KöMaL Problems in Mathematics, September 2023
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Problems with sign 'K'Deadline expired on October 10, 2023. |
K. 774. A train is passing through a tunnel at a uniform speed. It takes 20 seconds, from the front of the train entering the tunnel to the rear end of the train emerging from the tunnel. A lamp hanging from the ceiling of the tunnel remains above the train for exactly 5 seconds. How long is the train?
(5 pont)
solution (in Hungarian), statistics
K. 775. A confectioner constructed a large almond paste shape by sticking together four cubes: two cubes having a 2 cm edge, one of edge 6 cm, and one of edge 8 cm. In each surface of contact, a face of one cube was lying entirely along a face of the other. A rectangular block is to be cut out of the combined shape, but cutting is only allowed along planes that form some face of an original cube. What is the largest possible volume that the resulting almond paste block may have?
(5 pont)
solution (in Hungarian), statistics
K. 776. A printing press is preparing tickets to be sold for an event. When the tickets themselves have been printed out, they are fed in a machine that prints numbers on them, always incrementing the number by 1. Unfortunately, owing to a disorder of the numbering machine each number divisible by 3 was printed twice in a row. In this way, a total of 3672 digits were used. (The starting number was 1.) How many tickets need to be numbered again when the numbering machine is fixed and operating correctly?
(5 pont)
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Problems with sign 'K/C'Deadline expired on October 10, 2023. |
K/C. 777. The arithmetic mean of the numbers \(\displaystyle a\) and \(\displaystyle b\) is 10, the arithmetic mean of \(\displaystyle b\) and 10 is \(\displaystyle c/2\). What is the arithmetic mean of \(\displaystyle a\) and \(\displaystyle c\)?
(5 pont)
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K/C. 778. A rectangle is divided into nine small rectangles with lines parallel to its sides as shown in the figure. The areas of five rectangles are known, the remaining four are unknown (as shown). Determine the area of the four rectangles. (The figure is not to scale.)
(5 pont)
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Problems with sign 'C'Deadline expired on October 10, 2023. |
C. 1773. Determine the value of the integer \(\displaystyle p\) such that the value of the real solution \(\displaystyle x\) of the equation \(\displaystyle (p-3)x+p+5=(2-p)x\) should be at least \(\displaystyle 2\). Find the solution of the equation for every possible \(\displaystyle p\).
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1774. \(\displaystyle AB\parallel{CD}\) in a trapezium \(\displaystyle ABCD\). The midpoints of sides \(\displaystyle AB\) and \(\displaystyle CD\) are \(\displaystyle E\) and \(\displaystyle F\), and line segments \(\displaystyle DE\), \(\displaystyle DB\), \(\displaystyle FB\) intersect diagonal \(\displaystyle AC\) at points \(\displaystyle P\), \(\displaystyle Q\), \(\displaystyle R\), respectively. Prove that \(\displaystyle \frac{CP}{PA}\cdot \frac{CQ}{QA} \cdot \frac{CR}{RA}= \left(\frac{CD}{AB}\right)^{\!3}\).
Proposed by B. Bíró, Eger
(5 pont)
solution (in Hungarian), statistics
C. 1775. \(\displaystyle P\) is an interior point of rectangle \(\displaystyle ABCD\). Determine the length of line segment \(\displaystyle PC\), given that \(\displaystyle PA=4\), \(\displaystyle PB=6\) and \(\displaystyle PD=9\).
(Vietnamese problem)
(5 pont)
solution (in Hungarian), statistics
C. 1776. A natural number has exactly \(\displaystyle 2023\) positive divisors. How many positive divisors may its square have?
Proposed by K.\(\displaystyle \,\)A. Kozma, Győr
(5 pont)
solution (in Hungarian), statistics
C. 1777. The legs of a right-angled triangle are 36 cm and 77 cm long. The angle bisector is drawn to the longer leg. What is the total length of the angle bisector that does not lie in the inscribed circle?
Proposed by B. Bíró, Eger
(5 pont)
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Problems with sign 'B'Deadline expired on October 10, 2023. |
B. 5326. At the end of a meeting with English and Hungarian participants, everyone bid farewell to everyone else, one by one. The English participants said ``Goodbye'', and the Hungarian participants said ``Viszlát'' to everyone else. There were 198 occurrences of ``Goodbye'' and 308 occurrences of ``Viszlát''. How many participants of each nationality were there?
Proposed by B. Hujter, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5327. The heights of triangle \(\displaystyle ABC\) are \(\displaystyle m_a\), \(\displaystyle m_b\) and \(\displaystyle m_c\). Suppose that one can construct a triangle having sides \(\displaystyle m_a\), \(\displaystyle m_b\) and \(\displaystyle m_c\), and the heights of that triangle are \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\). Show that one can again construct a triangle having sides \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\).
