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KöMaL Problems in Mathematics, December 2007

Please read the rules of the competition.


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Problems with sign 'K'

Deadline expired on 10 January 2008.


K. 145. Charlie is planning to go on a trip around the world with his land rover. He is going to cover 42\;000 km on land. A tyre lasts 24\;000 km on any of the car's wheels. How many tyres does he need to buy, and how should they be placed on the wheels so that they last for the whole journey?

(6 pont)

solution (in Hungarian), statistics


K. 146. Kate runs twice as fast as she walks. On her way to school, she walks for twice the length of time as she runs. In this way she takes 20 minutes to get there. On her way home, she runs for twice the length of time as she walks. How long does she take to get home?

(6 pont)

solution (in Hungarian), statistics


K. 147. In a sale, a 4\;200\;000-forint (HUF, Hungarian currency) car was sold for 2\;310\;000 forints. In the same sale (that is, with a price reduced by the same percentage), another car was sold for 146\;000 forints less than 3/5 of its original price. What was the original price?

(6 pont)

solution (in Hungarian), statistics


K. 148. Anna, Bea, Cecilia, Dora, Emma and Fiona are going to the cinema. Their tickets are for six consecutive seats in a row. Anna and Bea insist on sitting next to each other, and Cecilia and Dora are not willing to sit next to each other at all. How many different seatings are there with these restrictions?

(6 pont)

solution (in Hungarian), statistics


K. 149. A square is divided into 30 triangles that do not overlap. Each side of the square is a side of a (different) triangle. The triangles meet at their vertices, that is, there is no vertex of a triangle lying in the interior of a side of another. How many vertices do the triangles have in the interior of the square altogether?

(6 pont)

solution (in Hungarian), statistics


K. 150. At what time instant after 8 o'clock do the short and long hands of a clock enclose the equal angles with the horizontal?

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on 15 January 2008.


C. 920. A field is bounded on one side by a 50-metre-long wall. Mehemed wants to surround the largest possible rectangular area with an electric fence for his grazing his cows. How can he achieve that if he has 44 metres of wire that he can fix to the ground with poles at

a) 1-metre,

b) 2-metre intervals.

What is the area of the surrounded pasture in each case?

(5 pont)

solution (in Hungarian), statistics


C. 921. The average of Steve's mathematics marks was between 4 and 5 before the last maths test of the first semester was given back to the class. (Marks in Hungarian schools are 1 to 5, 5 means excellent and 1 means fail. Teachers sometimes give half marks such as 3 and a half, too, to indicate that it is between a 3 and a 4, but final marks are whole numbers.) The teacher said to Steve, ``16 of you took this test. I also gave half marks. The range of your marks is 2, the mode is 4.5, and the median is 4. The average of the marks is the worst possible average that a group of 16 may have with these conditions. If you can tell me whether you may or may not have got a 5 on this test, then you will get a 5 for the first semester.'' Steve did get his 5. What was his answer to the question?

(5 pont)

solution (in Hungarian), statistics


C. 922. Find all integer solutions of the equation x2+12=y4.

(5 pont)

solution (in Hungarian), statistics


C. 923. The lengths of the parallel sides of a cyclic trapezium are a=10, c=15, the radius of the circumscribed circle is r=10. What may be the length of the legs? What is the area of the trapezium?

(5 pont)

solution (in Hungarian), statistics


C. 924. The areas of the rectangles obtained by intersecting a certain cuboid with a plane passing through two parallel edges may be t1=60, t_2=4\sqrt{153}, or t_3=12\sqrt{10}. Calculate the volume and surface area of the cuboid.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on 15 January 2008.


B. 4042. A 2007×2008 chessboard is covered by a few 2×2 and 1×4 dominoes without overlaps. One 2×2 domino of the set of dominoes used is replaced with a 1×4 domino. Prove that it is not possible to cover the chessboard with the modified set.

(4 pont)

solution (in Hungarian), statistics


B. 4043. For what pairwise different positive integers is the value of


\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}+\frac{d}{d+1}

an integer?

(4 pont)

solution (in Hungarian), statistics


B. 4044. Prove that the average distance of a point in the interior of a regular polygon from the side of the polygon equals the radius of the inscribed circle.

(3 pont)

solution (in Hungarian), statistics


B. 4045. An equilateral triangle of unit side is moved between the arms of a 120o angle XOY so that vertex A lies on the arm OX, vertex B lies on the arm OY, and the line AB separates the points C and O. Determine the locus of vertex C.

(4 pont)

solution (in Hungarian), statistics


B. 4046. Solve the following simultaneous equations:

a\sqrt a+b\sqrt b =183,

a\sqrt b+b\sqrt a  =182.

(3 pont)

solution (in Hungarian), statistics


B. 4047. A finite number of circular discs cover an area T on the plane together. Prove that it is possible to select a few discs that do not overlap and cover an area of at least T/9 together.

(4 pont)

solution (in Hungarian), statistics


B. 4048. The vertices of a regular polygon of 2n sides are A_1, A_2, \ldots, A_{2n} in this order. Prove that


\frac{1}{A_1A_2^2} + \frac{1}{A_1A_n^2} = \frac{4}{A_1A_3^2}.

(3 pont)

solution (in Hungarian), statistics


B. 4049. a, b, c are positive real numbers, and ab+bc+ca=\frac{1}{3}. Prove that


\frac{a}{a^2-bc+1}+\frac{b}{b^2-ca+1}+\frac{c}{c^2-ab+1}\ge\frac{1}{a+b+c}.

(5 pont)

solution (in Hungarian), statistics


B. 4050. Show that the equation x3-x2-2x+1=0 has two real roots a and b, such that a-ab=1.

Suggested by J. Pataki, Budapest

(5 pont)

solution (in Hungarian), statistics


B. 4051. Prove that every factor of \underbrace{111\ldots1}_{5^n} ends in a 1 for all positive integers n.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on 15 January 2008.


A. 440. Given an isosceles triangle ABC such that the midpoint of base BC is M, points D and E lie on sides AB and AC, respectively, and DE is parallel to BC.

Choose two points, P and Q on the extensions of segment BC beyond B and C, respectively, such that \frac 1{MP}+\frac 1{MQ} =\frac 1{MB}. Let PD and QE meet at R. What is the locus of point R?

(5 pont)

solution (in Hungarian), statistics


A. 441. For an arbitrary sequence of numbers A=(a_0,a_1,a_2,\ldots), let SA=(a_0,a_0+a_1,
a_0+a_1+a_2,\ldots) be the sequence of partial sums of the series a_0+a_1+a_2+\ldots. Is there any sequence A, not constant zero, for which the sequences A, SA, SSA, SSSA,... are all convergent?

Miklós Schweitzer Competition, 2007

(5 pont)

statistics


A. 442. The n-element subsets of a set of size n2+n-1 are distributed into two groups. Prove that one of those groups contains n pairwise disjoint sets.

Miklós Schweitzer Competition, 2007

(5 pont)

statistics


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

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