K. 67. In a television quiz, 10 people are asked questions. They all answer simultaneously by pressing the button assigned to the answer they think correct. Those who answer correctly get as many points each as the number of incorrect answers. After the fifth question answered, the players had 116 points altogether. 32 of these were collected by Géza. Show that Géza answered all the five questions correctly.
K. 68. Connect an interior point of a parallelogram to each vertex. Show that it is possible to construct a quadrilateral with the four segments hence obtained as sides such that its vertices are lying on the sides of the parallelogram.
K. 69. Out of the digits of a three-digit number of different digits, all the possible two-digit numbers of different digits are formed, and these two-digit numbers are added. Given that the sum is equal to the original three-digit number, find all such three-digit numbers.
K. 71. The diameter of the wheels on Jack's bike is 28'' (inches), and the diameter of the wheels of Jill's bike is 14'' (inches). Jack's wheel turns around twice while he makes 3 turns on the pedals. Jill's wheels turns around 3 times while she makes two turns on the pedals. Jack turns the pedals twice as many times a minute as Jill does. Cycling at a uniform speed, Jill covers 2 kilometers in 20 minutes. How long does Jack take to cover the same distance?
K. 72. The lengths of the edges in cm of a square-based prism are whole numbers. With a plane perpendicular to the base and parallel to one lateral face, a 4-cm-thick part is cut off the prism. The volume of the remaining solid is 126 cm3. How long are the edges of the original prism?
C. 836. In a Fun Shop selling party novelties, a packet of streamers costs p percent more than a packet of confetti. In other words, a packet of confetti is q percent cheaper than a packet of streamers. The difference of p and q is 90. How many packets of streamers can one buy for the price of 10 packets of confetti?
C. 838. Mihály Tímár (the hero of a famous Hungarian novel) is in a trouble. The red crescent that marked the sack containing the treasure has come off. He knows, however, that the heaviest of the four sacks hides the treasure. From three measurements, he has found out that the first sack together with the second one are lighter than the other two, the first and third together are equal in weight to the other two, and the first and fourth sacks together are heavier than the other two. Which sack hides the treasure?
B. 3875. There were 31 people at a party. For any 15 of them, there is a further member of the group who knows all of them. (Acquaintances are mutual.) Prove that there is a guest of the party who knows all the others.
B. 3877. The centroid of the triangle ABC is S, and the midpoint of AB is F. For an interior point P of the line segment AF, consider that point Q of the line PS for which QC and AB are parallel. Let R be the intersection of the lines QA and BC. Prove that the line segment PR halves the area of the triangle ABC.
B. 3879. An escribed circle of a convex polygon is defined as a circle that touches one side of the polygon externally, and also touches the extensions of the two adjacent sides. Let k denote the sum of the areas of the escribed circles of a polygon, and let t denote the total area of the circles drawn over the sides of the polygon as diameters. Assume that the polygon has an inscribed circle, and let K and T denote the perimeter and area of the inscribed circle, respectively. Show that .