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Problems in Mathematics, December 2009

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 11 January 2010.

A. 494. Let p1,...,pk be prime numbers, and let S be the set of those integers whose all prime divisors are among p1,...,pk. For a finite subset A of the integers let us denote by \mathcal{G}(A) the graph whose vertices are the elements of A, and the edges are those pairs a,b\inA for which a-b\inS. Does there exist for all m\ge3 an m-element subset A of the integers such that (a\mathcal{G}(A) is complete? (b\mathcal{G}(A) is connected, but all vertices have degree at most 2?

Miklós Schweitzer Memorial Competition, 2009

(5 points)

Solution, statistics

A. 495. In the acute triangle ABC we have \angleBAC=\alpha. The point D lies in the interior of the triangle, on the bisector of \angleBAC, and points E and F lie on the sides AB and BC, respectively, such that \angleBDC=2\alpha, \angleAED=90o+\frac{\alpha}{2}, and \angleBEF=\angleEBD. Determine the ratio BF:FC.

(5 points)

Solution, statistics

A. 496. Let a1,a2,...,a2k be distinct integers and let M be a set of k integers not containing 0 and s=a1+a2+...+a2k. A grasshopper is to jump along the real axis, starting at the point 0 and making 2k jumps with lengths a1,a2,...,a2k in some order. If ai>0 then the grasshopper jumps to the right; while if ai<0 then the grasshopper jumps to the left, to the point in the distance |ai| in the respective steps. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M.

(5 points)

Solution (in Hungarian)


Problems with sign 'B'

Deadline expired on 11 January 2010.

B. 4222. The students in a class of 30 organized 16 trips during the school year. Eight students went on the trip each time in a van. Show that there are two students in the class who went on at least two trips together.

(3 points)

Solution (in Hungarian)

B. 4223. Consider the number pairs (1,36), (2,35), ..., (12,25). Is it possible to select one number from each given pair, such that the sum of the numbers selected equals the sum of the numbers not selected? Will the answer change if the last two pairs are left out?

(Suggested by J. Pataki, Budapest)

(4 points)

Solution (in Hungarian)

B. 4224. The lengths of the diagonals of a rhombus with side 2 units are added. What integer values may the sum have?

(Suggested by G. Nyul)

(3 points)

Solution (in Hungarian)

B. 4225. Solve the simultaneous equations

\frac{\sqrt{x+z}+\sqrt{x+y}}{\sqrt{y+z}} +
\frac{\sqrt{y+z}+\sqrt{x+y}}{\sqrt{x+z}} &= 14 - 4\sqrt{x+z}- 4\sqrt{y+z},

\sqrt{x+z}+\sqrt{x+y} +\sqrt{z+y} &= 4.

(Suggested by B. Bíró)

(3 points)

Solution (in Hungarian)

B. 4226. a<b<c are the sides of a triangle H. Consider the three rhombuses, such that one vertex coincides with a vertex of H and the other three vertices lie on the sides of H. Given that two of these rhombuses have the same area, show that b2=ac.

(4 points)

Solution (in Hungarian)

B. 4227. Is it true that if there exists a quadrilateral with sides a, b, c and d then there exists such a cyclic quadrilateral, too?

(4 points)

Solution (in Hungarian)

B. 4228. The sequence pn is defined recursively: Let p1=2 and let pn+1 be the largest prime factor of p_1p_2\ldots p_n+1. Is 11 a term of the sequence?

(5 points)

Solution (in Hungarian)

B. 4229. In the parallelogram ABCD, 2BD2=BA2+BC2. Show that the circumscribed circle of triangle BCD goes through one of the points that trisect the diagonal AC.

(Suggested by L. Koncz, Budapest)

(4 points)

Solution (in Hungarian)

B. 4230. Each edge of a regular pyramid of square base has unit length. Determine the distance between the lines of two skew edges.

(4 points)

Solution (in Hungarian)

B. 4231. Show that the number of terms with coefficients not divisible by 3 in the binomial expansion of (x+y)n is not divisible by 5.

(5 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 11 January 2010.

C. 1010. Santa divides 53 Christmas candies in three bags, so that there is a different number of candies in each bag but the total number of candies in any two is greater than the number of candies in the third one. In how many different ways is that possible?

(5 points)

Solution (in Hungarian)

C. 1011. Prove that the value of the expression a3-3ab2+2b3 is non-negative if a and b are non-negative real numbers.

(5 points)

Solution (in Hungarian)

C. 1012. A circle passing through the centre of a square is drawn about each vertex. The circles intersect the sides of the square at 8 points altogether. Prove that the intersections form a regular octagon.

(5 points)

Solution (in Hungarian)

C. 1013. Represent the set of points (x,y) in the coordinate plane the coordinates of which satisfy the two conditions below: x2+y2\le25, -1\le \frac{x}{x+y}\le 1.

(5 points)

Solution (in Hungarian)

C. 1014. The number of persons who booked ticket for the New Year's concert is a perfect square. If 100 more persons booked ticket then the number of spectators would be a perfect square plus 1. If still 100 more persons booked ticket then the number of spectators would be again a perfect square. How many persons booked ticket for the concert?

(5 points)

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 11 January 2010.

K. 224. There are dice of six and four faces (cubes and tetrahedra) on a table. Their faces are numbered with dots, 1 to 6 and 1 to 4, respectively. The number of all dots on the dice is 323. If we had as many six-sided dice as we have of the four-sided dice and vice versa, the number of dots would be 185. How many dice of each kind are there on the table?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 229. Show that a given line segment AB can be divided into three equal parts by the following method: A 30o angle is drawn to each end of the line segment so that their other arms intersect at point C. Then the perpendicular bisectors of the line segments AC and BC are constructed. These will intersect AB at two points that cut it into three equal parts.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 230. The traditional way of numbering houses in a street uses odd numbers on one side and even numbers on the other side. Numberless Street is bounded by identical plots of land on both sides. The odd side is fully developed, there is one house on each plot of land. On the even side, there are a few consecutive plots that have no houses on them yet, but there are two houses on the first plot. The owners decided to use number plates of the same design in the whole street. The number plates they bought are sold for 50 forints a digit (HUF, Hungarian currency). They spent 4250 forints altogether. The numbers for the odd side cost 550 forints more than the numbers for the even side. The owner of the first house on the odd side considered 1 an unlucky number, so they started the numbering by 3. On the even side, the two houses on the first plot got the numbers 2 and 4. They did not buy number plates for the houses that would be built on the vacant plots, but they did take them into consideration in numbering: they did not give their numbers to other buildings. What is the largest number on the odd side and how many vacant plots are there on the even side?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 231. If the last digits of the products 1.2, 2.3, 3.4, ..., n(n+1) are added, the result is 2010. How many products are used?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 232. E is the midpoint of side BC and F is the midpoint of side CD of a parallelogram ABCD. Diagonal BD intersects line AE at P and line AF at Q. Find the ratio of the areas of triangles APQ and AEF.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 233. Mr. Bear's favourite honey jar is a right circular cylinder of diameter 16 cm. His favourite spoon is 23 cm long, he normally eats honey with that spoon. One day Mr. Bear accidentally dropped the spoon into the jar, and it submerged totally in the honey. What was the minimum possible amount of honey in the jar? (Ignore the volume of the spoon.)

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 234. A square is drawn on the outside of each side of a rectangle with given perimeter. What should be the dimensions of the rectangle to minimize the area of the dodecagon obtained?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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