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

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## Problems with sign 'K'

Deadline expired on April 14, 2020.

K. 654. There were 20 people at a meeting. It turned out that everyone knew exactly 13 of the other participants (acquaintance is mutual). What is the minimum possible number of acquaintances that an arbitrary pair of participants may have in common?

(6 pont)

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K. 655. The four-digit numbers $\displaystyle \overline{ABCD}$, $\displaystyle \overline{BCBA}$, $\displaystyle \overline{BDAB}$ and $\displaystyle \overline{DDAD}$ are distinct four-digit primes (different letters denote different digits). Which numbers are they? You can use website http://matek.com/szamok/primszamok to check if a particular four-digit number is a prime number.

(6 pont)

solution (in Hungarian), statistics

K. 656. Given a 21 cm by 29 cm rectangular sheet of paper, how can you use it to measure a distance of

$\displaystyle a)$ exactly 3 cm,

$\displaystyle b)$ exactly 1 cm,

without using anything else? (It is allowed to fold the sheet of paper.)

(6 pont)

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K. 657. Find all multiples of 99 from 1 to $\displaystyle 10\,000$ in which the sum of the digits is not divisible by 18.

(6 pont)

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K. 658. In each of two rectangular rooms of the same floor area, the floor is covered with $\displaystyle 25~\mathrm{cm}\times40~\mathrm{cm}$ tiles. No tile is cut. In one room, the 40-cm sides of the tiles are parallel to the longer side of the rectangle, and in the other room they are parallel to the shorter side. In one room, there are 9 fewer tiles along the longer wall than in the other room, and 6 more tiles along the shorter wall than in the other.

How long are the sides of the bases of the rooms?

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'C'

Deadline expired on April 14, 2020.

C. 1595. Find all pairs $\displaystyle (x,y)$ of positive integers such that

$\displaystyle \frac1x+\frac1y=\frac{2}{1893}.$

Could have been proposed by Zaphenath Paaneah, Thebes, Egypt

(5 pont)

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C. 1596. The sides of a triangle are 5 cm, 5 cm and 6 cm long. The sides, and the tangents drawn to the incircle parallel to the sides form a hexagon. What is the area of this hexagon?

(5 pont)

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C. 1597. How many different right-angled triangles are there in which the measures of the sides are integers, and one side is $\displaystyle 2^n$ units long? (Where $\displaystyle n$ is a positive integer: express your answer in terms of $\displaystyle n$.)

(5 pont)

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C. 1598. The length of the line segment $\displaystyle MN$ joining the midpoints of sides $\displaystyle AB$ and $\displaystyle CD$ in a convex quadrilateral $\displaystyle ABCD$ is the arithmetic mean of the lengths of sides $\displaystyle AD$ and $\displaystyle BC$. Show that the quadrilateral $\displaystyle ABCD$ is a trapezium.

(5 pont)

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C. 1599. Solve the following equation over the set of pairs of natural numbers:

$\displaystyle 2y^2-2x^2-3xy+3x+y=13.$

Proposed by T. Imre, Marosvásárhely

(5 pont)

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C. 1600. Solve the following equation over the set of real numbers:

$\displaystyle 4^x+9^x+36^x+\sqrt{\frac12-2x^2}=1.$

Proposed by B. Bíró, Eger

(5 pont)

solution (in Hungarian), statistics

C. 1601. The height of a lateral face of a right pyramid with a square base is twice as long as the base edge. At what percentage of this height (counting from the base) do we need to cut the pyramid with a plane parallel to the base so that the total area of the lateral surface plus top square of the resulting frustum is equal to half the lateral surface area of the original pyramid?

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on April 14, 2020.

B. 5086. Solve the equation $\displaystyle \big(x^3-y^2\big)^{2}= \big(x^2-y^3\big)^{2}$ over the set of pairs of integers.

Proposed by M. Szalai, Szeged

(4 pont)

solution (in Hungarian), statistics

B. 5087. The distances of an interior point $\displaystyle P$ of a square $\displaystyle ABCD$ from the vertices $\displaystyle A$, $\displaystyle B$, $\displaystyle D$ are $\displaystyle 1$, $\displaystyle \sqrt2\,$, and $\displaystyle 2$, respectively. Calculate the measure of the angle $\displaystyle APB$.

Proposed by B. Bíró, Eger

(4 pont)

solution (in Hungarian), statistics

B. 5088. With respect to a given set $\displaystyle G$ of numbers, the positive integer $\displaystyle k>1$ is called interesting if there exist $\displaystyle k$ distinct elements in set $\displaystyle G$ such that the arithmetic mean of these elements also belongs to set $\displaystyle G$.

Let $\displaystyle H=\{1;3;4;9;10;\ldots\}$ be the set of those numbers that can be represented as a sum of some different powers of 3.

$\displaystyle a)$ What numbers $\displaystyle k>1$ are interesting with respect to the set $\displaystyle H$?

$\displaystyle b)$ Let $\displaystyle c \notin H$ be an arbitrary positive integer. Prove that every number $\displaystyle k>1$ is interesting with respect to the set $\displaystyle H'=H \cup \{c\}$.

(5 pont)

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B. 5089. Two skew edges of a tetrahedron are perpendicular to each other, their lengths are 12 and 13, and the distance between their lines is 14 units. Determine the volume of the tetrahedron.

