Competitions Portal

Problems in Mathematics, May 2011

Please read the rules of the competition.

Problems sign 'A'

Deadline expired.

A. 536. The positive real numbers a, b, c, d satisfy a+b+c+d=abc+abd+acd+bcd. Prove that

${(a+b)(c+d)}+(a+d)(b+c) \ge 4\sqrt{(1+ac)(1+bd)}.$

(5 points)

A. 537. The edges of the complete graph on n vertices are labeled by the numbers $1,2,\dots,\binom{n}{2}$ in such a way that each number is used exactly once. Prove that if n is sufficiently large then there exists a (possible cyclic) path of three edges such that the sum of the numbers assigned to these edges is at most 3n-1000.

(Kolmogorov Cup, 2009; a problem by I. Bogdanov, G. Chelnokov and K. Knop)

(5 points)

A. 538. In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines a₁, a₂, r₁, r₂ in such positions that a₁ and a₂ are perpendicular to $\mathcal{P}$ , r₁ is coplanar with a₁, r₂ is coplanar with a₂, finally r₁ and r₂ intersect $\mathcal{P}$ at the same angle. Rotate r₁ around a₁ and rotate r₂ around a₂; denote by $\mathcal{S}_1$ and $\mathcal{S}_2$ the two surfaces of revolution they sweep out. Show that the common points of $\mathcal{S}_1$ and $\mathcal{S}_2$ lie in a plane.

(5 points)

Problems sign 'B'

Deadline expired.

B. 4362. Each vertex of a solid cube is cut off. The faces of the resulting polyhedron are 8 triangles and 6 heptagons. What are the possible numbers of vertices and edges of such polyhedra?

(3 points)

Solution (in Hungarian)

B. 4363. The reciprocals of the natural numbers 2 to 2011 are written on a blackboard. In each step, two numbers x and y are erased and replaced with the number $\frac{xy}{xy + (1-x)(1-y)}$ . This step is repeated 2009 times, until a single number remains. What may the remaining number be?

(Suggested by B. Kovács, Szatmárnémeti)

(4 points)

Solution (in Hungarian)

B. 4364. Let a $\ge$ b $\ge$ c>0. Prove that $\frac{a^{2}-b^{2}}{c} +\frac{c^{2}-b^{2}}{a} +\frac{a^{2}-c^{2}}{b}\ge 3a-4b+c$ .

(Suggested by J. Mészáros, Jóka)

(4 points)

Solution (in Hungarian)

B. 4365. Find all positive integers n such that 2ⁿ-1 and 2ⁿ⁺²-1 are both primes, and 2ⁿ⁺¹-1 is not divisible by 7.

(Suggested by S. Kiss, Budapest)

(3 points)

Solution (in Hungarian)

B. 4366. Let M denote the orthocentre of the acute-angled triangle ABC, and let A₁, B₁, C₁, respectively, denote the circumcentres of triangles BCM, CAM, ABM. Prove that the lines AA₁, BB₁ és CC₁ are concurrent.

(4 points)

Solution (in Hungarian)

B. 4367. Solve the following equation: $\frac{3x+3}{\sqrt{x}} =4+\frac{x+1}{\sqrt{x^{2}-x+1}}$ .

(Suggested by J. Mészáros, Jóka)

(4 points)

Solution (in Hungarian)

B. 4368. Let D, E and F, respectively, be points on sides AB, BC, CA of a triangle ABC such that AD:DB=BE:EC=CF:FA $\ne$ 1. The lines AE, BF, CD intersect one another at points G, H, I, respectively. Prove that the centroids of triangles ABC and GHI coincide.

(Suggested by Sz. Miklós, Herceghalom)

(3 points)

Solution (in Hungarian)

B. 4369. Each of the circles k₁, k₂ and k₃ passes through a point P, and the circles k_i and k_j also pass through the point M_i,j. Let A be an arbitrary point of circle k₁. Let k₄ be an arbitrary circle passing through A and M_1,2, and let k₅ be an arbitrary circle passing through A and M_1,3. Show that if the other intersections of the pairs of circles k₄ and k₂, k₅ and k₃, k₄ and k₅ are B, C and D, respectively, then the points M_2,3, B, C, D are either concyclic or collinear.

(4 points)

Solution (in Hungarian)

B. 4370. Let a, b, c denote the lengths of the sides of a triangle, and let u, v, w, respectively, be the distances of the centre of the incircle from the vertices opposite to the sides. Prove that $(a+b+c) \left(\frac{1}{u}+\frac{1}{v}+\frac{1}{w}\right)\le 3\left(\frac{a}{u}+\frac{b}{v}+\frac{c}{w}\right)$ .

(Suggested by J. Mészáros, Jóka)

(5 points)

Solution (in Hungarian)

B. 4371. Prove that

$\frac{1}{\sin^2{\frac{\pi}{14}}} + \frac{1}{\sin^2{\frac{3\pi}{14}}} + \frac{1}{\sin^2{\frac{5\pi}{14}}} = 24.$

(Suggested by B. Kovács, Szatmárnémeti)

(5 points)

Solution (in Hungarian)

Problems sign 'C'

Deadline expired.

C. 1080. A steerable airship has two engines and a given supply of fuel. If both engines are operated, the airship covers 88 kilometres per hour. If the first engine were used only, the fuel would last 25 hours longer, but the distance covered per hour would only be 45 km. If the second engine were used only, the fuel would last 16 hours longer than with two engines, and the distance covered per hour would only be 72 km. In which case can the airship travel the longest distance?

(5 points)

Solution (in Hungarian)

C. 1081. Two regular polygons are said to be matching if the double of the interior angle of one of them equals the triple of the exterior angle of the other. Find all pairs of matching polygons.

(5 points)

Solution (in Hungarian)

C. 1082. The first digit of a six-digit number is transferred to the end of the number. Then the first digit of the resulting six-digit number is, again, transferred to the end of the number. The six-digit number obtained in this way is three times the previous number and $\frac 35$ times the original number. What is the original six-digit number?

(5 points)

Solution (in Hungarian)

C. 1083. The length of one side of a triangle is 8 cm, one of the angles lying on it is 60^o, and the radius of the inscribed circle is $\sqrt 3$ cm. How long are the other two sides?

(5 points)

Solution (in Hungarian)

C. 1084. The point P(8;4) divides a chord of the parabola y²=4x in a 1:4 ratio. Find the coordinates of the endpoints of the chord.

(5 points)

Solution (in Hungarian)

Send your solutions to the following address:

or by e-mail to:

solutions@komal.hu

Our web pages are supported by:

ELTE