K. 260. In a certain class, more than 93% of the students are girls but there are boys, too. What is the smallest possible number of students in the class if the number of girls is an integer percentage of it?
K. 262. The entrance to the Happy Holidays Campsite lies right next to a straight road. The entrance to the Sweet Summer Youth Camp lies 300 metres off the road. The distance between the two entrances is 500 as the crow flies. A restaurant is to be built right next to the road. Where should it be built so that it is equidistant from the entrances of the campsite and the youth camp?
K. 263. Charles writes the numbers 3, 5, 6 on three cards, and Charlotte writes the numbers 8, 9, 10 on three cards. Each of them selects two cards at random out of the three of their own. Charles multiplies his two numbers, and Charlotte adds hers. What is the probability that Charles will get a greater number than Charlotte?
C. 1045. Three regular dice are rolled. The three numbers obtained on top are listed in an arbitrary order. Then the list is continued with the numbers obtained on the bottom, in the same order. The resulting six-digit number is divided by 111, 7 is subtracted from the ratio, and the difference is divided by 9. Prove that the result is a three-digit number whose digits are the numbers rolled.
C. 1047. A fair coin is tossed ten times in a row. Every time a head is tossed, a digit of 2 is written down. When a tail is tossed, a digit of 3 is written down. What is the probability that the resulting ten-digit number is divisible by a) 3, b) 4?
C. 1049. Consider the circles drawn on two adjacent sides of a square of side 2 units. Find the radius of the circle that touches one of the sides and one circle from the inside and the other circle from the outside.
B. 4292. The feet of the perpendiculars dropped from vertex C of triangle ABC onto the interior angle bisectors from vertices A and B are E and F, respectively. The inscribed circle of the triangle touches side AC at point D. Prove that EF=CD.
B. 4295. A 13×13 table is filled in with numbers, such that the sum of the numbers is the same in each of the 13 rows and 13 columns. What is the smallest possible number of fields in which the entries should be changed so that the 26 sums are all different?
(Based on a problem of the "Kavics Kupa" Competition, 2010)
B. 4299. Parallelogram CDEF is inscribed in triangle ABC, with vertices D, E, F lying on sides CA, AB and BC, respectively. Given the length of the line segment DF, construct the point E. For what point E will the length of diagonal DF be minimal?
A. 515. There is given a triangle ABC. For every 0<t<1 let U(t) and V(t) be the points which divide the line segments AB and BC in the ratio t:(1-t), respectively. Prove that there is a parabola which is tangent to all lines U(t)V(t).
A. 517. Let m3 be a positive integer, and let m(x) be the mth cyclotomic polynomial, and denote by m(x) the polynomial with integer coefficients for which . Prove that for every integer a, any prime divisor of the number m(a) either divides m or is of the form mk1.