(3 pont)

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B. 5090. The inscription on one side of a fair coin is $\displaystyle +1$, and $\displaystyle -1$ is on the other side. The coin is tossed $\displaystyle n$ times in a row, and the $\displaystyle n$ results are written down in a row. Then the product of every pair of consecutive items is written below them, resulting in a new list of numbers that only consists of $\displaystyle (n-1)$ items. The procedure is repeated until a single number remains. What is the expected value of the sum of the $\displaystyle \frac{n(n+1)}{2}$ numbers written down in the resulting triangular arrangement of numbers?

(3 pont)

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B. 5091. In a regular dodecagon $\displaystyle A_1A_2\ldots A_{12}$, let $\displaystyle P$ denote the intersection of the diagonals $\displaystyle A_1A_8$ and $\displaystyle A_6A_{11}$, and let $\displaystyle R$ denote the intersection of lines $\displaystyle A_7A_8$ and $\displaystyle A_9A_{11}$. Show that the line $\displaystyle PR$ divides diagonal $\displaystyle A_1A_4$ in a $\displaystyle 2:1$ ratio.

Based on the idea of B. Bíró, Eger

(5 pont)

solution (in Hungarian), statistics

B. 5092. The sum of the elements is calculated for each subset of the set $\displaystyle \{0,1,\dots,n-1\}$. What may the value of $\displaystyle n$ be if exactly one $\displaystyle n$th of the resulting $\displaystyle 2^n$ sums is divisible by $\displaystyle n$?

(6 pont)

solution (in Hungarian), statistics

B. 5093. The intersection of two congruent regular pentagons is a decagon with sides of $\displaystyle a_1, a_2, \ldots, a_{10}$ in this order. Prove that

$\displaystyle a_1a_3+a_3a_5+a_5a_7+a_7a_9+a_9a_1 = a_2a_4+a_4a_6+a_6a_8+a_8a_{10}+a_{10}a_2.$

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on April 14, 2020.

A. 772. Each of $\displaystyle N$ people chooses a random integer number between 1 and 19 (including 1 and 19, and not necessarily with the same distribution). The random numbers chosen by the people are independent from each other, and it is true that each person chooses each of the 19 numbers with probability at most $\displaystyle 99\%$. They add up the $\displaystyle N$ chosen numbers, and take the remainder of the sum divided by 19. Prove that the distribution of the result tends to the uniform distribution exponentially, i.e. there exists a number $\displaystyle 0<c<1$ such that the mod 19 remainder of the sum of the $\displaystyle N$ chosen numbers equals each of the mod 19 remainders with probability between $\displaystyle 1/19-c^N$ and $\displaystyle 1/19+c^N$.

Submitted by Dávid Matolcsi, Budapest

(7 pont)

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A. 773. Let $\displaystyle b \ge 3$ be a positive integer and let $\displaystyle \sigma$ be a nonidentity permutation of the set $\displaystyle \{0, 1, \ldots, b - 1\}$ such that $\displaystyle \sigma(0) = 0$. The substitution cipher $\displaystyle C_\sigma$ encrypts every positive integer $\displaystyle n$ by replacing each digit $\displaystyle a$ in the representation of $\displaystyle n$ in base $\displaystyle b$ with $\displaystyle \sigma(a)$. Let $\displaystyle d$ be any positive integer such that $\displaystyle b$ does not divide $\displaystyle d$. We say that $\displaystyle C_\sigma$ complies with $\displaystyle d$ if $\displaystyle C_\sigma$ maps every multiple of $\displaystyle d$ onto a multiple of $\displaystyle d$, and we say that $\displaystyle d$ is cryptic if there is some $\displaystyle C_\sigma$ such that $\displaystyle C_\sigma$ complies with $\displaystyle d$.

Let $\displaystyle k$ be any positive integer, and let $\displaystyle p = 2^k + 1$.

$\displaystyle a)$ Find the greatest power of $\displaystyle 2$ that is cryptic in base $\displaystyle 2p$, and prove that there is only one substitution cipher that complies with it.

$\displaystyle b)$ Find the greatest power of $\displaystyle p$ that is cryptic in base $\displaystyle 2p$, and prove that there is only one substitution cipher that complies with it.

$\displaystyle c)$ Suppose, furthermore, that $\displaystyle p$ is a prime number. Find the greatest cryptic positive integer in base $\displaystyle 2p$, and prove that there is only one substitution cipher that complies with it.

Submitted by Nikolai Beluhov, Bulgaria

(7 pont)

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A. 774. Let $\displaystyle O$ be the circumcenter of triangle $\displaystyle ABC$, and $\displaystyle D$ be an arbitrary point on the circumcircle of $\displaystyle ABC$. Let points $\displaystyle X$, $\displaystyle Y$ and $\displaystyle Z$ be the orthogonal projections of point $\displaystyle D$ onto lines $\displaystyle OA$, $\displaystyle OB$ and $\displaystyle OC$, respectively. Prove that the incenter of triangle $\displaystyle XYZ$ is on the Simson-Wallace line of triangle $\displaystyle ABC$ corresponding to point $\displaystyle D$.

Submitted by Lajos Fonyó, Keszthely

(7 pont)

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