KöMaL Problems in Mathematics, September 2024
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Problems with sign 'K'Deadline expired on October 10, 2024. |
K. 819. Kati has written the number \(\displaystyle +1\) ten times on the blackboard. In each move she can change the sign of five numbers on the board. She can repeat this move an arbitrary number of times. Is it possible that after a series of moves she will end up with nine \(\displaystyle +1\)'s and a single \(\displaystyle -1\) on the board? If the answer is yes, find the minimum number of moves needed.
(5 pont)
solution (in Hungarian), statistics
K. 820. In the family Tóth, the number of children is 6. The average of the ages of the boys is 20 years, and the average of the ages of the girls is 12 years. Interestingly, each child has a twin sibling of the same sex. Find the age of each child.
(5 pont)
K. 821. A cylindrical, open-top container was placed inside a cubic, open-top container with a side length of 1 meter, and it was fixed to the bottom of the cubic container. Water flows uniformly from a tap into the cubic container. We observe that the water level on the wall of the cube rises steadily for 10 minutes, then stops rising for 10 minutes, and then when it starts to rise again, it takes another 20 minutes for the cubic container to fill completely. Find the radius of the base and the height of the cylindrical container.
(5 pont)
Problems with sign 'K/C'Deadline expired on October 10, 2024. |
K/C. 822. Kati has to calculate the area of the cyclic pentagon in the diagram. The lengths of the sides measured in cm are given in the diagram. Kati obtained the result \(\displaystyle 30+10{,}5\sqrt{30}~\textrm{cm}^2\). Has she calculated correctly?
(5 pont)
solution (in Hungarian), statistics
K/C. 823. We translate each side line of a convex 2024-gon perpendicularly to the given side by 4 units to obtain another convex 2024-gon. Prove that the perimeter of the new 2024-gon is longer by at least 25 units than the perimeter of the original polygon.
(5 pont)
Problems with sign 'C'Deadline expired on October 10, 2024. |
C. 1818. Let \(\displaystyle ABCD\) be a unit square, and let \(\displaystyle k\) be the circle with center \(\displaystyle A\) and radius \(\displaystyle AC\). Let \(\displaystyle E\) and \(\displaystyle F\) be the points of intersection of circle \(\displaystyle k\) and rays \(\displaystyle AB\) and \(\displaystyle AD\), respectively. Let line \(\displaystyle EF\) intersect \(\displaystyle BC\) at point \(\displaystyle G\), and let \(\displaystyle H\) be the reflection of point \(\displaystyle B\) across line \(\displaystyle AG\). Find the length of line segment \(\displaystyle HE\) in the given unit.
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1819. Let \(\displaystyle ABCD\) be a unit square, and let \(\displaystyle k\) be the circle with center \(\displaystyle A\) and radius \(\displaystyle AC\). Let \(\displaystyle E\) and \(\displaystyle F\) be the points of intersection of circle \(\displaystyle k\) and rays \(\displaystyle AB\) and \(\displaystyle AD\), respectively. Let line \(\displaystyle EF\) intersect \(\displaystyle BC\) at point \(\displaystyle G\), and let \(\displaystyle H\) be the reflection of point \(\displaystyle B\) across line \(\displaystyle AG\). Find the length of line segment \(\displaystyle HE\) in the given unit.
Proposed by Dániel Hegedűs, Gyöngyös
(5 pont)
solution (in Hungarian), statistics
C. 1820. Prove that if \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c>0\) and \(\displaystyle a+b+c=1\), then
\(\displaystyle a)\) \(\displaystyle {\frac{1-a^2}{b+c}+\frac{1-b^2}{c+a}+\frac{1-c^2}{a+b }=4}\),
\(\displaystyle b)\) \(\displaystyle {\frac{1-a^3}{b+c}+\frac{1-b^3}{c+a}+\frac{1-c^3}{a+b }\geq \frac{13}{3}}\).
Proposed by Mihály Bencze, Brașow
(5 pont)
solution (in Hungarian), statistics
C. 1821. Jules and Jim play with a fair die. If the result of the roll is a composite number, Jim gets a point, otherwise Jules gets a point. The game ends when one of the players collected six points. Find the probability that the result will be \(\displaystyle 6:3\) for the winning player.
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1822. The diagonals \(\displaystyle AC\) and \(\displaystyle BD\) of convex quadrilateral \(\displaystyle ABCD\) intersect each other at point \(\displaystyle M\). Let positive integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) be the areas of the resulting triangles \(\displaystyle ABM\), \(\displaystyle BCM\), \(\displaystyle CDM\) and \(\displaystyle DAM\), respectively.
\(\displaystyle a)\) Prove that the product \(\displaystyle a\cdot b\cdot c\cdot d\) is a perfect square.
\(\displaystyle b)\) Suppose that there are exactly two distinct odd primes among \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\). Find the values of \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) such that the area of the quadrilateral \(\displaystyle ABCD\) is the smallest possible perfect square.
Proposed by Bíró Bálint, Eger
(5 pont)
Problems with sign 'B'Deadline expired on October 10, 2024. |
B. 5398. In trapezoid \(\displaystyle ABCD\) the bases are \(\displaystyle AB \parallel CD\) and \(\displaystyle \angle ADC - \angle CBA = 90^{\circ}\). Prove that the sum of the squares of the legs equals the difference of the squares of the bases.
Proposed by Miklós Oláh, Szilágykraszna
(3 pont)
solution (in Hungarian), statistics
B. 5399. A five-digit perfect square does not contain the digit 9. If we increase each of its digits by 1, we get another perfect square. Find all possible perfect squares with this property.
