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

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## Problems with sign 'K'

Deadline expired on April 10, 2019.

K. 619. What is the largest possible number of primes such that the sum of any three of them is also a prime?

(6 pont)

solution (in Hungarian), statistics

K. 620. The sum of five positive integers is 20. The absolute values of their pairwise differences are 1, 2, 3, 3, 4, 5, 6, 7, 9, 10. Find all such sets of five numbers.

(6 pont)

solution (in Hungarian), statistics

K. 621. Nine members of a math club are designing a $\displaystyle 3\times3$ square flag as shown in the figure. In the nine fields, they arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that the sum of the numbers in each row, each column, and each diagonal is divisible by 3. How many different flags may they make?

(6 pont)

solution (in Hungarian), statistics

K. 622. The 16 tokens in the game of QUARTO are all different from each other in some property. The tokens can be categorized into two sets of the same number of elements in four different ways:

– tall or flat;

– black or white;

– round or square;

– with or without a hole on the top.

Is it possible to arrange the 16 tokens in a circle so that adjacent ones should have exactly two properties in common?

(6 pont)

solution (in Hungarian), statistics

K. 623. The front side of a square sheet of paper $\displaystyle ABCD$ is red, and the back side is white. $\displaystyle E$ and $\displaystyle F$ divide diagonal $\displaystyle AC$ into three equal parts, with $\displaystyle E$ lying closer to $\displaystyle A$. The sheet is folded along lines perpendicular to $\displaystyle AC$ by folding the back side towards the front (that is, making the back of the sheet appear on top). During the first folding, point $\displaystyle A$ is moved to cover $\displaystyle F$, and during the second folding, point $\displaystyle C$ is moved to cover $\displaystyle E$. What will be the ratio of the red area to the white area on the front side of the sheet in the end?

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'C'

Deadline expired on April 10, 2019.

C. 1532. Show that if $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ are positive numbers and

$\displaystyle a+b+c\ge\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac},$

then one of them is at least 1.

(5 pont)

solution (in Hungarian), statistics

C. 1533. The perimeter of a right-angled triangle is $\displaystyle k$, one of the legs is $\displaystyle b$, and the opposite angle is $\displaystyle \beta$. Consider the triangle in which there are two sides of lengths $\displaystyle k$ and $\displaystyle b\cdot\sqrt{2}\,$, and they enclose an angle of $\displaystyle 45^\circ$. Find the smallest angle of this triangle.

(5 pont)

solution (in Hungarian), statistics

C. 1534. Find all real pairs $\displaystyle (x,y)$ satisfying $\displaystyle 5x^2+y^2-4xy+24 \le 10x-1$.

(5 pont)

solution (in Hungarian), statistics

C. 1535. Prove that if the area of a convex quadrilateral is halved by each diagonal, then the quadrilateral is a parallelogram.

(5 pont)

solution (in Hungarian), statistics

C. 1536. Find all real pairs $\displaystyle (x,y)$ satisfying

$\displaystyle xy =x+y+5,$

$\displaystyle x^2+y^2 =5.$

(5 pont)

solution (in Hungarian), statistics

C. 1537. A circle $\displaystyle k_1$ of radius $\displaystyle 6$ and a circle $\displaystyle k_2$ of radius $\displaystyle 3$ touch each other on the outside, and each of them touches a circle $\displaystyle k$ of radius 9 on the inside. One common exterior tangent of $\displaystyle k_1$ and $\displaystyle k_2$ intersects circle $\displaystyle k$ at points $\displaystyle P$ and $\displaystyle Q$. Determine the length of the line segment $\displaystyle PQ$.

(Croatian problem)

(5 pont)

solution (in Hungarian), statistics

C. 1538. Six pairs of twin brothers participated in one of the practices of the Twins' Table Tennis Club. The coaches did not want any brothers to play at the same table.

$\displaystyle a)$ In how many different ways may they divide the players to play round-the-table games at two different tables?

$\displaystyle b)$ In how many different ways is it possible to divide the players into sets of four to play doubles at three different tables? (The position of the players at the tables does not matter.)

(Based on an English problem)

(5 pont)

solution (in Hungarian), statistics

## Problems with sign 'B'

Deadline expired on April 10, 2019.

B. 5014. After the elections in Nowhereland, there are $\displaystyle 50 < n < 100$ representatives in the parliament, all from a single party called the Blue Party. (The Blue Party has a single president.) According to the law, a party in the parliament may be divided into two parties as long as the following conditions are met:

• The president of the old party is not allowed to become a member of the newly formed parties. His or her parliament mandate will terminate, thereby reducing the total number of representatives.
• Every other member may decide which new party to join.
• Each of the new parties must have at least one member among the representatives.
• Each of the new parties must elect a president from their representatives.

If at least one such splitting of a party results in all parties in the parliament having the same number of members, the parliament will be dissolved. What should be the value of $\displaystyle n$ so that this could never happen?

