KöMaL Problems in Mathematics, April 2011
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Problems with sign 'C'Deadline expired on May 10, 2011. |
C. 1075. Find the three-digit number that is twelve times the sum of its digits.
(5 pont)
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C. 1076. The roots of a quadratic equation are the squares of two consecutive integers. The geometric mean of the two roots is 1 greater than the difference of the roots. Write down the quadratic equation in question.
(5 pont)
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C. 1077. E is the point dividing side AC of triangle ABC in a 3:1 ratio, such that EC is the shorter part. The line passing through E and the midpoint F of side BC intersects the line AB at D. What percentage is the area of triangle ADE of the triangle ABC?
(5 pont)
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C. 1078. Define ``words'' as strings of at most ten characters made out of the 26 letters of the English alphabet. (A ``word'' may as well consist of a single letter, and two ``words'' are considered different if they differ in at least one letter.) Prove that the number of all possible ``words'' obtained in this way is divisible by 27.
(5 pont)
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C. 1079. We have a champagne glass shaped like a truncated cone. The radius of the base circle is 1 cm, the radius of the top circle is 4 cm and its height is 6 cm (see the figure). To what height do we need to fill the glass to have half a glass of drink?
(5 pont)
Problems with sign 'B'Deadline expired on May 10, 2011. |
B. 4352. What is the minimum possible number of convex pentagons needed to put together a convex 2011-sided polygon out of them?
(Suggested by J. Mészáros, Jóka)
(3 pont)
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B. 4353. Let A be a positive integer, and let B denote the number obtained by writing the digits of A in reverse order. Show that at least one of the numbers A+B and A-B is divisible by 11.
(Suggested by J. Mészáros, Jóka)
(3 pont)
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B. 4354. Given four distinct points A, B, C, D on the plane, construct two circles touching each other on the outside such that one of them passes through A and B, the other passes through C and D, and their point of tangency lies on the line segment BC.
(4 pont)
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B. 4355. Prove that if the product of the positive numbers x, y and z is 1, then
(Based on the idea of J. Szoldatics, Dunakeszi)
(4 pont)
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B. 4356. On the same side of a given line segment AB, there are at most six similar triangles one side of which is AB. Prove that the third vertices of all these triangles are concyclic.
(4 pont)
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B. 4357. Let n>1 be a positive integer. Show that n^{3}-n^{2} is a factor of the binomial coefficient .
(Suggested by G. Holló, Budapest)
(3 pont)
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B. 4358. Solve the following equation: .
(Suggested by B. Kovács, Szatmárnémeti)
(5 pont)
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B. 4359. A regular tetrahedron is intersected with a plane perpendicular to one of its faces in line e. The lines of intersection of this plane with the other three faces enclose angles of _{1}, _{2}, _{3} with e. Determine the possible values of he expression .
(5 pont)
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B. 4360. Let S(n) denote the sum of the digits for any positive integer n. For what integers k greater than 1 is there a positive real number c_{k} such that S(kn)c_{k}^{.}S(n) for all positive integers n?
(Suggested by P. Erben, Budapest)
(5 pont)
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B. 4361. Let a and b denote the semi-axes of an ellipse. Let O denote the centre, and let be points on the curve such that each of the angles P_{i}OP_{i+1} is . Prove that if n>1 then .
(5 pont)
Problems with sign 'A'Deadline expired on May 10, 2011. |
A. 533. Determine all triples (a,b,c) of positive integers for which abc+1 divides a^{2}+b^{2}.
(5 pont)
A. 534. The sides of a triangle are a, b and c, the lengths of the corresponding medians are s_{a}, s_{b} and s_{c}, respectively. Prove that .
(Proposed by: Donát Nagy, Szeged)
(5 pont)
A. 535. There is given a simple graph G with vertices v_{1},...,v_{n}. H_{1},...,H_{n} are sets of nonnegative integers such that for every i=1,2,...,n, the cardinality of H_{i} is at most half of the degree of v_{i}. Prove that G contains a subgraph G' with the same vertices such that the degree of v_{i} in G' is not an element of H_{i} for any i.
(Proposed by: László Miklós Lovász, Budapest)
(5 pont)
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