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KöMaL Problems in Mathematics, October 2009

Please read the rules of the competition.


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Problems with sign 'K'

Deadline expired on 10 November 2009.


K. 217. Three kinds of bombs can be placed on a squared playing field. When a bomb explodes, it will destroy its own cell and the cells around, including the contents of those cells. The diagram shows the ranges of the three types of bombs. (The numbers indicate the location and type of the bomb.) If a cell destroyed by a bomb contains another bomb, that bomb will also explode, together with all cells in its range. Place 2 of each kind of bomb on the squared field so that the largest possible number of cells are destroyed if any of the bombs is exploded.

(6 pont)

solution (in Hungarian), statistics


K. 218. A rectangle is divided into smaller rectangles with lines parallel to its sides. The distance of the dividing lines varies (not shown by the diagram). The numbers written in some rectangles show their areas. What is the area of the rectangle marked y?

14 6 20
21
52 39
20 30
40 y

(6 pont)

solution (in Hungarian), statistics


K. 219. Five different positive integers were chosen and pairwise multiplied together. The results are as follows: 6, 10, 15, 24, 36, 42, 60, 105, 63, 252. What were the five numbers?

(6 pont)

solution (in Hungarian), statistics


K. 220. Find all possible sets of five not necessarily different positive integers, such that their sum equals their product.

(6 pont)

solution (in Hungarian), statistics


K. 221. If Sunny does not wear any sunscreen, her skin will get burnt after 12 minutes of sunbathing. One time she started sunbathing with a non-waterproof cream of factor 12. Then she took a shower. For the rest of the time she put on a cream factor 20. Thus she was able to spend 208 minutes in the sun altogether without getting burnt. For how many minutes did she wear each cream? (The sun protection factor shows what fraction of the harmful [UV-B] rays reach the skin through the cream. For example factor 20 lets through one-twentieth of the radiation.)

(6 pont)

solution (in Hungarian), statistics


K. 222. The diagram below shows the nets of four cubes unfolded in a plane. There are line segments painted on the surface of each cube, and these are also shown in the diagram. (No line is drawn along the edges of the cubes.) For which cube will the lines form a closed loop?

(6 pont)

solution (in Hungarian), statistics


Problems with sign 'C'

Deadline expired on 10 November 2009.


C. 1000. There are 30 people sitting at a round table. Some of them are liars, the others tell the truth. We know that out of the two neighbours of every liar, exactly one is a liar. Each of the 30 people is asked how many liars are sitting next to them. 12 say exactly one and the others say that both of their neighbours are liars. How many liars are there around the table? (Based on a problem from Subcarpathia)

(5 pont)

solution (in Hungarian), statistics


C. 1001. A certain integer has two prime factors. The number of its divisors is 6, and the sum of the divisors is 28. Which number is it?

(5 pont)

solution (in Hungarian), statistics


C. 1002. Find those right-angled triangles of integer sides for which the measures of the perimeter and area are equal.

(5 pont)

solution (in Hungarian), statistics


C. 1003. A wholesale merchant sells drugstore products and stationery. He has a container of volume 12 m3 that has a capacity of 5 tonnes of goods. One tonne of drugstore products fill up a volume of 1 m3, and a tonne of stationery products take up 3 m3. He makes a profit of 100,000 forints (HUF, Hungarian currency) per tonne on drugstore products and 200,000 forints per tonne on stationery. What is the maximum profit that the merchant can make by selling a container of goods?

(5 pont)

solution (in Hungarian), statistics


C. 1004. An arbitrary line is drawn through vertex A of a square ABCD, and perpendiculars are dropped onto the line from the vertices B and D. The feet of the perpendiculars are B1 and D1, respectively. Prove that AB12+AD12=BB12+DD12.

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'B'

Deadline expired on 10 November 2009.


B. 4202. The numbers 1 to 2009 are written on a sheet of paper. In the second step, the double of each number is also written on the sheet and then all those numbers are erased that occur twice on it. This step is repeated as follows: in step i, every number on the sheet is multiplied by i, the results are written down and then all those numbers are erased that occur twice. Prove that there will be at least 2009 numbers on the sheet after every step.

(5 pont)

solution (in Hungarian), statistics


B. 4203. A common tangent of two intersecting circles touches them at the points A and B, and the line segment connecting their centres intersects them at C and D, respectively. Prove that ABCD is a cyclic quadrilateral.

(4 pont)

solution (in Hungarian), statistics


B. 4204. a, b, c, d are four positive numbers. Five of the products ab, ac, ad, bc, bd, cd are known: they are 2, 3, 4, 5 and 6. What is the value of the sixth product?

(3 pont)

solution (in Hungarian), statistics


B. 4205. A, B, C, D are moving points in the plane, such that AD=BC=2 and AC=BD=4 remain valid, and the line segments AC and BD intersect each other for all positions of the points. How does the distance CD depend on the distance AB?

(3 pont)

solution (in Hungarian), statistics


B. 4206. Let p>3 be a prime number and let k and m be non-negative integers. Prove that pk+pm cannot be a perfect square. (Suggested by P. Kutas)

(3 pont)

solution (in Hungarian), statistics


B. 4207. Is it true that every polygon has a vertex from which it is possible to draw a diagonal in the inside of the polygon to a non-adjacent vertex that is closest to it? (Suggested by P. Maga)

(4 pont)

solution (in Hungarian), statistics


B. 4208. Let n be a positive integer. Determine the first digit following the decimal point in the number


\sum_{k=1}^{n}{\frac{\sqrt{k(k+1)}}{n}}.

(Suggested by M. Bencze, Brasov)

(4 pont)

solution (in Hungarian), statistics


B. 4209. In an acute triangle ABC, the feet of the altitudes drawn from A and B are A1 and B1, respectively, and the orthocentre of the triangle is M. The median drawn from B intersects line A1B1 at point P. Prove that BPM\sphericalangle is a right angle if and only if B1C=3AB1.

(4 pont)

solution (in Hungarian), statistics


B. 4210. An acute-angled triangle has sides a, b, c and area t. The sides satisfy the equality abc=a+b+c. Prove that


1<t\le \frac{3\sqrt{3}}{4}\,.

(4 pont)

solution (in Hungarian), statistics


B. 4211. Show that there is no polynomial of rational coefficients that takes a non-integer value at exactly one integer. Is there a polynomial of this property with real coefficients? (Suggested by P. Maga)

(5 pont)

solution (in Hungarian), statistics


Problems with sign 'A'

Deadline expired on 10 November 2009.


A. 488. Let P1P2P3 be a triangle with circumcenter O, the point Q is in the triangle. Denote ti and Oi the area and the circumcenter of the triangle QPi+1Pi+2, respectively, where i=1,2,3 (the vertices are counted cyclically: P4=P1 and P5=P2). Prove that t_1\cdot \overrightarrow{OO_1} + t_2\cdot \overrightarrow{OO_2} + t_3\cdot
\overrightarrow{OO_3} = 0.

(5 pont)

statistics


A. 489. Does there exist a non-constant polynomial which always takes non-square integers at integer values of the variable?

(5 pont)

statistics


A. 490. The two base faces of a prism are equilateral triangles and the other three faces are squares. At the beginning it stands on its triangle face. Then it is rolled around one of its edges that lays on the table. After some rollings, the prism will stand in the original position. Prove that then all vertices will be in the same position as at the beginning. (Suggested by L. Csirmaz, Budapest)

(5 pont)

statistics


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

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