C. 1350. Define the sequence \(\displaystyle (a_n)\) as follows: \(\displaystyle a_1=1\), and \(\displaystyle a_{n+1}=a_n+4n\) for \(\displaystyle n>0\). Prove that each term of sequence \(\displaystyle (a_n)\) can be expressed as a sum of two consecutive square numbers.

C. 1351. Trapezium \(\displaystyle ABCD\) has an inscribed circle that touches the sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\) and \(\displaystyle DA\) at points \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle G\) and \(\displaystyle H\), respectively. The interior angle at vertex \(\displaystyle B\) is \(\displaystyle 60^\circ\). Let \(\displaystyle I\) denote the intersection of lines \(\displaystyle AD\) and \(\displaystyle FG\), and let \(\displaystyle K\) denote the midpoint of \(\displaystyle FH\). Prove that if \(\displaystyle HE\) is parallel to \(\displaystyle BC\) then \(\displaystyle IK\) is also parallel to them.

C. 1352. Consider a line segment of unit length lying on the diagonal of a unit square. The perpendiculars drawn at the endpoints of this line segment intersect the sides of the square. What is the maximum possible area of the convex hexagon obtained in this way?

C. 1354. In a circle of unit radius, \(\displaystyle n\) identical small circles (\(\displaystyle n>2\)) are drawn such that each of them touches the unit circle on the inside, and also touches both of the adjacent small circles. What fraction of the area of the unit circle is covered by the small circles? For \(\displaystyle n=3\), 4 and 6, calculate the numerical value of the ratio.

C. 1355. All positive even integers are written down, in increasing order. The list is broken down into rows such that the \(\displaystyle n\)th row consists of \(\displaystyle n\) consecutive even numbers. What is the sum of the numbers in the 2016th row?

C. 1356. In a cube \(\displaystyle ABCDEFGH\) let \(\displaystyle K\), \(\displaystyle L\) and \(\displaystyle M\) denote the midpoints of edges \(\displaystyle AB\), \(\displaystyle CG\) and \(\displaystyle EH\), respectively. What is the ratio of the volume of the tetrahedron \(\displaystyle FKLM\) to the volume of the cube?

B. 4786. Let \(\displaystyle p\) and \(\displaystyle q\) be positive integers. Prove that at least one of \(\displaystyle p^2+q\) and \(\displaystyle p+q^2\) is not a perfect square.

B. 4787. The area of a certain right-angled trapezium is equal to the half of the product of the legs. At which point of the perpendicular leg does the other leg subtend a maximum angle?

B. 4789. The interior angle bisectors drawn from vertices \(\displaystyle A\) and \(\displaystyle B\) in a triangle \(\displaystyle ABC\) intersect the circumscribed circle again at the points \(\displaystyle G\) and \(\displaystyle H\), respectively. The points of tangency of the inscribed circle of triangle \(\displaystyle ABC\) on sides \(\displaystyle BC\) and \(\displaystyle AC\) are \(\displaystyle D\) and \(\displaystyle E\), respectively. Let \(\displaystyle K\) denote the circumcentre of triangle \(\displaystyle DCE\). Show that the points \(\displaystyle G\), \(\displaystyle H\) and \(\displaystyle K\) are collinear.

B. 4790. In a scalene triangle \(\displaystyle ABC\), a Thales circle is drawn over each median. The Thales circles of the medians drawn from vertices \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) intersect the circumscribed circle of the triangle \(\displaystyle ABC\) again at points \(\displaystyle A_1\), \(\displaystyle B_1\), and \(\displaystyle C_1\), respectively. Prove that the perpendiculars drawn to \(\displaystyle AA_1\) at \(\displaystyle A\), to \(\displaystyle BB_1\) at \(\displaystyle B\) and to \(\displaystyle CC_1\) at \(\displaystyle C\) are concurrent.

B. 4791. Altitudes \(\displaystyle AD\) and \(\displaystyle CE\) of triangle \(\displaystyle ABC\) intersect at point \(\displaystyle M\). Line \(\displaystyle DE\) intersects the line of side \(\displaystyle AC\) at \(\displaystyle P\). Prove that line \(\displaystyle PM\) is perpendicular to the median drawn from vertex \(\displaystyle B\) of the triangle.

B. 4792. Is it true that by deleting infinitely many appropriate digits out of the decimal representation of any positive irrational number, we can always get back the original number?

B. 4793. How many permutations do the numbers \(\displaystyle 1, 2, 3, \ldots, n\) have in which there are exactly \(\displaystyle a)\) one, \(\displaystyle b)\) two occurrences of a number being greater than the adjacent number on the right of it?

A. 668. There is given a positive integer \(\displaystyle k\), some distinct points \(\displaystyle A_1,A_2,\ldots,A_{2k+1}\) and \(\displaystyle O\) in the plane, and a line \(\displaystyle \ell\) passing through \(\displaystyle O\). For every \(\displaystyle i=1,\ldots,2k+1\), let \(\displaystyle B_i\) be the reflection of \(\displaystyle A_i\) about \(\displaystyle \ell\), and let the lines \(\displaystyle OB_i\) and \(\displaystyle A_{i+k}A_{i+k+1}\) meet \(\displaystyle C_i\). (The indices are considered modulo \(\displaystyle 2k+1\): \(\displaystyle A_{2k+2}=A_1\), \(\displaystyle A_{2k+3}=A_2\), ..., and it is assumed that these intersections occur.) Show that if the points \(\displaystyle C_1,C_2,\ldots,C_{2k}\) lie on a line then that line passes through \(\displaystyle C_{2k+1}\) also.

A. 669. Determine whether the set of rational numbers can be ordered to in a sequence \(\displaystyle q_1,q_2,\ldots\) in such a way that there is no sequence of indices \(\displaystyle 1\le
i_1<i_2<\dots<i_6\) such that \(\displaystyle q_{i_1},q_{i_2},\ldots,q_{i_6}\) form an arithmetic progression.

Proposed by: Gyula Károlyi, Budajenő and Péter Komjáth, Budapest

A. 670. Let \(\displaystyle a_1,a_2,\ldots\) be a sequence of nonnegative integers such that

\(\displaystyle
\sum_{i=1}^{2n} a_{id} \le n
\)

holds for every pair \(\displaystyle (n,d)\) of positive integers. Prove that for every positive integer \(\displaystyle K\), there are some positive integers \(\displaystyle N\) and \(\displaystyle D\) such that