K. 188. The base AB of an isosceles triangle ABC is extended beyond vertex B by the length of the leg. The endpoint obtained is C_{1}. A perpendicular is erected onto the base AB at vertex A. C_{2} is the point on the perpendicular that lies in the same half plane as vertex C, at a distance from A that equals the length of the leg. Given that the points C_{1}, C and C_{2} are collinear, find the angles of triangle ABC.

K. 189. In a mathematics lesson, the teacher gave the students five kinds of problems to practise. There were three problems of each kind. Students got 1 mark for solving a single problem of a kind, 4 marks per problem if they solved two of the same kind, and 9 marks per problem if they solved all three of a kind. The students worked on the problems and received marks in teams. At the end of the lesson, every team had a different number of marks, but all their numbers of marks were found to be divisible by 3. What was the maximum possible number of teams?

K. 191. Two kinds of windscreen wiper are investigated. Each of them has a 31-cm blade attached to an also 31-cm arm at the middle. The arm rotates through a 90^{o} angle when the wiper is in operation. One kind of wiper is rigid, the blade encloses 45^{o} with the lower edge of the windscreen and points in the direction of the arm throughout the wiping process. In the other wiper, the blade is attached to the arm by a hinge, and it remains perpendicular to the lower edge of the windscreen all the way long. Which type of wiper sweeps through a larger area?

K. 192. Ann, Beth and Cathy all filled out the same test of six true-or-false questions. Ann's answers (in this order) were F, F, T, T, T, T; Beth's answers were (in this order) T, F, F, T, T, T, and Cathy's answers (in this order) were T, T, F, F, T, T. Ann only got two answers wrong and Beth got only two of them right. How many correct answers may Cathy have got?

C. 966. Hungarian licence plates of vehicles contain (three letters and) three digits. The digits do not need to be different. Someone observed that on the average, the licence plates of nearly 3 vehicles out of 10 contain identical digits. Is he right?

C. 968. As shown, the square ABCD is divided, with lines parallel to its sides, into the squares NPLD and KBMP, and two congruent rectangles. Let P denote the intersection of KL and MN, and let Q be the intersection of BN and DK. Show that the points C, P and Q are collinear.

C. 969. The arms of a pair of compasses need to be opened through an angle twice as wide to draw a circle of radius 6.5 cm as for a circle of radius 3.3. How long are the arms of the compasses?

B. 4132. Annie and Bonnie are playing the following game on a 2008×2008 chessboard: Annie selects a few fields of the board but does not tell Bonnie which. Then she writes in each field the number of selected fields that it is connected to by an edge or vertex. Is that enough information for Bonnie to tell which fields were selected? What is the answer if they play on a 2009×2009 board?

B. 4134. Let denote the number of three term arithmetic progressions that can be selected from the terms of a sequence a_{1}<a_{2}<...<a_{k}. Prove that .

B. 4136. Given a convex quadrilateral, construct a rhombus with vertices lying on the sides of the quadrilateral, and sides parallel to the diagonals of the quadrilateral.

B. 4140.E is the midpoint of the arc BC on the circumscribed circle of a cyclic quadrilateral ABCD. The midpoint of arc DA is F. Let P and Q be the centres of the inscribed circles of triangles ABC and ABD, respectively. Show that PQ is parallel to EF.

A. 467. Let ABCD be a circumscribed trapezoid such that the lines AD and BC intersect at point R. Denote by I the incenter of the trapezoid, and let the incircle touch the sides AB and CD at points P and Q, respectively. Let the line through P, which is perpendicular to PR, meet the lines angle bisector AI and BI at points A_{1} and B_{1}, respectively. Similarly, let the line through Q, perpendicular to QR, meet CI and DI at C_{1} and D_{1}, respectively. Show that A_{1}D_{1}=B_{1}C_{1}.

A. 468. We are given two triangles. Their side lengths are a,b,c and A,B,C respectively, the areas are t and T, respectively. Prove that -a^{2}A^{2}+a^{2}B^{2}+a^{2}C^{2}+b^{2}A^{2}-b^{2}B^{2}+b^{2}C^{2}+c^{2}A^{2}+c^{2}B^{2}-c^{2}C^{2}16tT.

A. 469. Let 0kn and m2 be integers. Consider the k-element subsets of ; in every such subset, compute the residue of the sum of elements, modulo m. Prove that if the m residues are distributed uniformly - i.e. every residue occurs exactly times - then nm.