KöMaL Problems in Mathematics, April 2005
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Problems with sign 'C'Deadline expired on May 17, 2005. |
C. 805. Find all sets of three integers, such that their product is four times their sum, and one number is twice the sum of the other two.
(5 pont)
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C. 806. Find all positive integers divisible by 7 that end in 5 in the decimal notation and the rest of their digits are 1.
(5 pont)
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C. 807. The lengths of two adjacent sides of a quadrilateral are 2 units and 1 unit, and they enclose an angle of 60^{o}. The quadrilateral is cyclic and it is also a tangent quadrilateral. What are the lengths of the other two sides?
(5 pont)
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C. 808. Solve the equation {3x}^{2}+{x}^{2}=1.
(5 pont)
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C. 809. The midpoint of the edge AE of the unit cube ABCDEFGH is P, and the midpoint of the face BCGF is R.
a) Find the area of the intersection of the cube with the plane through the points P, B, R.
b) The above plane cuts the cube into two solids. What is the ratio of the volumes of the two parts?
(5 pont)
Problems with sign 'B'Deadline expired on May 17, 2005. |
B. 3812. Find all positive integers n, such that
a) 7^{399}|n! , but , and
b) 7^{400}|n! , but .
(4 pont)
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B. 3813. Prove that there is exactly one single point M inside an arbitrary triangle ABC, such that
MA+BC=MB+AC =MC+AB.
(4 pont)
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B. 3814. n and k are positive integers, such that 2kn|k^{2}+n^{2}-k. Prove that k is a square number.
(4 pont)
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B. 3815. A plane intersects each line segment of a closed polygon P_{1}P_{2}...P_{n}P_{1} of the space. The intersection with the line segment P_{i}P_{i+1} is the interior point Q_{i}. Prove that
(3 pont)
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B. 3816. The centre of the inscribed circle of the triangle ABC is O. The extensions of the line segments AO, BO, CO beyond O intersect the circumscribed circle at A_{1}, B_{1}, C_{1}, respectively. Prove that the area of the triangle A_{1}B_{1}C_{1} is
where R is the radius of the circumscribed circle, and , , are the angles of the original triangle ABC.
(4 pont)
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B. 3817. The sequence a_{n} is defined as follows:
a_{1}2, , , if n2.
Prove that the inequality
a_{1}+2a_{2}+3a_{3}+...+na_{n}=a_{1}^{.}a_{2}^{.}a_{3}^{.}...^{.}a_{n}
is true for all n.
(5 pont)
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B. 3818. The length of all three edges from a certain vertex of a tetrahedron is unity, and they pairwise enclose 45 angles. Find the volume of the tetrahedron. (based on the idea of
(4 pont)
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B. 3819. Show that if A_{1}B_{1}, A_{2}B_{2} and A_{3}B_{3} are three parallel chords of a circle, then the perpendiculars dropped from the point A_{1}, A_{2} and A_{3} onto the line B_{2}B_{3}, B_{3}B_{1} and B_{1}B_{2}, respectively, are concurrent.
(5 pont)
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B. 3820. Solve the following equation:
(5^{x}-2^{x-2})^{2}+ 2lg (5^{x}+2^{x-2})=x.
(4 pont)
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B. 3821. a, b, c are positive numbers, such that a^{2}+b^{2}+c^{2}=1. Find the smallest possible value of the sum
(5 pont)
Problems with sign 'A'Deadline expired on May 17, 2005. |
A. 371. Given that a+bc+1, b+c a+1, c+ab+1 for the numbers a,b,c 0, show that a^{2}+b^{2}+c^{2}2abc+1.
(5 pont)
A. 372. An equilateral triangle of side n is divided into equilateral triangles of unit side. How many castles can be placed on the lattice points of the triangular lattice obtained so that no two of them attack each other? Castles can move parallel to the sides of the triangle, in six directions altogether. (Suggested by A. Egri, Hajdúszoboszló)
(5 pont)
A. 373. P is a point in the interior of the quadrilateral A_{1}A_{2}A_{3}A_{4} that does not lie on either diagonal. The points B_{i} lie in the interior of each line segment A_{i}P. Let C_{ij} be the intersection of the lines A_{i}B_{j} and A_{j}B_{i} (1i<j 4). Prove that the line segments C_{12}C_{34}, C_{13}C_{24}, C_{14}C_{23} are all concurrent.
(5 pont)
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