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

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## Problems with sign 'K'

Deadline expired on November 10, 2020.

K. 664. We have six coins, four of which weigh 100 grams each, and the remaining two weigh 99 grams each. With the help of an equal-arm balance and no weights, what is the minimum number of measurements that are sufficient to identify one of the lighter coins?

(6 pont)

solution (in Hungarian), statistics

K. 665. Some toy robots are lining up on one side of a street. In one move, we can instruct exactly three robots to cross the street. For what number of robots can we make all the robots line up on the opposite side?

(6 pont)

solution (in Hungarian)

K. 666. How many six-digit multiples of 182 are there in which the three-digit number formed by the first three digits is equal to the three-digit number formed by the last three digits?

(6 pont)

solution (in Hungarian)

K. 667. Start with a positive integer. In each move, take the half of the number if it is even, or add 1 to the number if it is odd. The sequence of moves terminates if it reaches the number 1.

$\displaystyle a)$ Is it true that whatever the starting number is, it is always possible to reach 1 sooner or later (with a finite number of moves)?

$\displaystyle b)$ Is it true that at most 30 moves are sufficient to reach 1 if we start from a four-digit number?

(6 pont)

solution (in Hungarian)

K. 668. $\displaystyle a)$ How many isosceles triangles are there for which the length of the legs is 13 cm and the area is 60 cm$\displaystyle {}^2$?

$\displaystyle b)$ How many right-angled triangles are there for which the legs are even integers, and the area is 60 cm$\displaystyle {}^2$?

(6 pont)

solution (in Hungarian)

## Problems with sign 'C'

Deadline expired on November 10, 2020.

C. 1623. Let $\displaystyle m$ be a positive integer. Show that

$\displaystyle a)$ there exist three $\displaystyle m$-digit powers of 2;

$\displaystyle b)$ there exist at most four $\displaystyle m$-digit powers of 2.

(Brazilian problem)

(5 pont)

solution (in Hungarian)

C. 1624. Point $\displaystyle P$ of side $\displaystyle AB$ in a square $\displaystyle ABCD$ is connected to $\displaystyle D$, and point $\displaystyle Q$ of side $\displaystyle BC$ is connected to $\displaystyle A$. The intersection of the resulting line segments is denoted by $\displaystyle R$. The area of triangle $\displaystyle ARD$ is 1200, the area of triangle $\displaystyle APR$ is 600, and the area of quadrilateral $\displaystyle PBQR$ is $\displaystyle 3380-240\sqrt{95}$ units of area. What is the area of quadrilateral $\displaystyle RQCD$?

Proposed by L. Németh, Fonyód

(5 pont)

solution (in Hungarian)

C. 1625. Prove that every selection of five one-digit positive integers contains a few numbers whose sum is divisible by 10.

(5 pont)

solution (in Hungarian)

C. 1626. Let $\displaystyle F$ denote the midpoint of side $\displaystyle BC$ in an acute-angled triangle $\displaystyle ABC$, and let $\displaystyle T$ be the foot of the altitude drawn from $\displaystyle B$. Prove that if $\displaystyle \angle FAC =30^{\circ}$ then $\displaystyle AF=BT$.

Based on the idea of S. Róka, Nyíregyháza

(5 pont)

solution (in Hungarian)

C. 1627. Prove that if $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ are real numbers, such that $\displaystyle a+b+c>0$, $\displaystyle ab+bc+ca>0$ and $\displaystyle abc>0$, then $\displaystyle a>0$, $\displaystyle b>0$ and $\displaystyle c>0$.

Proposed by S. Róka, Nyíregyháza

(5 pont)

solution (in Hungarian)

C. 1628. Find two distinct positive integers $\displaystyle n$ for which $\displaystyle 4^n+4^9+4^{100}$ is a perfect square.

(5 pont)

solution (in Hungarian)

C. 1629. A sphere passes through four vertices of one face of a cube, and is tangent to the opposite face. Determine the radius of the sphere if the edge of the cube is 8 units long.

(Croatian problem)

(5 pont)

solution (in Hungarian)

## Problems with sign 'B'

Deadline expired on November 10, 2020.

B. 5118. Is it possible that $\displaystyle x$, $\displaystyle \frac{14x+5}{9}$ and $\displaystyle \frac{17x-5}{12}$ are all integers?

(3 pont)

solution (in Hungarian)

B. 5119. In an acute-angled triangle $\displaystyle ABC$, a tangent is drawn to the inscribed circle, parallel to side $\displaystyle BC$. The tangent intersects side $\displaystyle AC$ at point $\displaystyle D$. $\displaystyle F$ is the orthogonal projection of point $\displaystyle D$ onto the side $\displaystyle BC$. Show that $\displaystyle AB=AD+BF$.

(3 pont)

solution (in Hungarian)

B. 5120. The positive integers are coloured in the following manner: the colour of $\displaystyle a+b$ is always uniquely determined by the colours of $\displaystyle a$ and $\displaystyle b$; that is, if the colour of $\displaystyle a$ and $\displaystyle a'$ is the same, and the colour of $\displaystyle b$ and $\displaystyle b'$ is the same, then $\displaystyle a+b$ and $\displaystyle a'+b'$ also have the same colour. Prove that if there is a colour that is used more than once then the colouring becomes periodic from some number onwards.

