KöMaL Problems in Mathematics, May 2023
Please read the rules of the competition.
Show/hide problems of signs:
 |
Problems with sign 'K'Deadline expired on June 12, 2023. |
K. 769. In Burger Burner Restaurant, it turned out that the soup is too salty. The chef decided to dilute it with the leftover soup from yesterday, which did not have enough salt in it. Given that 5 percent of the salty soup is salt while there is only 1.2 percent salt in the leftover soup, how much of each should the chef mix in order to obtain 72 decilitres of soup with a salt content of 3.48 percent?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
)
(5 pont)
solution (in Hungarian), statistics
K. 770. What is the ratio of the number of chessboard fields from which a knight may move to at least four fields to the number of fields from where the knight may move to eight fields?
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
K. 771. Frankie cut a rectangle into exactly nine squares. On inspection, he observed that the area of one square was \(\displaystyle 64~\mathrm{cm}^2\), the areas of two other squares were \(\displaystyle 16~\mathrm{cm}^2\), and the rest of them were \(\displaystyle 4~\mathrm{cm}^2\) each. What was the perimeter of the original rectangle?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
(5 pont)
 |
Problems with sign 'K/C'Deadline expired on June 12, 2023. |
K/C. 772. How many four-digit natural numbers are there in decimal notation in which the first three digits (from the left) are different, all four digits are prime numbers, but the four-digit number is not divisible by any of its digits?
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
K/C. 773. Is there a right-angled triangle in which the measures of the sides are integers, the lengths of exactly two sides are prime numbers and the area is also a prime number?
Proposed by B. Bíró, Eger
(5 pont)
 |
Problems with sign 'C'Deadline expired on June 12, 2023. |
C. 1768. Show that the simultaneous equations
$$\begin{align*} 8x^3+27y^3 & =-6\cdot 5^3, \\ \frac{3}{x}+\frac{2}{y} & =\frac{xy}{5} \end{align*}$$have no solution if \(\displaystyle x\), \(\displaystyle y\) are real numbers.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1769. The orthocentre of an acute-angled triangle \(\displaystyle ABC\) is \(\displaystyle M\), and \(\displaystyle AB\ge BC \ge CA\) for the sides. The perpendicular bisector of line segment \(\displaystyle AM\) intersects side \(\displaystyle AC\) at \(\displaystyle D\), and the perpendicular bisector of line segment \(\displaystyle BM\) intersects side \(\displaystyle BC\) at \(\displaystyle E\). Find the angles of triangle \(\displaystyle ABC\), given that points \(\displaystyle D\), \(\displaystyle M\), \(\displaystyle E\) are collinear.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1770. Solve the equation
\(\displaystyle \sqrt{7+\frac{3}{\sqrt{x}}}=7-\frac{9}{x} \)
over the set of real numbers.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1771. In isosceles right-angled triangle \(\displaystyle ABC\), the midpoint of leg \(\displaystyle BC\) is \(\displaystyle D\), and the point closer to vertex \(\displaystyle B\) that divides the hypotenuse in a one to two ratio is \(\displaystyle E\). Prove that \(\displaystyle AD\) and \(\displaystyle CE\) are perpendicular.
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1772. How many at most three-digit numbers are there in decimal notation that become a palindrome number if converted to binary notation? (A number is called a palindrome if its digits read the same left to right and right to left.)
Proposed by L. Koncz, Budapest
(5 pont)
 |
Problems with sign 'B'Deadline expired on June 12, 2023. |
B. 5318. All positive divisors of a positive integer are written down on a sheet of paper. There are two numbers on the sheet that leave a remainder of 2 when divided by 8, and there are four numbers that leave a remainder of 4. How many numbers may there be on the sheet that leave a remainder of 6 when divided by 8?
Proposed by B. Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5319. Is it true that every acute-angled triangle has at least one altitude whose foot lies in the middle third of the side?
