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Problems in Mathematics, September 2011

Please read the rules of the competition.


Problems with sign 'A'

Deadline expired on 10 October 2011.

A. 539. Find all prime numbers p\ge3 for which 1+k(p-1) is prime for every integer 1\le
k\le\frac{p-1}2.

Kolmogorov Cup, 2010

(5 points)

Solution, statistics

A. 540. Let A1A2A3 be a non-equilateral triangle, let point M be its orthocenter, let F be its Feuerbach point, and let k be the circumcircle of the triangle. For i=1,2,3 denote by ki the circle that is internally tangent to k and tangent to the sides AiAi+1 and AiAi+2. (The indices are considered modulo 3, i.e. A4=A1 and A5=A2.) Let Ti be the point of tangency between k and ki. Prove that the lines A1T1, A2T2, A3T3 and MF are concurrent.

Proposed by: Gábor Damásdi and Márton Mester, Budapest

(5 points)

Solution, statistics

A. 541. The elements of the set H are finite sequences whose elements are from {1,2,3}. Suppose that no element of H is a subsequence of any other element of H. Show that H is finite.

Proposed by: András Pongrácz, Budapest

(5 points)

Solution, statistics


Problems with sign 'B'

Deadline expired on 10 October 2011.

B. 4372. Show that if there are three collinear points among any four selected from a set of n points then at least n-1 points are collinear.

(3 points)

Solution (in Hungarian)

B. 4373. An international conference of polypodal creatures is organized on the top of the Glass Mountain. The numbers of legs of the participants, a1,...,an, all are positive even numbers. The side of the Glass Mountain is slippery. Each of the creatures gathering at the base of the mountain can only get to the top if they wear special climbing shoes on at least half their feet. At least how many shoes are needed altogether to get all participants up to the top if a shoe must always be worn on a foot when it travels up or down the mountain?

Suggested by G. Mészáros, Kemence

(4 points)

Solution (in Hungarian)

B. 4374. Pacworm the cheese mite is sitting in the middle of a cheese cube of 5 cm edges. He eats a tunnel through the cheese. He always proceeds 1 cm parallel to an edge of the cube and then changes direction. While turning, he makes sure that he makes a 90-degree turn and that facing in the new direction he has more than 1 cm of cheese in front. He always chooses a new direction among the possibilities with equal probability. What is the probability that after travelling 5 cm he ends up in a position where there is exactly one edge at a distance of at most 0.8 cm?

(4 points)

Solution (in Hungarian)

B. 4375. Let a and b be the legs of a right-angled triangle, and let m be the height drawn to the hypotenuse c. Which line segment is longer, a+b or m+c?

Suggested by P. Székely, Budapest

(3 points)

Solution (in Hungarian)

B. 4376. Prove that if x, y are non-negative numbers then

x^4 + y^3 + x^2 + y + 1 > \frac{9}{2}xy .

Suggested by J. Szoldatics, Dunakeszi

(4 points)

Solution (in Hungarian)

B. 4377. Regular triangles ABD, BCE, CAF are drawn to the sides of a triangle ABC on the outside. Let the midpoints of line segments DE, EF, FD be G, H, I, respectively. Prove that BG=CH=IA.

Suggested by Sz. Miklós, Herceghalom

(4 points)

Solution (in Hungarian)

B. 4378. Let p denote a positive prime number. Solve the equation

x3y3+x3y2-x2y3+x2y2-x+y=p+2

on the set of integers.

Suggested by B. Bíró, Eger

(5 points)

Solution (in Hungarian)

B. 4379. Given the points different from the vertices where the exterior angle bisectors of a non-isosceles triangle intersect the circumscribed circle, construct the triangle.

(4 points)

Solution (in Hungarian)

B. 4380. m points are chosen on the sides of a convex k-sided polygon, such that the resulting point set \mathcal{P} of n=k+m elements is centrally symmetric. Prove that there may be at most 2n-3 line segments of equal length among the \binom{n}{2} line segments determined by the points of the set \mathcal{P}. For what point sets \mathcal{P} are there 2n-3 line segments of equal length?

(5 points)

Solution (in Hungarian)

B. 4381. Given three pairwise skew lines, determine the possible positions of the centre of a parallelepiped that has an edge lying on each line.

(4 points)

Solution (in Hungarian)


Problems with sign 'C'

Deadline expired on 10 October 2011.

C. 1085. There are n coins on the table, with their tails side facing upwards. In each step, n-1 coins are turned over. Is it possible to achieve that all coins have heads facing upwards?

(5 points)

Solution (in Hungarian)

C. 1086. In a right-angled triangle the length of the angle bisector of the right angle is 2\sqrt{10} and it divides the hypotenuse in a 1:2 ratio. Calculate the length of the altitude drawn to the hypotenuse.

(5 points)

Solution (in Hungarian)

C. 1087. The first term of an arithmetic progression is 1, its second term is n, and the sum of the first n terms is 33n. Find n.

(5 points)

Solution (in Hungarian)

C. 1088. The base of a right prism is a right-angled triangle. The lengths of the edges of the prism are integers. The prism has two faces with areas equal to 30 and 13 units. Find the volume of the prism.

(5 points)

Solution (in Hungarian)

C. 1089. The area of a quadrilateral ABCD is 20. The vertices are labelled in counterclockwise order. The coordinates of three vertices are A(-2;0), B(2;0) and C(2;4). What is the minimum possible perimeter of the quadrilateral ABCD?

(5 points)

Solution (in Hungarian)


Problems with sign 'K'

Deadline expired on 10 October 2011.

K. 295. How many digits does the number 2011201020092008...10987654321 have? Is it divisible by 3?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 296. The first term of a sequence is 2011. From the second term onwards, each term is equal to (-2) times the reciprocal of the number 2 greater than the previous term. What is the 2011th term of the sequence?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 297. A square is cut out of the upper right corner of the rectangle in the diagram. The area of the resulting figure is 2011 cm2. What was the area of the original rectangle?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 298. Three consecutive sides of a quadrilateral have the same length. The two interior angles enclosed by the three sides are 60o and 70o. What is the measure of the largest angleThree consecutive sides of a quadrilateral have the same length. The two interior angles enclosed by the three sides are 60o and 70o. What is the measure of the largest angle of the quadrilateral? of the quadrilateral?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 299. There are 2011 pairs of twins living in a city. Among the 4022 people, there are 1900 males, and the number of pairs of twin sisters is 11 greater than the number of sister-brother pairs. How many pairs of twin brothers are there?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)

K. 300. A few girls went on holiday together. Each of them has blond, brown or black hair. All but 3 girls are blondes, all but 4 have brown hair and all but 5 have black hair. How many girls went on holiday?

(6 points)

This problem is for grade 9 students only.

Solution (in Hungarian)


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