(4 pont)

solution (in Hungarian)

B. 5121. Solve the following simultaneous equations, where $\displaystyle x_1, x_2,\ldots,x_n$ are positive real numbers, and $\displaystyle n$ is a positive integer:

\begin{align*} x_1+x_2+\ldots +x_n & =9,\\ \frac1{x_1}+\frac1{x_2}+\ldots +\frac1{x_n} & =1. \end{align*}

(4 pont)

solution (in Hungarian)

B. 5122. ErWin Layup is the best penalty taker of all times in the basketball league of Nowhereland. Although he missed the very first penalty throw of his career, altogether he has only missed 2020 out of his total of $\displaystyle 222\,222$ throws.

Statisticians in Nowhereland consider a basketball penalty throw interesting if the ratio of successful penalty throws to all penalty throws, calculated immediately after the throw and expressed as a percentage, is a positive integer. (For example, if a player scores 12 out of a total of 40 throws then his last throw is interesting, since $\displaystyle \frac{12}{40} \cdot 100 = 30 \in \mathbb{N}^+$, while the following throw, which is the 41st, cannot be interesting, whether successful or not.)

What is the minimum number of interesting penalty throws that ErWin Layup may have had?

(5 pont)

solution (in Hungarian)

B. 5123. Ann and Barbara divided between themselves the 81 cards of the game of SET. Ann received 40 cards and Barbara received 41. Each girl counted the number of ways they can form a SET of three cards out of the cards held by her. What may be the sum of the numbers they obtained?

(6 pont)

solution (in Hungarian)

B. 5124. The base of a right pyramid is a square $\displaystyle ABCD$, and the apex of the pyramid is $\displaystyle E$. The skew edges $\displaystyle AB$ and $\displaystyle CE$ are connected by a transversal that is normal to both of them. The feet of the normal transversal are point $\displaystyle P$ on the line segment $\displaystyle AB$, and point $\displaystyle Q$ on the line segment $\displaystyle CE$. Given that $\displaystyle Q$ bisects the edge $\displaystyle CE$, determine the ratio $\displaystyle AP: PB$, and calculate the angle enclosed between the lateral faces and the base of the pyramid.

(5 pont)

solution (in Hungarian)

B. 5125. The centre of the circumscribed circle of a cyclic quadrilateral $\displaystyle ABCD$ is $\displaystyle O$. The rays $\displaystyle AB$ and $\displaystyle DC$ intersect at point $\displaystyle E$. In the circle $\displaystyle BCE$, the point diametrically opposite to $\displaystyle E$ is $\displaystyle F$. Show that the lines $\displaystyle AC$, $\displaystyle BD$ and $\displaystyle OF$ are concurrent.

(6 pont)

solution (in Hungarian)

## Problems with sign 'A'

Deadline expired on November 10, 2020.

A. 783. A polyomino is a figure which consists of unit squares joined together by their sides. (A polyomino may contain holes.) Let $\displaystyle n \ge 3$ be a positive integer. Consider a grid of unit square cells which extends to infinity in all directions. Find, in terms of $\displaystyle n$, the greatest positive integer $\displaystyle C$ which satisfies the following condition: For every colouring of the cells of the grid in $\displaystyle n$ colours, there is some polyomino within the grid which contains at most $\displaystyle n - 1$ colours and whose area is at least $\displaystyle C$.

Submitted by Nikolai Beluhov, Stara Zagora, Bulgaria and Stefan Gerdjikov, Sofia, Bulgaria

(7 pont)

solution (in Hungarian)

A. 784. Let $\displaystyle n$, $\displaystyle s$, $\displaystyle t$ be positive integers and $\displaystyle 0<\lambda<1$. A simple graph on $\displaystyle n$ vertices with at least $\displaystyle \lambda n^2$ edges is given. We say that $\displaystyle (x_1,\ldots,x_s,y_1,\ldots y_t)$ is agood insertion, if letters $\displaystyle x_i$ and $\displaystyle y_j$ denote not necessarily distinct vertices and every $\displaystyle x_iy_j$ is an edge of the graph ($\displaystyle 1~\le i \le s$, $\displaystyle 1\le j\le t$). Prove that the number of good insertions is at least $\displaystyle \lambda^{st}n^{s+t}$.

(7 pont)

A. 785. Let $\displaystyle k\ge t\ge 2$ positive integers. For integers $\displaystyle n\ge k$ let $\displaystyle p_n$ be the probability that if we choose $\displaystyle k$ from the first $\displaystyle n$ positive integers randomly, any $\displaystyle t$ of the $\displaystyle k$ chosen integers have greatest common divisor 1. Let $\displaystyle q_n$ be the probability that if we choose $\displaystyle k-t+1$ from the first $\displaystyle n$ positive integers the product is not divisible by a perfect $\displaystyle t$-th power that is greater then 1.

Prove that sequences $\displaystyle p_n$ and $\displaystyle q_n$ converge to the same value.

Submitted by Dávid Matolcsi, Budapest

(7 pont)

statistics