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Problems in Mathematics, May 2013

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 June 2013.

A. 590. Find all integer numbers a for which there exists a polynomial p(x) with integer coefficients satisfying


p\big(\big(\root3\of {a}\,\big)^2 + \root3\of {a}\,\big) =
\big(\root3\of {a}\,\big)^2 - \root3\of {a}.

Based on a problem of the Vojtech Jarník competition

(5 points)

Solution (in Hungarian)

A. 591. There is given a convex quadrilateral ABCD and some points P, Q, R and S on the line segments AB, BC, CD and DA, respectively. The line segments PR and QS meet at T. Suppose that each of the the quadrilaterals APTS, BQTP, CRTQ and DSTR have an inscribed circle. Prove that the quadrilateral ABCD also has an inscribed circle.

(5 points)

Solution (in Hungarian)

A. 592. Color the vertices of a binary tree of depth 22n as follows. In the initial position let all vertices be white. Take the vertices one by one in a random order. Color the current vertex to red, except if from that vertex there starts a path of length n downwards (away from the root), whose other vertices are already red. Let p(n) be the probability that during the procedure, the root remains white. Determine \lim_{n\to\infty} p(n).

Proposed by Endre Csóka

(5 points)

Solution (in Hungarian)


Problems with sign 'B'

Deadline expired on 10 June 2013.

B. 4542. The right-angled vertex of a right angled triangle is projected orthogonally onto the angle bisector of one of the acute angles. Prove that the projection lies on the midline parallel to the hypotenuse.

F. Olosz, Szatmárnémeti

(3 points)

Solution (in Hungarian)

B. 4543. Some vertices of a regular nonagon are coloured red, the rest of them are coloured black. A triangle is said to be ``boring'' if all of its vertices are the same colour. Prove that there are two congruent ``boring'' triangles.

Matlap, Kolozsvár

(4 points)

Solution (in Hungarian)

B. 4544. Solve the simultaneous equations below.

x+y+z=3,

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{5}{12},

x3+y3+z3=45.

Competition problem from the Felvidék

(4 points)

Solution (in Hungarian)

B. 4545. Is it possible for the sum of the reciprocals of 2013 different positive integers to be

a) 2.013;

b) 20.13?

Matlap, Kolozsvár

(5 points)

Solution (in Hungarian)

B. 4546. What is the largest possible number of rays from a common starting point in the space such that they pairwise enclose obtuse angles?

(5 points)

Solution (in Hungarian)

B. 4547. Determine the minimum value of the expression \sqrt{1-x+x^{2}}+\sqrt{1-\sqrt{3}\cdot x+x^{2}}\,.

(5 points)

Solution (in Hungarian)

B. 4548. ABC is an isosceles right angled triangle of unit legs. A1 is a point on side AB, B1 is a point on side BC, and C1 is a point on hypotenuse CA. What is the minimum possible length of A1B1 if the triangles ABC and A1B1C1 are similar?

Kvant

(4 points)

Solution (in Hungarian)

B. 4549. Prove that the sum of the sines of the angles in a triangle cannot be smaller than the sum of the sines of the doubles of the angles.

Suggested by G. Holló, Budapest

(5 points)

Solution (in Hungarian)

B. 4550. Prove that every power of 2 that is larger than 4 can be represented in the form a2+7b2, where a and b are positive odd numbers.

Suggested by K. Williams Kada, Szeged

(6 points)

Solution (in Hungarian)

B. 4551. P is the point on base AB of a symmetric trapezium ABCD for which AP-BP=AC-BC. The perpendicular drawn to AB at P intersects the lines CD, AC and BD at the points Q, R and S, respectively. Let k1 be the circle that touches the lines AC and BD at points R and S, and let k2 be the circle of diameter PQ. Prove that the circles k1 and k2 touch each other.

(6 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 June 2013.

C. 1170. Agnes was born on 25 March and her brother Peter was born on 9 February. They would like to construct a third-degree function f of integer coefficients, such that f(9)=2 and f(25)=3. Is there such a function?

(5 points)

Solution (in Hungarian)

C. 1171. One diagonal of a trapezium is 7 cm long. It divides the other diagonal into pieces of lengths 4.5 cm and 6 cm. The length of the shorter leg of the trapezium is 5 cm. Find the area of the trapezium.

(5 points)

Solution (in Hungarian)

C. 1173. Consider the line y=\frac72 x. Find the distance from the line to those points of integer coordinates that are the closest to the line but do not lie on the line.

(5 points)

Solution (in Hungarian)

C. 1174. The edges of a cube are coloured in three different colours such that parallel edges are the same colour. The points dividing the edges 1:2 are marked. Two marked points of each colour are selected at random. What is the probability that they are all coplanar? (It is not required to prove that the six points of a selection are coplanar.)

(5 points)

Solution (in Hungarian)


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