KöMaL Problems in Mathematics, May 2020
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Problems with sign 'C'Deadline expired on June 10, 2020. |
C. 1609. Solve the following simultaneous equations over the set of real numbers:
$$\begin{align*} x+y+\frac xy & =19,\\ \frac{x(x+y)}{y} & =60. \end{align*}$$(5 pont)
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C. 1610. In a unit circle, the diameter \(\displaystyle AB\) and the chord \(\displaystyle AC\) enclose a \(\displaystyle 30^\circ\) angle. Let \(\displaystyle B'\) denote the reflection of \(\displaystyle B\) about the point \(\displaystyle C\). Determine the distances between \(\displaystyle B\) and the points where the tangents drawn from \(\displaystyle B'\) to the circle intersect the line \(\displaystyle AB\).
(5 pont)
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C. 1611. Some numbers are selected from the set of the first 21 positive integers such that the absolute values of the differences of all pairs of selected numbers should be different. What is the largest possible number of different absolute values obtained? Give an example of a case when this occurs.
(5 pont)
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C. 1612. The convex heptagon \(\displaystyle A_1A_2A_3A_4A_5A_6A_7\) has a circumscribed circle centred at an interior point of the heptagon. Prove that the sum of the interior angles at the vertices \(\displaystyle A_1\), \(\displaystyle A_3\) and \(\displaystyle A_5\) is less than \(\displaystyle 450^{\circ}\).
(5 pont)
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C. 1613. There were \(\displaystyle n\) teams participating in a basketball championship. Every team played every other team exactly once, and there was no draw. At the end of the championship, the \(\displaystyle i\)th team had \(\displaystyle x_i\) games won and \(\displaystyle y_i\) games lost (\(\displaystyle i=1,2,\ldots,n\)). Prove that
\(\displaystyle x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2. \)
(Croatian problem)
(5 pont)
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C. 1614. 12 round muffins of diameter 9 cm are arranged along the edge of a round tray of radius 30 cm such that they all touch the edge of the tray, and the neighboring muffins are separated by the same distance from each other. What is this equal distance?
(5 pont)
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C. 1615. The text was changed. The correct text:
Julie's grandmother bakes cookies every Monday. She always selects out of her infinite number of recipes at random. \(\displaystyle 60\%\) of her recipes contain chocolate chips. Julie is quite picky about cookies: she only likes \(\displaystyle 90\%\) of grandma's chocolate chip cookies, and only \(\displaystyle 30\%\) of the other kinds of cookies. On a special Monday, grandmother is making two different kinds of cookies. Find the probability that Julie will like exactly one of them.
(5 pont)
Problems with sign 'B'Deadline expired on June 10, 2020. |
B. 5102. There are \(\displaystyle n\) distinct points in the plane, which are not all collinear. Show that there exists a closed polygon with these vertices that does not cut through itself. (A polygon is allowed to have angles equal to \(\displaystyle 180^{\circ}\), too.)
(3 pont)
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B. 5103. Let \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle z\) be positive numbers that satisfy the equalities \(\displaystyle a^2+b^2=c^2\) and \(\displaystyle x^2+y^2=z^2\). Prove that \(\displaystyle {(a+x)}^2+{(b+y)}^2\le {(c+z)}^2\), and determine the condition for equality.
Proposed by S. Kiss, Nyíregyháza
(3 pont)
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B. 5104. Let \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\) denote the points of tangency of the incircle of triangle \(\displaystyle ABC\) on the sides, and let \(\displaystyle R\) and \(\displaystyle r\) be the radii of the circumscribed and inscribed circles, respectively. Prove that the ratio of the areas of triangles \(\displaystyle A_1B_1C_1\) and \(\displaystyle ABC\) is \(\displaystyle r:2R\).
(4 pont)
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B. 5105. Let \(\displaystyle n\) denote a positive integer. Determine the smallest number of colours \(\displaystyle k\) that are sufficient for colouring the edges of any directed simple graph of \(\displaystyle n\) vertices without producing a circuit of the same colour.
