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Problems in Mathematics, December 2010

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 January 2011.

A. 521. A. 521. There are given a positive integer m and an infinite sequence a1<a2<... of positive integers such that ak\lemk holds for infinitely many indices k. Prove that there exist positive integers b1,...,bm with the property that every integer can be written as ai-aj+bk where i,j are positive integers and 1\lek\lem.

(5 points)

Solution (in Hungarian)

A. 522. The vertices and side arcs of the spherical hexagon ABCDEF are located on a hemisphere. The arc AB is perpendicular to the arc BC, the arc AF is perpendicular to the arc EF, moreover AB=AF, BC=CD and DE=EF. Prove that the great arcs AD and CE are perpendicular.

(5 points)

Solution (in Hungarian)

A. 523. There are given a simple graph and a positive integer n. Show that there are two, not necessary distinct vertices a and b in the graph for which the number of directed walks of length n, starting point a and end-point b, is even.

(5 points)

Statistics


Problems with sign 'B'

Deadline expired on 10 January 2011.

B. 4312. In a company, everyone knows 5 other people. (Acquaintances are mutual.) Two members of the company are appointed captains. The captains take turns selecting members for their teams, until everyone is selected. Prove that at the end of the selection process there are the same number of acquaintances within each team.

Suggested by T. Hubai and Z. Király

(3 points)

Solution (in Hungarian)

B. 4313. A, B, C, D, E and F are a group of six people. n bars of chocolate given to the group in the following way: Everyone gets at least one, A gets less than B, B gets less than C, C gets less than D, D gets less than E, and finally, F gets the most. The members of the group know these conditions, they know the value of n, and of course, they know how many bars of chocolate they were given themselves. They have no other information available for them. What is the smallest possible value of n for which it is possible to give them the bars of chocolate so that no one can tell how many bars of chocolate everyone has?

Based on a Kavics Kupa competition problem, 2010

(3 points)

Solution (in Hungarian)

B. 4314. The radii of three concentric circles are 1, 2, and 3 units. A point is marked on each circle such that they form a regular triangle. What may be the length of the side of the triangle?

(4 points)

Solution (in Hungarian)

B. 4315. Given that r is a positive rational number and rr is also rational, prove that r is an integer.

(4 points)

Solution (in Hungarian)

B. 4316. Let E denote the point closer to vertex B that divides side BC of a square ABCD in a 1:4 ratio. Let F be the point obtained by reflecting about C the point closer to D that divides side CD in a 1:2 ratio. Prove that the lines AE and BF intersect on the circumscribed circle of the square ABCD.

Suggested by J. Szászné Simon, Budapest

(4 points)

Solution (in Hungarian)

B. 4317. Solve the following simultaneous equations: \frac{1}{\sqrt{1-x^2}} +
\frac{1}{\sqrt{1-y^2}} =\frac{35}{12}, \frac{x}{\sqrt{1-x^2}} -
\frac{y}{\sqrt{1-y^2}} =\frac{7}{12}.

Suggested by M. Szombathy, Eger

(4 points)

Solution (in Hungarian)

B. 4318. Let P and Q be arbitrary points on the edges AB and CD of a given tetrahedron ABCD, respectively. Determine the locus of the midpoint of the line segment PQ.

(4 points)

Solution (in Hungarian)

B. 4319. A triangle is drawn on a rectangular sheet of paper. Unfortunately, the vertices are off the sheet, but there is a segment of each side that lies on the sheet. Given that the orthocentre of the triangle is on the sheet, find the orthocentre by doing construction on the sheet.

(4 points)

Solution (in Hungarian)

B. 4320. Write down the numbers x_k=\big[k\sqrt2\,\big] (k=1,2,\ldots) in a row, and underneath, in a second row list the integers 0<y1<y2<... that do not occur among the numbers xk. Express the difference yk-xk as a function of k.

Suggested by L. László, Budapest

(5 points)

Solution (in Hungarian)

B. 4321. Prove that the following inequality is true in every triangle: \frac{b}{\sin
\left(\gamma + \dfrac{\alpha}{3}\right)} +
\frac{c}{\sin \left(\beta + \dfrac{\alpha}{3}\right)} > \frac{2}{3} \cdot
\frac{a}{\sin \left(\dfrac{\alpha}{3}\right)}.

From the journal Ifjúsági Matematikai Lapok, Kolozsvár

(5 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 January 2011.

C. 1055. How many eight-digit numbers of the form \overline{abcdabcd} are there that are divisible by 18 769?

(5 points)

Solution (in Hungarian)

C. 1056. There are 10 spherical Christmas ornaments of the same size in two boxes. 5 spheres are red, 3 are gold and 2 are silver. One box contains 4 of them, the other contains 6. In how many different ways may the spheres be placed in the boxes if those of the same colour are not distinguished and the order of the spheres within a box does not count?

(5 points)

Solution (in Hungarian)

C. 1057. Some points in the plane are marked with the colours red, white and green (the national colours of Hungary). No three points are collinear. The number of lines connecting points of different colours is 213, and the number of lines connecting those of the same colour is 112. How many points are marked in the plane?

(5 points)

Solution (in Hungarian)

C. 1058. Two vertices of a triangle are fixed, and the third vertex is moving along a curve such that the sum of the squares of the sides equals 8 times the area of the triangle. What curve is it?

(5 points)

Solution (in Hungarian)

C. 1059. The two diagonals from a vertex of a regular pentagon are drawn. The resulting figure can be folded to form the lateral faces of a pyramid with a regular triangular base. Find the volume of the pyramid.

(5 points)

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 10 January 2011.

K. 271. Alex visited his relatives. He took the train and then changed for a bus. The train and the bus travelled at an average speed of 80 km/h and 45 km/h, respectively. The journey by train took the same time as the journey by bus, but Alex covered a 140 km longer distance by train than by bus. Find the total distance travelled by Alex in kilometres.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 272. P is a point on the hypotenuse AB of a right-angled triangle ABC, such that AC=AP. Q is a point on the line segment AP, such that PCQ\sphericalangle
=45^\circ. Prove that triangle CQB is isosceles.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 273. There is a mouse in a long straight tube, at 3/8 of the length. A cat is sitting at a point along the extension of the line of the tube, closer to the end of the tube that the mouse is also closer to. The cat notices the mouse, and starts to run towards the tube. At the same instant, the mouse also starts to run towards one of the ends of the tube. (Both of them run at uniform speeds.) However, the mouse cannot escape: No matter which end of the tube he chooses, the cat will just catch him at the endpoint of the tube. By what factor does the cat run faster than the mouse?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 274. In a city, 2/3 of the men and 3/5 of the women are married. (Everyone has one spouse and the spouses live in the same city.) What fraction of the inhabitants of the city are married?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 275. A flying saucer with aliens on board is travelling at a uniform altitude above the ground (that is, at a constant distance from the Earth) at a uniform speed of 800 km/h. At 8 a.m., it was over London, and after 1 hour and 24 minutes it is already over Berlin. Assume that the Earth is a sphere of radius 6370 km. The distance between London and Berlin is 929 km measured along the surface. The saucer travels the shortest path between the two points under the given conditions. Calculate its distance from the surface of the Earth.

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 276. Given eight three-digit numbers, we write them next to each other in pairs to form six-digit numbers in all possible ways. We observe that there is always a six-digit number among them that is divisible by 7. Why is that so?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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