Mathematical and Physical Journal
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KöMaL Problems in Mathematics, December 2020

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Problems with sign 'K'

Deadline expired on January 11, 2021.


K. 674. In the backyard of aunt Ann, there are 120 animals: brown hens, white ducks, brown pigs and white rabbits. The number of white animals is 64, and the number of two-legged animals is 84. There are twice as many brown hens as white rabbits. How many of each species of animal live in aunt Ann's backyard?

(6 pont)

solution (in Hungarian), statistics


K. 675. A large company was giving a Christmas party to its employees. Some of them brought their spouses along. There were five times as many men present at the party as women. At 10 p.m., some husbands left for home with their wives, and thus the number of women dropped to one seventh of the number of men remaining. What fraction of the men left at 10 p.m.?

(6 pont)

solution (in Hungarian), statistics


K. 676. A \(\displaystyle 6\times6\) chessboard is tiled with eighteen \(\displaystyle 1\times2\) dominoes without overlaps. Show that it is possible to cut the chessboard with a straight line that will not cut across any domino stone.

(6 pont)

solution (in Hungarian), statistics


K. 677. The five elements of a number set \(\displaystyle S\) are pairwise added to produce the sums 0, 6, 11, 12, 17, 20, 23, 26, 32 and 37. Find the elements of \(\displaystyle S\).

(Texas Mathematical Olympiad)

(6 pont)

solution (in Hungarian), statistics


K. 678. There are 2020 coins lying on the table, lined up and showing heads, tails, heads, tails, ... alternating. In one move, it is allowed to reverse any three consecutive coins. With an appropriate sequence of such moves, is it possible to achieve that every coin should show tails?

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on January 11, 2021.


C. 1637. In Dragonland, every seven-headed dragon blows fire, but not all seven-headed, fire-blowing creatures are dragons. According to the latest census figures, the number of dragons in Dragonland is equal to the number of fire-blowing creatures. Is it true that every dragon has seven heads?

(5 pont)

solution (in Hungarian), statistics


C. 1638. Determine those non-regular triangles for which the orthocentre, the circumcentre, the incentre and two vertices are concyclic.

(5 pont)

solution (in Hungarian), statistics


C. 1639. We have five numbers in mind. By selecting three numbers in every possible way and adding them together, we got the following sums: 41, 42, 44, 51, 52, 53, 54, 54, 55, 64. What are the five numbers?

Proposed by S. Kiss, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics


C. 1640. In a quadrilateral \(\displaystyle ABCD\), let \(\displaystyle S\) denote the centroid of triangle \(\displaystyle ABC\), and let \(\displaystyle P\) denote the centroid of triangle \(\displaystyle ACD\). Prove that the line segment connecting the midpoints of diagonals \(\displaystyle AC\) and \(\displaystyle BD\) bisects the line segment \(\displaystyle SP\).

(5 pont)

solution (in Hungarian), statistics


C. 1641. In the expansion of the power \(\displaystyle {(a+b+c)}^{10}\), determine the coefficient of the term in \(\displaystyle a^3b^2c^5\).

Proposed by S. Kiss, Nyíregyháza

(5 pont)

solution (in Hungarian), statistics


C. 1642. The opposite sides of a convex hexagon \(\displaystyle ABCDEF\) are parallel, the distances separating the parallel pairs of sides are equal, and the angles at vertices \(\displaystyle A\) and \(\displaystyle D\) are right angles. Prove that the diagonals \(\displaystyle BE\) and \(\displaystyle CF\) enclose an angle of \(\displaystyle 45^{\circ}\).

(5 pont)

solution (in Hungarian), statistics


C. 1643. Without using a calculator, evaluate the expression

\(\displaystyle (\log_{10}11)\cdot (\log_{11}12)\cdot (\log_{12}13)\cdot\ldots \cdot(\log_{99}100). \)

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on January 11, 2021.


B. 5134. Find all integers \(\displaystyle n\) for which the number \(\displaystyle \sqrt{\lfrac{3n-5}{n+1}}\) is also an integer.

