KöMaL Problems in Mathematics, April 2008
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Problems with sign 'C'Deadline expired on May 15, 2008. |
C. 940. Prove if n is a positive integer then 2^{4n}-1 or 2^{4n}+1 is divisible by 17.
(5 pont)
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C. 941. There are two lamps on the ceiling of a rectangular classroom of length 10 m. They emit conical light beam of apex angles of 90^{o}. One is situated in the middle of the ceiling, and illuminates a circle of diameter 6 m on the floor. The lampshade of the other lamp is turned so that the illuminated region on the floor is long enough to draw a line segment of length 10 m on it in the direction parallel to the length of the room, but no light falls on the walls at the ends of the room. Find the distance between the two lamps.
(5 pont)
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C. 942. In an arithmetic progression of common difference d, a_{1}=1 and a_{n}=81. In a geometric progression of common ratio q, b_{1}=1 and b_{n}=81. Given that , find all such sequences.
(5 pont)
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C. 943. The area of a sector of a circle is 100. what is the radius if the perimeter is a minimum?
(5 pont)
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C. 944. Rick, Dan and Alan play table tennis with two players at one side of the table and the third one at the other side. Dan and Alan playing together beat Rick three times as often as Rick beats them, Dan wins the game against Rick and Alan as often as he loses it, and Alan wins the game against Rick and Dan twice as often as he loses it. Last time they played six games during the afternoon, two in each arrangement of players. What is the probability that Rick won at least once?
(5 pont)
Problems with sign 'B'Deadline expired on May 15, 2008. |
B. 4082. The diagram shows a plane figure that cannot cover a unit semicircle but two congruent copies of the figure can cover a unit circle. Is there a convex figure that has the same property?
Kvant
(4 pont)
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B. 4083. Each of the digits 0 to 9 is written in 10 fields of a 10×10 table. Is it possible that every row and every column has at most 4 different digits in it?
Kvant
(4 pont)
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B. 4084. Find all sequences (a_{n}) of positive integers, such that (a_{i},a_{j})=(i,j) for all ij.
(3 pont)
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B. 4085. Prove that if a symmetrical trapezium has an inscribed circle then its height is the geometric mean of the bases.
(3 pont)
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B. 4086. Four points are selected at random on a sphere, independently of each other. What is the probability that the resulting tetrahedron contains the center of the sphere?
(5 pont)
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B. 4087. Prove that if the lengths of the sides of a triangle are 2, 3 and 4 then it has angles and , such that 2+3=180^{o}.
Suggested by Z. Varga, Siófok
(3 pont)
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B. 4088. P is an interior point of side B_{1}B_{2} of an acute-angled triangle AB_{1}B_{2}. Let Q denote the reflection of P about point A. Let D_{i} denote the perpendicular projection of P onto the line segment AB_{i}, and let F_{i} be the midpoint of the line segment D_{i}B_{i}. Prove that if QD_{1} is perpendicular to PF_{1}, then QD_{2} is perpendicular to PF_{2}.
Suggested by J. Bodnár, Budapest
(5 pont)
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B. 4089. Solve the equation x^{4}-7x^{3}+13x^{2}-7x+1=0.
(4 pont)
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B. 4090. The angle bisector f drawn from vertex A of an acute-angled triangle ABC intersects side BC at D and the circumscribed circle at E. The altitude drawn from C intersects f at M and the circumscribed circle at Q. The altitude drawn from B intersects f at N and the circumscribed circle at P. Prove that
(4 pont)
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B. 4091. Prove that
for arbitrary positive integers m, t.
(5 pont)
Problems with sign 'A'Deadline expired on May 15, 2008. |
A. 452. A simple graph on 2n points has n^{2}+1 edges. Prove that the edges form at least n triangles.
(5 pont)
A. 453. n points are selected at random on a sphere, independently of each other. What is the probability that the convex hull of the selected points contains the center of the sphere?
(5 pont)
A. 454. Is there any polynomial p having degree at least 1 over the reals that satisfies p^{2}(x)-1=p(x^{2}+1) for all real numbers x?
(5 pont)
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