Proposed by B. Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5320. Given that \(\displaystyle \frac{a_{n+3}}{a_{n+1}} + \frac{a_n}{a_{n+2}} = 2\) for each term of a sequence \(\displaystyle a_n\), and the first three terms are \(\displaystyle a_1=1\), \(\displaystyle a_2=4\) and \(\displaystyle a_3=2\), prove that \(\displaystyle \frac{2^{2021}}{a_{2023}}\) is an integer.
Proposed by A. Eckstein, Temesvár
(5 pont)
solution (in Hungarian), statistics
B. 5321. Show that the sum of the squares of the medians of a triangle is smaller than one and a half times the square of the semi-perimeter.
Proposed by L. Németh, Fonyód
(4 pont)
solution (in Hungarian), statistics
B. 5322. Prove that if
\(\displaystyle \frac{\cos\alpha}{s-b}-\frac{\cos\beta}{s-a}=\frac{\cos\alpha-\cos\beta}{s-c} \)
in a triangle, with conventional notations, then the triangle is right-angled or isosceles. (As usual, \(\displaystyle s\) denotes the semi-perimeter of the triangle.)
Proposed by G. Holló, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5323. We are playing the following game: arbitrary real numbers from the interval \(\displaystyle [0, 100]\) are written on 2023 cards. Then the cards are dropped in an urn, and one card is pulled out at random. If the number on the card equals 2/3 of the mean of the numbers on all the cards then we will win as much money as the number on the card pulled out. Otherwise we do not win anything. What numbers should be written on the cards so that the expected value of the money gained is a maximum?
Proposed by B. Dura-Kovács, Garching
(5 pont)
solution (in Hungarian), statistics
B. 5324. Arthur and Barbara are playing the following game: they take turns in writing down a digit, proceeding left to right, until they obtain a 2023-digit number. Arthur starts the game, with a nonzero digit. Arthur will win if the resulting number has a divisor of the form \(\displaystyle 1\overbrace{7\ldots7}^{\text{\(\displaystyle n\) pcs 7}}\) (\(\displaystyle n\ge 1\)). Otherwise Barbara wins. Which of them has a winning strategy?
Based on the idea of G. Kós, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5325. Determine all bounded convex polyhedra such that the planes of the faces divide the space into \(\displaystyle c+e+\ell+1\) regions, where \(\displaystyle c\), \(\displaystyle e\) and \(\displaystyle \ell\), respectively, stand for the number of vertices, edges and faces.
Proposed by V. Vígh, Sándorfalva
(6 pont)
 |
Problems with sign 'A'Deadline expired on June 12, 2023. |
A. 854. Prove that
\(\displaystyle \sum_{k=0}^n \frac{2^{2^k}\cdot 2^{k+1}}{2^{2^k}+3^{2^k}}<4 \)
holds for all positive integers \(\displaystyle n\).
Submitted by Béla Kovács, Szatmárnémeti
(7 pont)
A. 855. In scalene triangle \(\displaystyle ABC\) the shortest side is \(\displaystyle BC\). Let points \(\displaystyle M\) and \(\displaystyle N\) be chosen on sides \(\displaystyle AB\) and \(\displaystyle AC\), respectively, such that \(\displaystyle BM=CN=BC\). Let \(\displaystyle I\) and \(\displaystyle O\) denote the incenter and circumcentre of triangle \(\displaystyle ABC\), and let \(\displaystyle D\) and \(\displaystyle E\) denote the incenter and circumcenter of triangle \(\displaystyle AMN\). Prove that lines \(\displaystyle IO\) and \(\displaystyle DE\) intersect each other on the circumcircle of triangle \(\displaystyle ABC\).
Submitted by Luu Dong, Vietnam
(7 pont)
A. 856. In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player.
Prove that at most \(\displaystyle 1024\) contestants participated in the tournament.
Submitted by Dávid Matolcsi, Budapest
(7 pont)
Upload your solutions above.