KöMaL Problems in Mathematics, November 2024
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Problems with sign 'K'Deadline expired on December 10, 2024. |
K. 829. Find the two smallest consecutive positive integers, adding the digits of which we get exactly 2024.
(5 pont)
solution (in Hungarian), statistics
K. 830. In swimming competitions, there are mixed-gender 4x100 m medley relay races. This means that the team consists of two men and two women, and in each stroke 100 meters has to be completed. It's up to the team to decide which stroke the male and female members will compete in. The best times achieved by the swimmers for the 100-meter segments is the following. In backstroke Botond 52.30; Csaba 52.52; András 52.59; Xénia 57.33; Zóra 57.72; Vera 57.98. In breaststroke Ferenc 59.03; Géza 59.11; Hugó 59.30; Sára 65.28; Tímea 65.54; Róza 65.59. In butterfly András 49.90; Botond 49.99; Dénes 50.21; Yvette 55.58; Ursula 55.63; Vera 56.21. In freestyle stroke András 47.40; Csaba 48.48; Elemér 48.50; Xénia 52.36; Yvette 52.49; Piroska 52.73. Supposing that the swimmers can achieve the same results in a competition, find the composition of the team that provides the best possible result.
(5 pont)
solution (in Hungarian), statistics
K. 831. We have arranged four congruent rectangles according to the figure obtaining a large outer and a small inner square. The ratio of the areas of the large square and a single rectangle is \(\displaystyle 25:6\), and the area of the inner small square is \(\displaystyle 144~\mathrm{cm}^2\). Find the lengths of the sides of the rectangles.
(5 pont)
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Problems with sign 'K/C'Deadline expired on December 10, 2024. |
K/C. 832. Nine English lords are planning to establish clubs. They want each club to have exactly three members from among them, but no two clubs should have more than one member in common. What is the maximum number of clubs they can establish?
(5 pont)
solution (in Hungarian), statistics
K/C. 833. A rectangle-shaped piece of paper is given with dimensions \(\displaystyle 20~\mathrm{cm}\times 30~\mathrm{cm}\), according to the figure. Folding vertex \(\displaystyle A\) over vertex \(\displaystyle C\), the line across we folded intersects sides \(\displaystyle AB\) and \(\displaystyle CD\) in points \(\displaystyle P\) and \(\displaystyle Q\). Prove that quadrilateral \(\displaystyle APCQ\) is a rhombus, and compute its area.
(5 pont)
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Problems with sign 'C'Deadline expired on December 10, 2024. |
C. 1828. Anna added the positive integers from \(\displaystyle 1\) to \(\displaystyle 500\), however, she accidentally skipped a three digit number. Find the number of possibilities for this to happen, if she obtained a sum that is divisible by \(\displaystyle 3\), and also ends in a digit \(\displaystyle 3\)?
Proposed by Katalin Abigél Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1829. Points \(\displaystyle E\) and \(\displaystyle F\) are chosen on sides \(\displaystyle AB\) and \(\displaystyle BC\) of unit square \(\displaystyle ABCD\), respectively, such that \(\displaystyle \angle EDF=45^{\circ}\). Find the exact value of the perimeter of triangle \(\displaystyle EBF\).
Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1830. The elements of set \(\displaystyle A=\{x;y;z;u;v\}\) are natural numbers satisfying \(\displaystyle x+2y=3v\) and \(\displaystyle z+u=2v\). Prove that the elements of \(\displaystyle A\) cannot be consecutive natural numbers.
Matlap, Kolozsvár
(5 pont)
solution (in Hungarian), statistics
C. 1831. Solve the following system of equations on the set of real numbers:
$$\begin{gather*} 2x^3-3x^2y+2xy^2-y^3+1=0,\tag*{(1)}\\ x^3+2x^2y-xy^2-2y^3+3=0.\tag*{(2)} \end{gather*}$$Proposed by Bálint Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1832. The area of triangle \(\displaystyle ABC\) is \(\displaystyle 15\sqrt{7}\), and the lengths of its sides are integers. Find all possible lengths of the three sides.
Proposed by Gergely Szmerka, Budapest
(5 pont)
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Problems with sign 'B'Deadline expired on December 10, 2024. |
B. 5414. Let \(\displaystyle ABCD\) be a rectangle. Points \(\displaystyle P\) and \(\displaystyle Q\) are chosen such that the circumcenter of triangle \(\displaystyle ABP\) is \(\displaystyle Q\) and the circumcenter of triangle \(\displaystyle BCQ\) is \(\displaystyle P\). Find the magnitude of angle \(\displaystyle PDQ\).
Proposed by Bálint Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5415. Each of the three kids, Beni, Lili and Domi completes three laps on the athletics track. The referee keeps track of the names of each kid completing a lap, and thus gets a list of nine names. Find the number of different lists the referee can get supposing that no two kids complete a lap at the same time and the speed of each kid remains constant for all three laps.
Proposed by Péter Pál Pach, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5416. Real numbers \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\) satisfy \(\displaystyle x+y+z=8\) and \(\displaystyle xy+yz+zx=5\). Find the biggest possible value of \(\displaystyle z\).
