**A. 542.** We have 1000 coins, but 100 of them are counterfeit. We know the weight of the real coins and it is known that the counterfeit coins are lighter than the real coins, but they may have different weights. Using a scale we want to find a counterfeit coin. Every time we can measure the total weight of a subset of the coins; the result indicates whether that set contains a counterfeit coin or not. How many measures are needed to find a counterfeit coin for sure?

(Suggested by *Dömötör Pálvölgyi,* Budapest)

(5 points)

**A. 544.** A circle *k* with center *O* and four distinct fixed points *A*, *B*, *C*, *D* lying on it are given. The circle *k*' intersects *k* perpendicularly at *A* and *B*. Let *X* be a variable point on the line *OA*. Let *U*, other than *A*, be the second intersection of the circles *ACX* and *k*'. Let *V*, other than *A*, be the second intersection of the circles *ADX *and *k*'. Let *W*, other than *B*, be the second intersection of the circle *BDU* and the line *OB*. Finally, let *E*, other than *B*, be the second intersection of the circles *BVW *and *k*. Prove that the location of the point *E* is independent from the choice of the point *X*.

(5 points)

**B. 4384.** According to problem **B. 4283.** of this journal, if a 23×23 square is dissected into 1×1, 2×2 and 3×3 squares then at least one of the pieces must be a 1×1 square. Given that there is exactly one 1×1 piece, find its possible positions in the big square.

(Suggested by *Z. Gyenes,* Budapest)

(5 points)

**C. 1091.** In the lottery, 5 numbers are drawn out of the numbers 1 to 90. One time, the following numbers were drawn, listed in increasing order: , , , , . The sum of the five numbers is , the product of the third and second numbers is , and the product of the third and fifth numbers is . Determine the digits *a*, *b*, *c*, *d*, *e*.

(5 points)

**K. 305.** Satellite pictures of some dry regions of the United States show interesting circular features. (E.g. to view the area of ZIP code 79068 in Texas, enter TX 79068 for search in Google Earth.) The circles are formed owing to an irrigation technology involving a long rod revolving about its midpoint or an endpoint. Since the water does not reach the entire area of the field, only the interior of the circles, part of the field will not have plants growing on it. Three brothers, Joe, Jim and Jack each have a square field 1 km on a side. The irrigated areas on their fields are laid out as shown in the *figure.* In Jack's arrangement, the radius of the smaller circles is 210.5 m. Which of the brothers has the most productive arrangement of circles on his field, that is, which of them can use the greatest fraction of his field for growing plants?

(6 points)

This problem is for grade 9 students only.

**K. 306.** A 4×4-es square table is filled in with the digits of 2011, so that each row, column and diagonal contains 2, 0, 1, 1 in some order. In how many different ways is that possible?

(6 points)

This problem is for grade 9 students only.