**B. 4324.** In triangle *ABC*, the foot of the altitude from vertex *A* is *D*, the midpoint of the altitude from vertex *B* is *E*, and the midpoint of the altitude from vertex *C* is *F*. Show that .

(Suggested by *Sz. Miklós,* Herceghalom)

(4 points)

**C. 1060.** Let *n* denote a number divisible by 3. Every third term of the sequence *n*-1, *n*-2, ..., 2, 1 is cancelled. The subsequent pairs of terms of the remaining sequence are given alternating plus and minus signs: +(*n*-1), +(*n*-2), -(*n*-4), -(*n*-5), ... . Prove that the sum of the numbers obtained in this way is *n*.

(5 points)

**K. 277.** Steve thought of a positive integer. He gave the following information about his number: If the number is divisible by 3, then it is between 50 and 59. If the number is not divisible by 4, then it is between 60 and 69. If the number is not divisible by 6, then it is between 70 and 79. Given that all his statements above are true, find out which number Steve had in mind.

(6 points)

This problem is for grade 9 students only.

**K. 279.** Ann, Belle, Carol and Dora played with a deck of 52 cards. In one game, Dora was dealing out the cards one by one to the players, starting with Ann, followed by Belle, Carol and Dora in this order, when suddenly some of the cards she had not dealt out yet slipped out of her hands and fell on the floor. The girls noticed that the number of cards on the floor was 2/3 of the number of cards Ann had already got, and the number of cards that Carol had got was 2/3 of those in the remaining part of the deck in Dora's hand that she had not dealt out yet. How many cards had Dora dealt out altogether?

(6 points)

This problem is for grade 9 students only.

**K. 281.** Dominique has a set of dominoes in which the stone of smallest value is a double 1, the largest value is a double 6, and there is one stone with every intermediate pair of numbers. Dominique realized that she could also represent certain fractions with her domino stones: the number on the upper domino part is the numerator and the number on the lower part is the denominator. Dominique and her sister Naomi were sitting facing each other at a table. Dominique laid the divisions between them on the *table.* When they checked the dominoes lying face down, they observed that the arithmetic was correct from both of their points of view. What may be the two stones lying face down?

(6 points)

This problem is for grade 9 students only.