Proposed by Attila Sztranyák, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5417. Which number is bigger, \(\displaystyle \left(2^{1000}\right)!\) or \(\displaystyle 2^{1000!}?\)
Proposed by Máté Szalai, Szeged
(4 pont)
solution (in Hungarian), statistics
B. 5418. Let \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) denote the side lengths of acute triangle \(\displaystyle ABC\), and let \(\displaystyle R\) denote its circumradius. Prove that \(\displaystyle \frac{1}{-a^2+ b^2+c^2}+\frac{1}{a^2-b^2+c^2}+\frac{1}{a^2+b^2-c^2}\ge \frac{1}{R^2},\) and find those triangles where equality holds.
Proposed by Géza Kiss, Csömör
(5 pont)
solution (in Hungarian), statistics
B. 5419. Let \(\displaystyle q(n)\) denote the greatest odd divisor of positive integer \(\displaystyle n\). Let \(\displaystyle P(n)=q(1)+q(2)+\ldots+q(n)\) and \(\displaystyle S(n)=1+2+\ldots+n\). Prove that the ratio \(\displaystyle P(n)/S(n)\) is smaller than \(\displaystyle 2/3\) for infinitely many values of \(\displaystyle n\), and also greater than \(\displaystyle 2/3\) for infinitely many values of \(\displaystyle n\).
Proposed by Attila Sztranyák, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5420. Ádám, the famous conman signed up for the following game of luck. There is a rotating table with a shape of a regular 13-gon, and at each vertex there is a black or a white cap. (Caps of the same colour are indistinguishable from each other.) Under one of the caps 1000 dollars are hidden, and there is nothing under the other caps. The host rotates the table, and then Ádám chooses a cap, and take what is underneath. Ádám's accomplice, Béla is working at the company behind this game. Béla is responsible for the placement of the 1000 dollars under the caps, however, the colors of the caps are chosen by a different collegaue. After placing the money under a cap, Béla
a) has to change the color of the cap,
b) is allowed to change the color of the cap, but he is not allowed to touch any other cap.
Can Ádám and Béla find a strategy in part a) and in part b), respectively, so that Ádám can surely find the money? (After entering the casino, Béla cannot communicate with Ádám, and he also cannot influence his colleague choosing the colors of the caps on the table.)
Proposed by Gábor Damásdi, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5421. The incenter and the inradius of the acute triangle \(\displaystyle ABC\) is \(\displaystyle I\) and \(\displaystyle r\), respectively. The excenter and exradius relative to vertex \(\displaystyle A\) is \(\displaystyle I_a\) and \(\displaystyle r_a\), respectively. Let \(\displaystyle R\) denote the circumradius. Prove that if \(\displaystyle II_a=r_a+R-r\), then \(\displaystyle \angle BAC = 60^{\circ}\).
Proposed by Class 2024C of Fazekas M. Gyak. Ált. Isk. és Gimn., Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on December 10, 2024. |
A. 890. Bart, Lisa and Maggie play the following game: Bart colors finitely many points red or blue on a circle such that no four colored points can be chosen on the circle such that their colors are blue-red-blue-red (the four points do not have to be consecutive). Lisa chooses finitely many of the colored points. Now Bart gives the circle (possibly rotated) to Maggie with Lisa's chosen points, however, without their colors. Finally, Maggie colors all the points of the circle to red or blue. Lisa and Maggie wins the game, if Maggie correctly guessed the colors of Bart's points. A strategy of Lisa and Maggie is called a winning strategy, if they can win the game for all possible colorings by Bart. Prove that Lisa and Maggie have a winning strategy, where Lisa chooses at most \(\displaystyle c\) points in all possible cases, and find the smallest possible value of \(\displaystyle c\).
Proposed by Dömötör Pálvölgyi, Budapest
(7 pont)
A. 891. Let \(\displaystyle ABC\) be an acute triangle. Points \(\displaystyle B'\) and \(\displaystyle C'\) are located on the interior of sides \(\displaystyle AB\) and \(\displaystyle AC\), respectively. Let \(\displaystyle M\) denote the second intersection of the circumcircles of triangles \(\displaystyle ABC\) and \(\displaystyle AB'C'\), while let \(\displaystyle N\) denote the second intersection of the circumcircles of triangles \(\displaystyle ABC'\) and \(\displaystyle AB'C\). Reflect \(\displaystyle M\) across lines \(\displaystyle AB\) and \(\displaystyle AC\), and let \(\displaystyle l\) denote the line through the reflections.
a) Prove that the line through \(\displaystyle M\) perpendicular to line \(\displaystyle AM\), the line \(\displaystyle AK\), and \(\displaystyle l\) are either concurrent or all parallel.
b) Show that if the three lines are concurrent at \(\displaystyle S\), then triangles \(\displaystyle SBC'\) and \(\displaystyle SCB'\) have equal areas.
Proposed by Áron Bán-Szabó, Budapest
(7 pont)
A. 892. Given two integers, \(\displaystyle k\) and \(\displaystyle d\) such that \(\displaystyle d\) divides \(\displaystyle k^3-2\). Show that there exists integers \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) satisfying \(\displaystyle d=a^3+2b^3+4c^3-6abc\).
Proposed by Csongor Beke and László Bence Simon, Cambridge
(7 pont)
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