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Problems in Mathematics, May 2010

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 June 2010.

A. 509. Prove that there exists a real number c>0 with the following property: among arbitrary, pairwise distinct positive integers a_1,a_2,\ldots,a_n (n\ge3), there are three whose least common multiple is at least c.n2.99.

(5 points)

Solution (in Hungarian)

A. 510. There is given a positive integer n and some straight lines in the plane such that none of the lines passes through (0,0), and every lattice point (a,b), where 0\lea,b\len are integers and a+b>0, is contained by at least a+b+1 of the lines. Prove that the number of the lines is at least n(n+3).

(5 points)

Statistics

A. 511. Show that for arbitrary positive integers nk there exists a polynomial p(x), with degree at most 100\sqrt{nk}, such that p(0) >
\big(\big|p(1)\big|+\ldots+\big|p(n)\big|\big) +
\big(\big|p(-1)\big|+\ldots+\big|p(-k)\big|\big).

(5 points)

Statistics


Problems with sign 'B'

Deadline expired on 10 June 2010.

B. 4272. The terms of sequence (an) are positive integers, such that an+1=an2+5an+1 for all n\ge1. Is it possible that the terms of the sequence are all composite numbers?

(5 points)

Solution (in Hungarian)

B. 4273. Given six circles in the plane that have an interior point in common. Prove that there is a circle among them that contains the centre of another in its interior.

(4 points)

Solution (in Hungarian)

B. 4274. The lengths of the sides of a parallelogram of unit area are a and b, where a<b<2a. What is the area of the quadrilateral bounded by the interior angle bisectors of the parallelogram?

(3 points)

Solution (in Hungarian)

B. 4275. Solve the equation \(\displaystyle x^{6}-x^{3}-2x^{2}-1=2(x-x^{3}+1)\sqrt{x}\).

Suggested by F. Pintér, Nagykanizsa, J. Szoldatics, Budapest

(4 points)

Solution (in Hungarian)

B. 4276. Show that an altitude of a triangle cannot be longer than the geometric mean of the radii of the excircles touching the two adjacent sides of the triangle.

(4 points)

Solution (in Hungarian)

B. 4277. Solve the equation \(\displaystyle x^{3}+y^{3}+1=x^{2}y^{2}\) on the set of integers.

Suggested by L. Surányi Budapest

(5 points)

Solution (in Hungarian)

B. 4278. Solve the simultaneous equations x+y=a, tan x.tan y=b, where a and b are real parameters.

(3 points)

Solution (in Hungarian)

B. 4279. Is it true that if the sum of distances of any interior point of a tetrahedron from the faces is constant, then the tetrahedron is regular?

(4 points)

Solution (in Hungarian)

B. 4280. M is the midpoint of the arc AB containing vertex C on the circumscribed circle k of triangle ABC. J is the centre of the escribed circle drawn to side AB. The perpendicular drawn to angle bisector CJ at point J intersects line AC at D and line BC at E. Line MJ intersects circle k again at F. Prove that the circle passing through points D, E, F touches the lines AC and BC, and also touches the circle k.

(5 points)

Solution (in Hungarian)

B. 4281. Someone chose n not necessarily different integers, and formed sums out of them in all (2n-1) possible ways. Given that 0 was not obtained, is it possible to determine the original numbers from the sums?

(5 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 June 2010.

C. 1035. In a mathematics competition, there were three problems. 56 participants solved at least one problem. 2 participants solved all problems. Out of those solving the second problems, 10 more solved the third problem than the first one. The number of those solving both of the first and second problems was 10 larger than the number of those solving the third problem only. All participants solving the first and third problems solved the second problem, too. There were 14 participants altogether who solved the first problem only or the second problem only. How many participants solved the third problem?

(5 points)

Solution (in Hungarian)

C. 1036. How many 9-digit numbers (in decimal system) divisible by 11 are there in which every digit occurs except zero?

(5 points)

Solution (in Hungarian)

C. 1037. Find the isosceles triangle of minimum area circumscribed about a semicircle, such that its base lies on the line of the diameter of the semicircle and its legs are tangent to the semicircle.

(5 points)

Solution (in Hungarian)

C. 1038. Define the function \mathop{\rm lac}\, (x) on the set of real numbers as follows:


\mathop{\rm lac}\, (x)=x, {\rm if \ } x\in [2n; 2n+1], {
\rm \ where \ } n\in\mathbb{Z}, \quad {\rm or} \quad 
-x+4n+3,  {\rm if \ } x\in 
]2n+1; 2n+2[, {\rm \ where \ } n\in\mathbb{Z}.

Solve the equation \mathop{\rm lac}\, (2x^2 + x + 4)=\mathop{\rm lac}\, (x^2 + 7x -1) on the set of real numbers.

(5 points)

Solution (in Hungarian)

C. 1039. There are four unit spheres inside a larger sphere, such that each of them touches the large sphere and the other three unit spheres. What fraction of the volume of the large sphere is filled by the four unit spheres altogether?

(5 points)

Solution (in Hungarian)


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