Proposed by Géza Kiss, Csömör
(3 pont)
solution (in Hungarian), statistics
B. 5400. In a \(\displaystyle 3\times 3\) magic square we increased one of the nine entries by 1. Find the least number of additional entries that has to be changed to obtain a magic square again.
(The \(\displaystyle 3\times 3\) magic square is a number grid in which all three numbers in each of the rows, columns, and diagonals add up to the same number.)
Proposed by Máté Juhász
(4 pont)
solution (in Hungarian), statistics
B. 5401. Find the biggest possible value of product \(\displaystyle mn\), provided that \(\displaystyle m\), \(\displaystyle n\) and \(\displaystyle \sqrt{25+\sqrt{n+\sqrt m}} + \sqrt{25-\sqrt{n+\sqrt m}}\) are all positive integers.
Proposed by Attila Sztranyák, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5402. The side lengths \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) of a triangle satisfy \(\displaystyle a^2+b^2+c^2=a^2b^2c^2\). Prove that the area of the triangle is at most \(\displaystyle \frac34\), and if equality holds, then the triangle is equilateral.
Proposed by Mihály Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5403. Suppose that the edges of a simple, connected \(\displaystyle k\)-regular (\(\displaystyle k \ge 2\)) graph \(\displaystyle G\) can be colored with \(\displaystyle k\) colors such that the colors of the edges meeting at any given vertex are all different. Prove that if we delete any edge from \(\displaystyle G\), the resulting graph will still be connected.
Proposed by Bálint Hujter, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5404. The altitudes of the acute triangle \(\displaystyle ABC\) are \(\displaystyle AT_A\), \(\displaystyle BT_B\) and \(\displaystyle CT_C\). The midpoints of the sides \(\displaystyle BC\), \(\displaystyle CA\) and \(\displaystyle AB\) are \(\displaystyle F_A\), \(\displaystyle F_B\) and \(\displaystyle F_C\), respectively. Let \(\displaystyle r\) denote the radius of the inscribed circle and let \(\displaystyle P_A\) denote the point on altitude \(\displaystyle AT_A\) satisfying \(\displaystyle AP_A = r\). Points \(\displaystyle P_B\) and \(\displaystyle P_C\) are defined similarly. Prove that line segments \(\displaystyle F_AP_A\), \(\displaystyle F_BP_B\) and \(\displaystyle F_CP_C\) are concurrent.
Proposed by Géza Kiss, Csömör
(6 pont)
solution (in Hungarian), statistics
B. 5405. Positive integers \(\displaystyle a_1,a_2,\ldots,a_n\) and \(\displaystyle b_1,b_2,\ldots,b_n\) satisfy the following property: for all indices \(\displaystyle i < j \leq n\) the greatest common divisor of \(\displaystyle b_i\) and \(\displaystyle b_j\) does not divide difference \(\displaystyle (a_i-a_j)\). Prove that \(\displaystyle \displaystyle \sum_{i=1}^n \frac1{b_i} \leq 1\).
Proposed by Boldizsár Varga, Budapest
(6 pont)
Problems with sign 'A'Deadline expired on October 10, 2024. |
A. 884. We fill in an \(\displaystyle n\times n\) table with real numbers such that the sum of the numbers in each row and each coloumn equals 1. For which values of \(\displaystyle K\) is the following statement true: if the sum of the absolute values of the negative entries in the table is at most \(\displaystyle K\), then it's always possible to choose \(\displaystyle n\) positive entries of the table such that each row and each coloumn contains exactly one of the chosen entries.
Proposed by Dávid Bencsik, Budapest
(7 pont)
A. 885. Let triangle \(\displaystyle ABC\) be a given acute scalene triangle with altitudes \(\displaystyle BE\) and \(\displaystyle CF\). Let \(\displaystyle D\) be the point where the incircle of \(\displaystyle \triangle ABC\) touches side \(\displaystyle B C\). The circumcircle of \(\displaystyle \triangle B D E\) meets line \(\displaystyle A B\) again at point \(\displaystyle K\), the circumcircle of \(\displaystyle \triangle C D F\) meets line \(\displaystyle A C\) again at point \(\displaystyle L\). The circumcircle of \(\displaystyle \triangle B D E\) and \(\displaystyle \triangle CDF\) meet line \(\displaystyle KL\) again at \(\displaystyle X\) and \(\displaystyle Y\), respectively. Prove that the incenter of \(\displaystyle \triangle DXY\) lies on the incircle of \(\displaystyle \triangle ABC\).
Proposed by Luu Dong, Vietnam
(7 pont)
A. 886. Let \(\displaystyle k\) and \(\displaystyle n\) be two given distinct positive integers greater than 1. There are finitely many (not necessarily distinct) integers written on the blackboard. Kázmér is allowed to erase \(\displaystyle k\) consecutive elements of an arithmetic sequence with a difference not divisible by \(\displaystyle k\). Similarly, Nándor is allowed to erase \(\displaystyle n\) consecutive elements of an arithmetic sequence with a difference that is not divisible by \(\displaystyle n\). The initial numbers on the blackboard have the property that both Kázmér and Nándor can erase all of them (independently from each other) in a finite number of steps. Prove that the difference of biggest and the smallest number on the blackboard is at least \(\displaystyle \varphi(n)+\varphi(k)\), where \(\displaystyle \varphi\) denotes Euler's totient function, i.e., \(\displaystyle \varphi(n)\) is the number of positive integers not exceeding \(\displaystyle n\) that are coprime to \(\displaystyle n\).
Proposed by Boldizsár Varga, Budapest
(7 pont)
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