K. 151. In a survey carried out with a group of students, it turned out that 2/3 of the boys and 3/4 of the girls like raspberries. It also turned out that the same number of boys and girls liked raspberries. What fraction of the students taking part in the survey like raspberries?
K. 152. 3% of the 800 inhabitants of the planet Awr have two heads. Half the remaining inhabitants have three heads, and the other half have only one. A shipment of fruits is arriving from the neighbouring planet Bwr: one delicious cwr for each head. How many, exactly?
K. 153. Steve has a four-digit number in mind that Alex has to guess. Alex chooses a four-digit number, and Steve tells him how many digits it contains that stand in the appropriate position. [For example, if the number to find out is 1234 and the guess is 6231, then 2 digits are correct: 2 and 3. (The 1 does not count, since that is not in the right position.)] Alex had nine guesses: 2186, 5127, 6924, 4351, 5916, 8253, 4521, 6384, 8517. In each of the nine numbers, he guessed exactly one digit right, that is, each of the nine numbers contains exactly one digit that stands in the right position. What is the number that Steve has in mind?
C. 926. A lecture hall has 24 lamps. Each lamp may operate 4 light bulbs at most. When the maintenance staff has installed four bulbs in some of the lamps, it became clear that the number of bulbs available was not enough. Then they continued with three bulbs per lamp, then two bulbs per lamp, and finished the job with one bulb per lamp. How many bulbs were missing if there were twice as many lamps with one bulb as with four bulbs, and there remained no bulbs at all for half as many lamps as those with three bulbs?
C. 929. The base edges and the lateral edges of a truncated pyramid of square base are all 4 units long. The sides of the top face are 2. What is the maximum possible distance between two vertices of the truncated pyramid?
B. 4053. The sequence of integers has infinitely many positive terms and infinitely many negative terms. For every n, the remainders of divided by n are pairwise different. How many times does the number 2008 occur in the sequence?
B. 4054. The radius of a circle inscribed in a triangle is r. The tangents drawn to the circle parallel to the sides cut three small triangles off the original triangle. Prove that the sum of the radii of the inscribed circles of the small triangles is also r.
B. 4056. The orthocentre of an acute-angled triangle is M, the centre of its circumscribed circle is O, and its sides are a<b<c. The line of side c, the line of the altitude drawn to side b, and the line MO form a triangle that is similar to the original triangle but with opposite orientation. Find the angles of the triangle.