Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# KöMaL Problems in Mathematics, November 2018

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## Problems with sign 'K'

Deadline expired on December 10, 2018.

K. 599. Write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in the circles, so that the sum of the four numbers along any straight line should be the same, and the sum of the numbers at the six points of the star should also be the same number. A few numbers are already entered. Find all possible arrangements.

(6 pont)

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K. 600. If one digit of a three-digit number is omitted, a two-digit number will be obtained. By omitting one digit of that two-digit number, a one-digit number will result. What may be the initial three-digit number so that the sum of the three-digit number, the two-digit number and the final one-digit number is 1001?

(6 pont)

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K. 601. The sides of a square $\displaystyle PQRS$ inscribed in an acute-angled triangle $\displaystyle ABC$ are 4 cm long, vertices $\displaystyle P$ and $\displaystyle Q$ lie on side $\displaystyle AB$, vertex $\displaystyle R$ lies on side $\displaystyle BC$, and vertex $\displaystyle S$ lies on side $\displaystyle AC$. Given that the length of side $\displaystyle AB$ is 8 cm, what is the area of the triangle?

(6 pont)

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K. 602. Andrew and Paul are playing a game. The winner is always awarded $\displaystyle x$ points and the loser always gets $\displaystyle y$ points (where $\displaystyle x>y$ are integers). There is no draw. After a few rounds, we observe that Andrew has 30 points and Paul has 25 points since Paul has only won twice. How many points are awarded to the winner?

(6 pont)

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K. 603. I have a two-digit number in mind. Let $\displaystyle S$ denote the sum of the digits, and let $\displaystyle P$ denote their product. What may be my number if it is equal to $\displaystyle P+S$?

(6 pont)

solution (in Hungarian), statistics

## Problems with sign 'C'

Deadline expired on December 10, 2018.

C. 1504. A $\displaystyle 3\times3$ table is filled in as shown. If the greatest common divisor of any set of $\displaystyle n$ entries of the table is $\displaystyle n$, it is allowed to rearrange those entries so that none of them stay in place. With an appropriate succession of such steps, is it possible to achieve that the final arrangement of the numbers is a reflection of the original arrangement in one diagonal? In the other diagonal?

 1 3 4 6 8 9 10 12 20

(5 pont)

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C. 1505. Consider the circumscribed circles of all the black fields of a chessboard. What fraction of the total area of the 64 fields is covered by these disks altogether?

(5 pont)

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C. 1506. Solve the equation $\displaystyle p^q+1=q^p$, where $\displaystyle p$, $\displaystyle q$ denote positive prime numbers.

(5 pont)

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C. 1507. The perpendicular bisectors of the legs of an obtuse-angled isosceles triangle divide the base into three equal parts. Find the measures of the angles.

(5 pont)

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C. 1508. Determine the value of $\displaystyle xy$, given that $\displaystyle x+y=1$ and $\displaystyle x^3+y^3=\frac12$.

(5 pont)

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C. 1509. A company selling teabags has placed gift vouchers in 10% of the boxes. If 10 boxes are bought, what is the probability of finding more than 1 voucher?

(5 pont)

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C. 1510. The base radii of a right circular truncated cone are $\displaystyle 8$ cm and $\displaystyle 5$ cm. The slant height is $\displaystyle 12$ cm. If the truncated cone is laid on its side and rolled, it will trace out a circular ring in the plane. Determine the radii of the inner and outer circles of the ring, and find the number of times the truncated cone rotates about its axis while it rolls around and returns to its starting position.

(5 pont)

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## Problems with sign 'B'

Deadline expired on December 10, 2018.

B. 4982. The diagonals $\displaystyle AC$ and $\displaystyle BD$ of a convex kite $\displaystyle ABCD$ intersect at point $\displaystyle E$ such that $\displaystyle AE<CE$. The midpoint of diagonal $\displaystyle AC$ is $\displaystyle F$. The circles $\displaystyle ABE$ and $\displaystyle CDE$ intersect again at $\displaystyle M$. Show that $\displaystyle \angle EMF=90^\circ$.

(3 pont)

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B. 4983. Find the real solutions of the equation

$\displaystyle x^2+2x-3-\sqrt{\frac{x^2+2x-3}{x^2-2x-3}}=\frac{2}{x^2-2x-3}.$

Proposed by L. Laczkó and J. Szoldatics, Budapest

(4 pont)

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B. 4984. Prove that for any positive integer $\displaystyle x$, there exists a positive integer $\displaystyle y$ such that $\displaystyle x^3+y^3+1$ is divisible by the number $\displaystyle x+y+1$. Is there a positive integer $\displaystyle x$ for which there are infinitely many $\displaystyle y$ with this property?

Proposed by L. Surányi, Budapest

(4 pont)

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B. 4985. Given that any three out of four lines determine a triangle, prove that the orthocentres of the four triangles are concurrent.

