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