Proposed by V. Vígh, Sándorfalva
(4 pont)
solution (in Hungarian), statistics
B. 5328. The number 2023 is written on the first page of a notebook. On every following page, the positive divisors of the numbers on the previous page are written (each divisor is written down \(\displaystyle n\) times if it divides \(\displaystyle n\) numbers on the previous page). How many numbers will there be on page 4?
Proposed by P.\(\displaystyle \,\)P. Pach, Budapest
(3 pont)
solution (in Hungarian), statistics
B. 5329. A fair die is rolled. The game terminates when we decide to stop or a number 1 is rolled. The award is the value of the last roll. Is there a strategy to achieve an expected award of at least 4?
Proposed by P.\(\displaystyle \,\)P. Pach, Budapest
(4 pont)
solution (in Hungarian), statistics
B. 5330. Suppose that \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) is a primitive Pythagorean triple, i.e., \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) are relatively prime positive integers satisfying \(\displaystyle a^2+b^2=c^2\). Show an axially symmetrical polygon that can be decomposed into \(\displaystyle c\) right-angled triangles with sides \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\).
Proposed by G. Kós, Budapest
(5 pont)
solution (in Hungarian), statistics
B. 5331. Prove that it is possible to cover a regular tetrahedron of unit length with two spheres of unit diameter.
Proposed by V. Vígh, Sándorfalva
(5 pont)
solution (in Hungarian), statistics
B. 5332. What may be the positive integer \(\displaystyle n\) if any sequence of \(\displaystyle 2^n\) consecutive positive integers contains a term that can be represented as a sum of the \(\displaystyle n\)th powers of at most \(\displaystyle n\) non-negative integers?
Proposed by P.\(\displaystyle \,\)P. Pach, Budapest
(6 pont)
solution (in Hungarian), statistics
B. 5333. The foot of the altitude drawn from vertex \(\displaystyle A\) of an acute-angled triangle \(\displaystyle ABC\) is \(\displaystyle T_A\). The ray drawn from vertex \(\displaystyle A\) through the centre \(\displaystyle O\) of the circumcircle intersects side \(\displaystyle BC\) at point \(\displaystyle R_A\). Denote the midpoint of line segment \(\displaystyle AR_A\) by \(\displaystyle F_A\). Starting from vertices \(\displaystyle B\) and \(\displaystyle C\), obtain the points \(\displaystyle T_B\), \(\displaystyle R_B\), \(\displaystyle F_B\), \(\displaystyle T_C\), \(\displaystyle R_C\), \(\displaystyle F_C\) in the same way. Show that the lines \(\displaystyle T_AF_A\), \(\displaystyle T_BF_B\) and \(\displaystyle T_CF_C\) are concurrent.
Proposed by L.\(\displaystyle \,\)B. Simon, Budapest
(6 pont)
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Problems with sign 'A'Deadline expired on October 10, 2023. |
A. 857. Let \(\displaystyle ABC\) be a given acute triangle, in which \(\displaystyle BC\) is the longest side. Let \(\displaystyle H\) be the orthocenter of the triangle, and let \(\displaystyle D\) and \(\displaystyle E\) be the feet of the altitudes from \(\displaystyle B\) and \(\displaystyle C\), respectively. Let \(\displaystyle F\) and \(\displaystyle G\) be the midpoints of sides \(\displaystyle AB\) and \(\displaystyle AC\), respectively. \(\displaystyle X\) is the point of intersection of lines \(\displaystyle DF\) and \(\displaystyle EG\). Let \(\displaystyle O_1\) and \(\displaystyle O_2\) be the circumcenters of triangles \(\displaystyle EFX\) and \(\displaystyle DGX\), respectively. Finally, \(\displaystyle M\) is the midpoint of line segment \(\displaystyle O_1O_2\). Prove that points \(\displaystyle X\), \(\displaystyle H\) and \(\displaystyle M\) are collinear.
Proposed by Boldizsár Varga, Verőce
(7 pont)
A. 858. Prove that the only integer solution of the following system of equations is \(\displaystyle u=v=x=y=z=0\): \(\displaystyle uv=x^2-5y^2\), \(\displaystyle (u+v)(u+2v)=x^2-5z^2\).
Proposed by Barnabás Szabó, Budapest
(7 pont)
A. 859. Path graph \(\displaystyle U\) is given, and a blindfolded player is standing on one of its vertices. The vertices of \(\displaystyle U\) are labeled with positive integers between \(\displaystyle 1\) and \(\displaystyle n\), not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after \(\displaystyle k\) steps are the \(\displaystyle k\) degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?
Proposed by Márton Németh, Budapest
(7 pont)
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