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Problems in Mathematics, September 2013

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 October 2013.

A. 593. Let a, b, c be positive real numbers. Prove that


\root3\of{7a^2b+1}+\root3\of{7b^2c+1}+\root3\of{7c^2a+1} \le
\frac{23}{12}(a+b+c) +
\frac1{12} \left(\frac1{a^2}+\frac1{b^2}+\frac1{c^2}\right).

Based on problem 1 of MEMO 2013

(5 points)

Solution, statistics

A. 594. The convex quadrilateral ABCD has an inscribed circle which is tangent to the sides AB and BC at E and F, respectively. The diagonal AC meets the inscribed circle at P and Q; the point Q is located between A and P. Show that the lines BD, EP and FQ are concurrent.

(5 points)

Solution (in Hungarian)

A. 595. Let p be a positive prime number for which there is a positive integer a such that p divides 2a2-1. Prove that there exist integers b and c such that p=2b2-c2.

(5 points)

Solution (in Hungarian)


Problems with sign 'B'

Deadline expired on 10 October 2013.

B. 4552. In this sentence, the number of numbers occurring once is a1, the number of numbers occurring twice is a2, ..., and the number of numbers occurring 2013 times is a2013. Determine the values of a_1, a_2, \ldots, a_{2013} such that the resulting statement is true. In how many different ways can that be done?

Suggested by L. Kozma, Saarbrücken, Germany

(5 points)

Solution (in Hungarian)

B. 4553. For what positive integers k will 2.3k be a perfect number?

(3 points)

Solution (in Hungarian)

B. 4554. Three angles of a cyclic quadrilateral are \alpha, 2\alpha and 3\alpha. Determine the angles of the cyclic quadrilateral.

Suggested by Gy. Lakos, Budapest

(3 points)

Solution (in Hungarian)

B. 4555. There are four points on the plane such that the distances of point pairs have exactly two different values, a and b where a>b. Find the possible values of a/b.

Suggested by Gy. Lakos, Budapest

(4 points)

Solution (in Hungarian)

B. 4556. Solve the simultaneous equations

x3=5x+y,

y3=5y+x.

Russian entrance examination problem

(4 points)

Solution (in Hungarian)

B. 4557. Given five points in the plane, no three of which are collinear. Show that it is possible to select three points out of them that form an obtuse-angled triangle.

(4 points)

Solution (in Hungarian)

B. 4558. The base of a regular four-sided pyramid is the unit square ABCD, its apex is E. P is a point on base edge AB, and Q is a point on lateral edge EC such that PQ is perpendicular to both AB and EC. Furthermore, AP:PB=6:1. How long are the lateral edges?

(5 points)

Solution (in Hungarian)

B. 4559. The interior angle bisectors drawn from vertices A, B and C of triangle ABC intersect the circumscribed circle at the points D, E and F, respectively. The intersections of the sides of triangles DEF and ABC, starting from A, towards B, are G, H, I, J, K and L in this order. Show that the triangles DGL, EHI and FKJ are all similar.

Suggested by Sz. Miklós, Herceghalom

(6 points)

Solution (in Hungarian)

B. 4560. The network of roads in the city of Icosapolis corresponds to the edge graph of an icosahedron. The home of Iorgos is at one vertex of the icosahedron, while his favourite theater is situated at the opposite vertex. On his way home from the theatre after dark, he stops at every vertex he reaches, and after some hesitation decides which way to proceed. Assume that at any vertex the probability of meeting someone who will show him one possible direction to get him home along the least possible number of edges is p. Otherwise he will chose an edge at random. (He may as well turn back in the direction he came from.) For what value of p is there a 50% chance that he will get home before getting back to the theatre?

Suggested by M. E. Gáspár, Budapest

(6 points)

Solution (in Hungarian)

B. 4561. Are there two distinct non-constant polynomials p and q such that \big\{p(n)\colon n\in\mathbb{N}\big\} =\big\{q(n)\colon n\in\mathbb{N}\big\}?

(5 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 October 2013.

C. 1175. Define the operation \therefore as follows on the set of real numbers: a\therefore b=(a-2)(b-2). Is this operation associative?

(5 points)

This problem is for grade 1 - 10 students only.

Solution (in Hungarian)

C. 1176. a, b, c, d, e are five consecutive integers in increasing order. The dimensions of a cuboid are a, bc. For what values will the diagonal of the cuboid be the hypotenuse of a right-angled triangle with legs d and e?

(5 points)

This problem is for grade 1 - 10 students only.

Solution (in Hungarian)

C. 1177. For what positive integer n will the number 1!+3!+...+(2n-1)! be a perfect square?

(5 points)

Solution (in Hungarian)

C. 1178. Bill wants to buy a glass of soda from a vending machine for 60 forints (HUF, Hungarian currency). He has five 10-forint coins and four 20-forint coins in his pocket. He pulls out coins at random. What is the probability that by pulling out four coins in a row he will get exactly 60 forints from his pocket?

(5 points)

Solution (in Hungarian)

C. 1179. Lily is making pudding by her grandmother's recipe. She has a 42×36-cm baking tin and glass bowls of diameter 10 cm. She places the bowls in the tin, right next to each other, then fills the tin with water and bakes the pudding. Her grandmother always made 12 bowls of pudding in this way, but Lily can make more. How?

(5 points)

Solution (in Hungarian)

C. 1180. Investigate which of the squares inscribed in an acute-angled triangle has maximum side.

Suggested by R. Gyimesi

(5 points)

This problem is for grade 11 - 12 students only.

Solution (in Hungarian)

C. 1181. Prove that (sin \alpha+1)(cos \alpha+1)<3 for all angles \alpha.

(5 points)

This problem is for grade 11 - 12 students only.

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 10 October 2013.

K. 379. Kate sewed a button on her coat. The button has four holes in it as shown in the figure (the holes form the four vertices of a square). As the thread is pulled through the holes again and again, it produces various patterns, as viewed from the front. One such pattern is shown in the figure. How many different patterns may result, provided that at least two holes need to be used to fix the button to the coat?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 380. Strings are made out of 6 letters A and 7 letters B. How many such strings can be made that are palindromes, i.e. that read the same either from the beginning or from the end?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 381. A merchant is selling a certain product for a price reduced by 20%, but he is still making a 20% profit relative to the purchase price. What percentage of the purchase price was his profit before the reduction?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 382. Use the digits 9, 8, 8, 7, 7, 7 and one more digit of your choice to write down the largest seven-digit number divisible by 36.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 383. Base AB of the regular triangle ABC is extended beyond vertex A by two fifths of the length AB, to get point P. Point P is connected to the point Q that lies on side AC (closer to A) and divides it 2:3. The resulting line intersects line CB at point R. Given that AP=2684, find the length of CR.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 384. Triangle ABC has an obtuse angle at vertex A. Denote the centre of the inscribed circle by S. The line drawn through S parallel to AB intersects side AC at D and side BC at E. Prove that DE=AD+BE.

German competition problem

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


Send your solutions to the following address:

    KöMaL Szerkesztőség (KöMaL feladatok),
    Budapest 112, Pf. 32. 1518, Hungary
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