(5 pont)

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B. 4986. Consider the 64 points of the space for which each of the three coordinates is 1, 2, 3 or 4. Kate and Peter are playing a three-dimensional tic-tac-toe game on this set of points. Kate starts the game by selecting any point and colouring it blue. In the second step, Peter selects a different point and colours it red. Then they take turns by selecting further points and colouring them in blue or red. Whoever first completes a collinear set of four points of their own colour will win the game. Show that it makes no difference for Kate whether she starts by colouring the point $\displaystyle (1,1,2)$ or the point $\displaystyle (2,2,1)$ blue in the first step.

Proposed by D. Benkő, South Alabama

(5 pont)

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B. 4987. The circumcentre of an acute-angled scalene triangle $\displaystyle ABC$ is $\displaystyle O$, its orthocentre is $\displaystyle M$, the foot of the altitude drawn from vertex $\displaystyle A$ is $\displaystyle D$, and the midpoint of side $\displaystyle AB$ is $\displaystyle F$. The ray drawn from $\displaystyle F$ through $\displaystyle M$ intersects the circumcircle of triangle $\displaystyle ABC$ at $\displaystyle G$.

$\displaystyle a)$ Prove that the points $\displaystyle A$, $\displaystyle F$, $\displaystyle D$ and $\displaystyle G$ are concyclic.

$\displaystyle b)$ Let $\displaystyle K$ denote the circle in $\displaystyle a)$, and let $\displaystyle E$ be the midpoint of line segment $\displaystyle CM$. Prove that $\displaystyle EK=OK$.

Proposed by B. Bíró, Eger

(5 pont)

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B. 4988. In an $\displaystyle (m+2)\times(n+2)$ table, we cut out the four $\displaystyle 1\times1$ corners''. Arbitrary real numbers are written in each field of the first and last rows, and in the first and last columns of the truncated table obtained in this way. Prove that it is possible to fill in the remaining $\displaystyle m\times n$ interior'' of the table in a unique way with real numbers such that every number is the arithmetic mean of the four adjacent numbers.

(Competition problem from Iran)

(6 pont)

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B. 4989. The midpoints of sides $\displaystyle BC$, $\displaystyle CA$ and $\displaystyle AB$ of a triangle $\displaystyle ABC$ are $\displaystyle D$, $\displaystyle E$ and $\displaystyle F$, respectively. Let $\displaystyle S$ denote the centroid of the triangle. Assume that the perimeters of triangles $\displaystyle AFS$, $\displaystyle BDS$ are $\displaystyle CES$ equal. Show that triangle $\displaystyle ABC$ is equilateral.

(6 pont)

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## Problems with sign 'A'

Deadline expired on December 10, 2018.

A. 734. For an arbitrary positive integer $\displaystyle m$, not divisible by $\displaystyle 3$, consider the permutation $\displaystyle x\mapsto 3x\pmod{m}$ on the set $\displaystyle \{1,2,\ldots,m-1\}$. This permutation can be decomposed into disjoint cycles; for instance, for $\displaystyle m=10$ the cycles are $\displaystyle (1\mapsto3\mapsto9\mapsto7\mapsto1)$, $\displaystyle (2\mapsto6\mapsto8\mapsto4\mapsto2)$ and $\displaystyle (5\mapsto5)$. For which integers $\displaystyle m$ is the number of cycles odd?

(7 pont)

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A. 735. For any function $\displaystyle f\colon [0,1]\to[0,1]$, let $\displaystyle P_n(f)$ denote the number of fixed points of the function

$\displaystyle \underbrace{f\big(\ldots f}_n(x)\ldots\big),$

i.e., the number of points $\displaystyle x\in[0,1]$ satisfying $\displaystyle \underbrace{f\big(\ldots f}_{n}(x)\ldots\big)=x$. Construct a piecewise linear, continuous, surjective function $\displaystyle f\colon [0,1]\to[0, 1]$ such that for a suitable number $\displaystyle {2<A<3}$, the sequence $\displaystyle \frac{P_n(f)}{A^n}$ converges.

Based on the 8th problem of the Miklós Schweitzer competition, 2018

(7 pont)

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A. 736. Let $\displaystyle P$ be a point in the plane of triangle $\displaystyle ABC$. Denote the reflections of $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ about $\displaystyle P$ by $\displaystyle A'$, $\displaystyle B'$ and $\displaystyle C'$, respectively. Let $\displaystyle A''$, $\displaystyle B''$, $\displaystyle C''$ be the reflections of $\displaystyle A'$, $\displaystyle B'$, $\displaystyle C'$ over the lines $\displaystyle BC$, $\displaystyle CA$ and $\displaystyle AB$, respectively. Let the line $\displaystyle A''B''$ intersect $\displaystyle AC$ at $\displaystyle A_b$ and let $\displaystyle A''C''$ intersect $\displaystyle AB$ at a point $\displaystyle A_c$. Denote by $\displaystyle \omega_A$ the circle through the points $\displaystyle A$, $\displaystyle A_b$, $\displaystyle A_c$. The circles $\displaystyle \omega_B$, $\displaystyle \omega_C$ are defined similarly. Prove that $\displaystyle \omega_A$, $\displaystyle \omega_B$, $\displaystyle \omega_C$ are coaxial, i.e., they share a common radical axis.

Proposed by Navneel Singhal, Delhi and K. V. Sudharshan, Chennai, India

(7 pont)

solution (in Hungarian), statistics