Proposed by M. Szalai, Szeged

(3 pont)

solution (in Hungarian), statistics


B. 5135. The feet of the altitudes drawn from vertices \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\) of an acute-angled triangle \(\displaystyle ABC\) are \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\), respectively; and the midpoints of the altitudes \(\displaystyle AA_1\) and \(\displaystyle BB_1\) are \(\displaystyle G\) and \(\displaystyle H\), respectively. Prove that the circumscribed circle of triangle \(\displaystyle C_1GH\) passes through the midpoint \(\displaystyle F\) of side \(\displaystyle AB\).

Proposed by B. Bíró, Eger

(4 pont)

solution (in Hungarian), statistics


B. 5136. The population of an island consists of underdogs and overlords. When a stranger visited the island, he was invited for dinner with a company of inhabitants. At the end, he asked each member of the company how many overlords were present. The underdogs all gave figures less than the true value and the overlords all gave figures larger than the true value. Is it true that the number of overlords can always be determined from the answers?

Based on a problem of the Dürer Competition

(5 pont)

solution (in Hungarian), statistics


B. 5137. Solve the following simultaneous equations over the set of real numbers:

$$\begin{align*} x+y^2 & =z^3,\\ x^2+y^3 & =z^4,\\ x^3+y^4 & =z^5. \end{align*}$$

Proposed by S. Róka, Nyíregyháza

(4 pont)

solution (in Hungarian), statistics


B. 5138. Triangle \(\displaystyle ABC\) is not isosceles. The interior angle bisectors drawn from vertices \(\displaystyle A\) and \(\displaystyle B\) intersect the opposite sides at points \(\displaystyle A'\) and \(\displaystyle B'\), respectively. Prove that the perpendicular bisector of \(\displaystyle A'B'\) will pass through the centre of the inscribed circle if and only if \(\displaystyle AB' + BA' = AB\).

Proposed by L. Surányi, Budapest

(5 pont)

solution (in Hungarian), statistics


B. 5139. The diagonals of a convex quadrilateral \(\displaystyle ABCD\) intersect at \(\displaystyle M\). The area of triangle \(\displaystyle ADM\) is greater than that of triangle \(\displaystyle BCM\). The midpoint of side \(\displaystyle BC\) of the quadrilateral is \(\displaystyle P\), and the midpoint of side \(\displaystyle AD\) is \(\displaystyle Q\), \(\displaystyle AP+AQ=\sqrt2\,\). Prove that the area of quadrilateral \(\displaystyle ABCD\) is smaller than 1.

(5 pont)

solution (in Hungarian), statistics


B. 5140. There are 10 countries on an island, some of which share a border, and some do not. Each country uses a currency of its own. Every country operates a single exchange office, by the following rules: if you pay 10 units of the currency of that country, you will get 1 unit of each of the currencies of the bordering countries. Initially Arthur and Theodore each own 100 units of the currency of every country. Then each of them shops around the exchange offices of various countries in any order they like, and keeps exchanging money while they can (that is, while they have at least 10 units of a kind). Prove that Arthur and Theodore will have the same number of Bergengocian dollars at the end (the Bergengocian dollar is the currency of one of the countries on the island).

Based on the idea of G. Mészáros, Budapest

(6 pont)

solution (in Hungarian), statistics


B. 5141. Prove that

\(\displaystyle \sum_{i=0}^n\, \sum_{j=i}^n \binom{n}{i} \binom{n+1}{j+1} = 2^{2n}. \)

Proposed by Dávid Nagy, Cambridge

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on January 11, 2021.


A. 789. Let \(\displaystyle p(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots +a_1x+1\) be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than \(\displaystyle 1/3\), and all the coefficients of \(\displaystyle p(x)\) are lying in the interval \(\displaystyle [-2019a,2019a]\) for some positive integer \(\displaystyle a\). Prove that if this polynomial is reducible in \(\displaystyle \mathbb{Z}[x]\), then the coefficients of one its factors are less than \(\displaystyle a\).

Submitted by Navid Safaei, Tehran, Iran

(7 pont)

statistics


A. 790. Andrew and Barry plays the following game: there are two heaps with \(\displaystyle a\) and \(\displaystyle b\) pebbles, respectively. In the first round Barry chooses a positive integer \(\displaystyle k\), and Andrew takes away \(\displaystyle k\) pebbles from one of the two heaps (if \(\displaystyle k\) is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game.

Which player has a winning strategy?

Submitted by András Imolay, Budapest

(7 pont)

statistics


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