: My number is smaller than that.''; You are right.''; My number is greater than that.''

If the answer is My number is smaller than that'' or You are right'', then Second pays 10 forints (Hungarian currency) to First. If the answer is My number is greater than that'' then he pays 20 forints.

What is the minimum possible amount of money that Second needs to have in order to be certain that he can find out the number, and what strategy should he use?

(The game terminates with the first You are right'' answer, even if Second already knows the number before asking the last question.)

(5 pont)

solution (in Hungarian), statistics

B. 5099. The angle at vertex $\displaystyle A$ of a rhombus $\displaystyle ABCD$ is $\displaystyle 60^\circ$. An ellipse is inscribed in the rhombus, with the axes lying along the diagonals of the rhombus. The points of tangency of the ellipse on sides divide the sides in a ratio $\displaystyle 1:3$. On sides $\displaystyle AB$ and $\displaystyle AD$ it is the point closer to $\displaystyle A$, and on sides $\displaystyle BC$ and $\displaystyle CD$ it is the point closer to $\displaystyle C$. Let some point $\displaystyle P$ move along the ellipse. Draw lines through $\displaystyle P$, parallel to the midlines of the rhombus, and consider the intersections with the other midline. Let these point be $\displaystyle Q$ and $\displaystyle R$. Show that the length of the line segment $\displaystyle QR$ is independent of the position of $\displaystyle P$.

(5 pont)

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B. 5100. Show that it is always possible to select some numbers (at least one) out of $\displaystyle n$ consecutive integers such that their sum is divisible by $\displaystyle (1+2+\dots+n)$.

Based on the idea of B. Kovács and Zs. Várkonyi

(6 pont)

solution (in Hungarian), statistics

B. 5101. $\displaystyle ABCDO$ is a four-sided pyramid, and $\displaystyle P$ is a point in the interior of base $\displaystyle ABCD$. A plane not passing through $\displaystyle O$ cuts the lines $\displaystyle OA$, $\displaystyle OB$, $\displaystyle OC$, $\displaystyle OD$ and $\displaystyle OP$ at points $\displaystyle A'$, $\displaystyle B'$, $\displaystyle C'$, $\displaystyle D'$, and $\displaystyle P'$, respectively. Prove that

$\displaystyle \frac{t_{PAB}\cdot t_{PCD}}{t_{PBC}\cdot t_{PDA}} = \frac{t_{P'A'B'}\cdot t_{P'C'D'}}{t_{P'B'C'}\cdot t_{P'D'A'}},$

where $\displaystyle t_{XYZ}$ denotes the area of triangle $\displaystyle XYZ$.

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on May 11, 2020.

A. 775. Let $\displaystyle H \subseteq \mathbb{R}^3$ such that if we reflect any point in $\displaystyle H$ across another point of $\displaystyle H$, the resulting point is also in $\displaystyle H$. Prove that either $\displaystyle H$ is dense in $\displaystyle \mathbb{R}^3$ or one can find equidistant parallel planes which cover $\displaystyle H$.

Submitted by Árpád Kurusa, Szeged and Vilmos Totik, Szeged

(7 pont)

statistics

A. 776. Let$\displaystyle k>1$ be a fixed odd number, and for non-negative integers $\displaystyle n$ let

$\displaystyle f_n=\sum_{\substack{0\le i\le n\\ k\mid n-2i}}\binom{n}{i}.$

Prove that $\displaystyle f_n$ satisfy the following recursion:

$\displaystyle f_n^2=\sum_{i=0}^{n}\binom{n}{i}f_if_{n-i}.$

Submitted by András Imolay, Budapest

(7 pont)

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