KöMaL Problems in Mathematics, April 2009
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Problems with sign 'C'Deadline expired on May 15, 2009. |
C. 985. A two-digit number is multiplied by 4, and the original two-digit number is written behind the number obtained. The resulting number has exactly six divisors. What may be the original two-digit number?
(5 pont)
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C. 986. Find all integers that may be the measure, in degrees, of the angles of a regular polygon.
(5 pont)
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C. 987. The lengths of the sides of a triangle cut out of paper are 8 cm, 10 cm and 12 cm. The triangle is folded along a line through the common vertex so that the shortest side overlaps with the longest side. A double-layer part and a single-layer part are obtained. Prove that the single-layer part is an isosceles triangle.
(5 pont)
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C. 988. Given that 4 passengers in a metro train of 6 carriages have colds, what is the probability that there are at most two carriages in which there is a passenger who has a cold?
(5 pont)
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C. 989. A cubical playing die is made out of a spherical body by cutting off six identical spherical caps. Each of the circular sections is tangent to the four adjacent ones. What percentage is the total area of the six circles of the whole surface area of the resulting die?
(5 pont)
Problems with sign 'B'Deadline expired on May 15, 2009. |
B. 4172. Let n be a positive integer and let denote an arbitrary order of the integers 1 to n. What is the largest possible value of the expression
(1-k_{1})^{2}+(2-k_{2})^{2}+(3-k_{3})^{2}+...+(n-k_{n})^{2}
of n terms?
(4 pont)
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B. 4173. Determine those convex quadrilaterals ABCD that have an interior point P for which the triangles ABP, BCP, CDP, DAP have equal areas.
Suggested by P. Maga
(4 pont)
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B. 4174. Solve the following simultaneous equations on the set of real numbers:
(1) | 4a+bc=32, |
(2) | 2a-2c-b^{2}=0, |
(3) | a+12b-c-ab=6. |
(4 pont)
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B. 4175. Let A, B, C, D be any points in the plane. Prove that if the circles ABC and ABD intersect each other at right angles then the circles ACD and BCD also intersect each other at right angles.
(4 pont)
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B. 4176. Solve the following equation:
(sin x+sin 2x+sin 3x)^{2}+(cos x+cos 2x+cos 3x)^{2}=1.
(4 pont)
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B. 4177. The tangents drawn to the circumscribed circle of triangle ABC at the points B and C intersect each other at point M. The line drawn through M parallel to AB intersects line AC at N. Prove that AN=BN.
(4 pont)
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B. 4178. Prove that for all positive integers n and k, the greatest common divisor of the numbers is 1.
Miklós Schweitzer Memorial Competition, 1949
(5 pont)
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B. 4179. The vertex C of a parabola is the centre of a circle that passes through the focus F of the parabola. Let the intersections of the parabola and the circle be A and B, let AB and CF intersect at E, and let D denote the point of the circle that lies opposite to F. Show that DE is the geometric mean of FE and the diameter of the circle.
(3 pont)
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B. 4180. Prove that the sequence contains infinitely many integer powers of 3.
(5 pont)
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B. 4181. The opposite edges of a tetrahedron are equal in length and they pairwise enclose equal angles. Prove that the tetrahedron is regular.
(4 pont)
Problems with sign 'A'Deadline expired on May 15, 2009. |
A. 479. Decide whether there exists a positive integer n that is divisible by 103 and satisfies .
Dutch competition problem, composed by Hendrik Lenstra, Leiden
(5 pont)
A. 480. Let p(z) be a complex polynomial of degree n, and suppose that all (complex) roots of p(z) are of unit modulus. For every real number c0, show that the roots of the polynomial
2z(z-1)p'(z)+((c-n)z+(c+n))p(z)
are of unit modulus as well.
(5 pont)
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A. 481. Prove that there are infinitely many n, for which there exist simple graphs S_{1},...,S_{n} with the following properties:
(a) each S_{i} is a complete bipartite graph;
(b) the union of the graphs S_{1},...,S_{n} is a complete graph on 2n vertices;
(c) each edge of this complete graph is contained in an odd number of the graphs S_{i}.
(5 pont)
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