KöMaL Problems in Mathematics, April 2018
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Problems with sign 'C'Deadline expired on May 10, 2018. |
C. 1476. Prove that the inequality
\(\displaystyle \frac{{(y-6)}^2}{3xy}+x\cdot \frac{y+3}{y}\ge 4+x-\frac{4}{x}-\frac{xy}{12} \)
holds for all positive \(\displaystyle x\) and \(\displaystyle y\).
(5 pont)
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C. 1477. Prove that if there is a point \(\displaystyle E\) on base \(\displaystyle AD\) of a trapezium \(\displaystyle ABCD\) such that the perimeters of triangles \(\displaystyle ABE\), \(\displaystyle BCE\) and \(\displaystyle CDE\) are equal then \(\displaystyle BC=\frac12 AD\).
(5 pont)
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C. 1478. Given that a six-digit number is divisible by \(\displaystyle 37\), its digits are all different, and \(\displaystyle 0\) does not occur among them, show that at least six more numbers divisible by \(\displaystyle 37\) can be obtained by changing the order of the digits.
(5 pont)
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C. 1479. In a triangle \(\displaystyle ABC\), \(\displaystyle T\) is an interior point of side \(\displaystyle AC\) such that \(\displaystyle TA=BC\), and \(\displaystyle P\) is an interior point of side \(\displaystyle AB\) such that the triangles \(\displaystyle CBP\) and \(\displaystyle PAT\) are congruent. \(\displaystyle Q\) is an interior point of side \(\displaystyle BC\) such that \(\displaystyle TQ\) is not parallel to \(\displaystyle AB\) and triangle \(\displaystyle BPQ\) is similar to triangle \(\displaystyle TCQ\). Prove that \(\displaystyle PT=QT\).
The problem was deleted from the contest.
(5 pont)
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C. 1480. Solve the equation
\(\displaystyle \frac{x^3-7x+6}{x-2}=\frac{2x+14}{x+2} \)
on the set of integers.
(5 pont)
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C. 1481. The vertices of a regular octagon inscribed in a circle of radius \(\displaystyle 2\) are connected in three different ways, as shown in the figure: each vertex with the adjacent vertices, each vertex with the second adjacent vertices, and finally, each vertex with the third adjacent vertices. Prove that the product of the radii of the three inscribed circles is \(\displaystyle 2\).
(5 pont)
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C. 1482. Prove that
\(\displaystyle \big|2\sin x +\sin {(2x)}\big| < \frac{3+2\sqrt2}{2}\,. \)
(5 pont)
Problems with sign 'B'Deadline expired on May 10, 2018. |
B. 4948. The positive integer \(\displaystyle n\) is said to be chunky if it has a prime factor greater than \(\displaystyle \sqrt{n}\). For example, \(\displaystyle 2017\) (a prime number), \(\displaystyle 2018=2\cdot 1009\) and \(\displaystyle 2022=2\cdot3\cdot 337\) are chunky, while \(\displaystyle 2023=7\cdot 17^2\) is not. How many chunky numbers are there which only have prime factors less than 30?
Proposed by S. Róka, Nyíregyháza
(3 pont)
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B. 4949. The feet of the altitudes drawn from vertices \(\displaystyle B\) and \(\displaystyle C\) of an acute-angled triangle \(\displaystyle ABC\) are \(\displaystyle D\) and \(\displaystyle E\), respectively. Let \(\displaystyle P\) be an interior point of \(\displaystyle AD\), and let \(\displaystyle Q\) be an interior point of \(\displaystyle AE\) such that \(\displaystyle EDPQ\) is a cyclic quadrilateral. Show that the line segments \(\displaystyle BP\) and \(\displaystyle CQ\) intersect on the median drawn from \(\displaystyle A\).
(3 pont)
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B. 4950. Let \(\displaystyle F_n\) denote the \(\displaystyle n\)th Fibonacci number (\(\displaystyle F_1=F_2=1\), \(\displaystyle F_{n+2}= F_{n+1}+F_n\)), and define the sequence \(\displaystyle a_0,a_1,a_2,\dots\) with the following recurrence relation: let \(\displaystyle a_0=2018\), and for all \(\displaystyle k\ge 0\) let \(\displaystyle a_{k+1}=a_k+F_n\), where \(\displaystyle F_n\) is the largest Fibonacci number less than \(\displaystyle a_k\). Will there be any Fibonacci number in the sequence \(\displaystyle (a_k)\)?
(4 pont)
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B. 4951. The elements of a set \(\displaystyle V\) are \(\displaystyle n\)-dimensional vectors (ordered \(\displaystyle n\)-tuples of numbers) of which each coordinate is \(\displaystyle -1\), \(\displaystyle 0\) or \(\displaystyle 1\). No three different vectors of \(\displaystyle V\) add up to the zero vector. Show that \(\displaystyle |V|\le 2\cdot 3^{n-1}\).