Proposed by K. Szabó, 11th grade student of Fazekas Mihály Primary and Secondary School and Training Centre, Budapest
(4 pont)
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B. 5106. The numbers \(\displaystyle n+1, n+2, \ldots, 2n\) are written on a blackboard (\(\displaystyle n\ge2\)), and the following procedure is repeated: two numbers are selected (\(\displaystyle x\) and \(\displaystyle y\)) from the board, erased, and replaced with the numbers \(\displaystyle x+y+\sqrt{x^2+y^2}\) and \(\displaystyle x+y-\sqrt{x^2+y^2}\). Prove that there will never be a number less than 1.442 written on the board.
(5 pont)
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B. 5107. The diagonals of a cyclic quadrilateral \(\displaystyle ABCD\) intersect at \(\displaystyle F\), the lines of sides \(\displaystyle AB\) and \(\displaystyle CD\) intersect at \(\displaystyle E\), the midpoint of line segment \(\displaystyle EF\) is \(\displaystyle G\), the midpoint of line segment \(\displaystyle BF\) is \(\displaystyle H\), and the midpoint of side \(\displaystyle BC\) is \(\displaystyle I\). Show that \(\displaystyle \angle GFD=\angle GIH\).
(6 pont)
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B. 5108. The points \(\displaystyle A\), \(\displaystyle B_1\), \(\displaystyle B_2\), \(\displaystyle B_3\), \(\displaystyle C_1\), \(\displaystyle C_2\), \(\displaystyle C_3\), in this order, lie on the same line. On one side of this line, perpendicular rays \(\displaystyle b_i\) are drawn from the points \(\displaystyle B_i\), and semicircles \(\displaystyle c_i\) are drawn with diameters \(\displaystyle AC_i\) (\(\displaystyle i=1,2,3\)), as shown in the figure. Prove that if the region bounded by \(\displaystyle b_1\), \(\displaystyle c_1\), \(\displaystyle b_2\), \(\displaystyle c_2\) and the region bounded by \(\displaystyle b_2\), \(\displaystyle c_2\), \(\displaystyle b_3\), \(\displaystyle c_3\) both have inscribed circles then the region bounded by \(\displaystyle b_1\), \(\displaystyle c_1\), \(\displaystyle b_3\), \(\displaystyle c_3\) also has an inscribed circle.
(5 pont)
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B. 5109. Let
\(\displaystyle x_1 = 2, \quad x_2 = 7, \quad x_{n+1} = 4 x_n - x_{n-1} \quad (n=2,3,\ldots). \)
Is there a perfect square in this sequence?
Proposed by G. Stoica, Saint John, Canada
(6 pont)
Problems with sign 'A'Deadline expired on June 10, 2020. |
A. 777. Finite graph \(\displaystyle G(V,E)\) on \(\displaystyle n\) points is drawn in the plane. For an edge \(\displaystyle e\) of the graph let \(\displaystyle x(e)\) denote the number of edges that cross over edge \(\displaystyle e\). Prove that
\(\displaystyle \sum\limits_{e\in E} \frac{1}{x(e)+1}\le 3n-6. \)
Submitted by Dömötör Pálvölgyi, Budapest
(7 pont)
A. 778. Find all square-free integers \(\displaystyle d\) for which there exist positive integers \(\displaystyle x\), \(\displaystyle y\) and \(\displaystyle n\) satisfying \(\displaystyle x^2+dy^2 = 2^n\).
Submitted by Kada Williams, Cambridge
(7 pont)
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A. 779. Two circles are given in the plane, \(\displaystyle \Omega\) and inside it \(\displaystyle \omega\). The center of \(\displaystyle \omega\) is \(\displaystyle I\). \(\displaystyle P\) is a point moving on \(\displaystyle \Omega\). The second intersection of the tangents from \(\displaystyle P\) to \(\displaystyle \omega\) and circle \(\displaystyle \Omega\) are \(\displaystyle Q\) and \(\displaystyle R\). The second intersection of circle \(\displaystyle IQR\) and lines \(\displaystyle PI\), \(\displaystyle PQ\) and \(\displaystyle PR\) are \(\displaystyle J\), \(\displaystyle S\) and \(\displaystyle T\), respectively. The reflection of point \(\displaystyle J\) across line \(\displaystyle ST\) is \(\displaystyle K\).
Prove that lines \(\displaystyle PK\) are concurrent.
(7 pont)
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