**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.

(5 points)

**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

(5 points)

**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

\(\displaystyle
\sum_{i=1}^{2N} a_{iD} = N-K.
\)

(Chinese problem)

(5 points)

**B. 4788.** Solve the following simultaneous equations on the set of real numbers:

\(\displaystyle x^2+y^3 =x+1,\)

\(\displaystyle x^3+y^2 =y+1.\)

Proposed by *J. Szoldatics,* Budapest

(4 points)

**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.

Proposed by *Sz. Miklós,* Herceghalom

(4 points)

**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.

Proposed by *K. Williams,* Szeged, Radnóti M. Gimn.

(5 points)

**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.

(5 points)

This problem is for grade 1 - 10 students only.

**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?

Proposed by *A. Rókáné Rózsa,* Békéscsaba

(5 points)

This problem is for grade 11 - 12 students only.

**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?

(5 points)

This problem is for grade 11 - 12 students only.