(4 pont)
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B. 4952. Is it possible to dissect a cube with a finite number of straight cuts so that the pieces can be put together to form two smaller congruent cubes?
Proposed by Z. Gyenes, Budapest
(5 pont)
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B. 4953. Prove that
\(\displaystyle \ln n+\sqrt{\frac12}+\sqrt{\frac23}+\ldots +\sqrt{\frac{n-1}{n}}<\sqrt2+\sqrt{\frac32}+\sqrt{\frac43}+\ldots +\sqrt{\frac{n}{n-1}}\, \)
for all integers \(\displaystyle n>1\).
Proposed by G. Holló, Budapest
(5 pont)
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B. 4954. Line \(\displaystyle \ell\) passes through vertex \(\displaystyle A\) of a triangle \(\displaystyle ABC\), and it is parallel to \(\displaystyle BC\). Let \(\displaystyle \ell\) intersect the interior angle bisectors of angles \(\displaystyle ABC\) and \(\displaystyle ACB\) at \(\displaystyle K\) and \(\displaystyle L\), respectively. The inscribed circle touches \(\displaystyle BC\) at point \(\displaystyle D\). Show that the circumscribed circle intersects the Thales circle of line segment \(\displaystyle KL\) at two points, and these two points are collinear with \(\displaystyle D\).
(6 pont)
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B. 4955. Let \(\displaystyle n\) be a positive integer. What is the largest possible number of ordered triples of non-negative integers \(\displaystyle (x_1,y_1,z_1),(x_2,y_2,z_2),\ldots\) such that the following conditions hold:
(1) For all \(\displaystyle i\), \(\displaystyle x_i + y_i + z_i = n\).
(2) The numbers \(\displaystyle x_1,x_2,\ldots\) are all different, the numbers \(\displaystyle y_1,y_2,\ldots\) are all different, and the numbers \(\displaystyle z_1,z_2,\ldots\) are also all different.
Give an example for such a sequence of maximum length with the required property.
Proposed by P. Erben, Budapest
(6 pont)
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B. 4956. A transformation of central similitude is applied to a tetrahedron \(\displaystyle ABCD\) with each vertex as centre. The four diminished tetrahedra obtained are \(\displaystyle AA_bA_cA_d\), \(\displaystyle B_aBB_cB_d\), \(\displaystyle C_aC_bCC_d\) and \(\displaystyle D_aD_bD_cD\). Given that these small tetrahedra are pairwise disjoint, prove that the volumes of tetrahedra \(\displaystyle {A_b B_c C_d D_a}\), \(\displaystyle {A_b B_d D_c C_a}\), \(\displaystyle {A_c C_b B_d D_a}\), \(\displaystyle {A_c C_d D_b B_a}\), \(\displaystyle {A_d D_b B_c C_a}\) and \(\displaystyle {A_d D_c C_b B_a}\) are equal.
Proposed by Sz. Kocsis, Budapest
(6 pont)
Problems with sign 'A'Deadline expired on May 10, 2018. |
A. 722. The Hawking Space Agency operates \(\displaystyle n-1\) space flights between the \(\displaystyle n\) habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely \(\displaystyle 1,2,\dots,\binom n2\) monetary units in some order. Prove that \(\displaystyle n\) or \(\displaystyle n-2\) is a square number.
(5 pont)
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A. 723. Let \(\displaystyle f\colon \mathbb{R}\to \mathbb{R}\) be a continuous function such that the limit
\(\displaystyle g(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2} \)
exists for all real \(\displaystyle x\). Prove that \(\displaystyle g(x)\) is constant if and only if \(\displaystyle f(x)\) is a polynomial function whose degree is at most \(\displaystyle 2\).
(5 pont)
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A. 724. A sphere \(\displaystyle \mathcal{G}\) lies within tetrahedron \(\displaystyle ABCD\), touching faces \(\displaystyle ABD\), \(\displaystyle ACD\), and \(\displaystyle BCD\), but having no point in common with plane \(\displaystyle ABC\). Let \(\displaystyle E\) be the point in the interior of the tetrahedron for which \(\displaystyle \mathcal{G}\) touches planes \(\displaystyle ABE\), \(\displaystyle ACE\), and \(\displaystyle BCE\) as well. Suppose the line \(\displaystyle DE\) meets face \(\displaystyle ABC\) at \(\displaystyle F\), and let \(\displaystyle L\) be the point of \(\displaystyle \mathcal{G}\) nearest to plane \(\displaystyle ABC\). Show that segment \(\displaystyle FL\) passes through the centre of the inscribed sphere of tetrahedron \(\displaystyle ABCE\).
(5 pont)
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