(3 pont)

solution (in Hungarian), statistics

B. 5015. The second intersections of three concurrent unit circles are $\displaystyle A$, $\displaystyle B$ and $\displaystyle C$. What is the radius of the circle $\displaystyle ABC$?

Proposed by J. Szoldatics, Budapest

(3 pont)

solution (in Hungarian), statistics

B. 5016. In a convex quadrilateral $\displaystyle ABCD$, point $\displaystyle E_1$ lies on side $\displaystyle AD$, point $\displaystyle F_1$ lies on side $\displaystyle BC$, $\displaystyle E_2$ lies on diagonal $\displaystyle AC$, and $\displaystyle F_2$ lies on diagonal $\displaystyle BD$. Given that

$\displaystyle AE_1:E_1D=BF_1:F_1C=AE_2:E_2C=BF_2:F_2D=AB:CD.$

and no pair of points coincide, prove that the lines $\displaystyle E_1F_1$ and $\displaystyle E_2F_2$ are perpendicular.

(4 pont)

solution (in Hungarian), statistics

B. 5017. Is there a function $\displaystyle f\colon \mathbb{R}\to \mathbb{R}$ with the following properties:

(1) if $\displaystyle x_1\ne x_2$ then $\displaystyle f(x_1)\ne f(x_2)$,

(2) there exist appropriate constants $\displaystyle a,b>0$ such that

$\displaystyle f(x^2)- \big(f(ax+b)\big)^2\ge \frac14.$

for all $\displaystyle x\in\mathbb{R}$?

(4 pont)

solution (in Hungarian), statistics

B. 5018. The sultan imprisoned all the 1024 mathematicians of his empire. They were not allowed to keep any of their possessions except for a single copper coin each. The mathematicians know that there are 1024 of them, but they are not able to communicate with one another in any way.

On his birthday the sultan offered them the following game: they are taken to the prison yard one by one. Each of them may say either 0 or 1 when taken there. If the sum of the numbers they say is 1, then he will let them all go free.

(The mathematicians cannot signal to each other, they do not know how many others have been to the yard before them, or what those before them have done in the yard.)

What are the chances that they can get out of the prison?

(5 pont)

solution (in Hungarian), statistics

B. 5019. The quadrilateral $\displaystyle ABCD$ is cyclic. Given that $\displaystyle AB+BC=AD+DC$ and $\displaystyle BA+AC=BD+DC$, show that $\displaystyle ABCD$ is a rectangle.

(6 pont)

solution (in Hungarian), statistics

B. 5020. A parabola is reflected in a line that passes through its focus and encloses an angle $\displaystyle \alpha$ with its axis. Show that the parabola and its reflection intersect at an angle of $\displaystyle \alpha$.

Proposed by L. Németh, Fonyód

(5 pont)

solution (in Hungarian), statistics

B. 5021. A positive integer $\displaystyle n$ is not divisible by 3. The sum of its positive divisors that leave a remainder of 1 when divided by 3 is $\displaystyle A(n)$, and the sum of its positive divisors that leave a remainder of 2 when divided by 3 is $\displaystyle B(n)$. Find those numbers $\displaystyle n$ for which $\displaystyle \big|A(n)-B(n)\big|<\sqrt{n}\,$.

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'A'

Deadline expired on April 10, 2019.

A. 746. Let $\displaystyle p$ be a prime number. How many solutions does the congruence $\displaystyle x^2+y^2+z^2+1\equiv0\pmod{p}$ have among the modulo $\displaystyle p$ remainder classes?

Proposed by: Zoltán Gyenes, Budapest

(7 pont)

solution (in Hungarian), statistics

A. 747. In a simple graph on $\displaystyle n$ vertices, every set of $\displaystyle k$ vertices has an odd number of common neighbours. Prove that $\displaystyle n+k$ must be odd.

Proposed by: András Imolay, Dávid Matolcsi, Ádám Schweitzer and Kristóf Szabó, Budapest

(7 pont)

statistics

A. 748. The circles $\displaystyle \Omega$ and $\displaystyle \omega$ in its interior are fixed. The distinct points $\displaystyle A$, $\displaystyle B$, $\displaystyle C$, $\displaystyle D$, $\displaystyle E$ move on $\displaystyle \Omega$ in such a way that the line segments $\displaystyle AB$, $\displaystyle BC$, $\displaystyle CD$ and $\displaystyle DE$ are tangents to $\displaystyle \omega$. The lines $\displaystyle AB$ and $\displaystyle CD$ meet at point $\displaystyle P$, the lines $\displaystyle BC$ and $\displaystyle DE$ meet at $\displaystyle Q$. Let $\displaystyle R$ be the second intersection of the circles $\displaystyle BCP$ and $\displaystyle CDQ$, other than $\displaystyle C$. Show that $\displaystyle R$ moves either on a circle or on a line.

Proposed by: Carlos Yuzo Shine, Sao Paolo

(7 pont)

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