**A. 695.** We are given \(\displaystyle 2k\) lines, \(\displaystyle e_1, \ldots, e_{2k}\) in a plane \(\displaystyle S\), further given a line \(\displaystyle g\) which has an angle \(\displaystyle \alpha\) with \(\displaystyle S\). Show that the sum of the pairwise angles between the lines \(\displaystyle e_1, \ldots,e_{2k}, g\) is at most

\(\displaystyle
(k^2+k)\cdot \dfrac \pi 2 + k\alpha.
\)

(5 points)

**A. 697.** For all primes \(\displaystyle p\ge3\), let

\(\displaystyle
S(p) = \sum_{k=1}^{\frac{p-1}2} \tan \frac{k^2\pi}{p}.
\)

\(\displaystyle (a)\) Show that \(\displaystyle p\equiv1\pmod4\) implies \(\displaystyle S(p)=0\).

\(\displaystyle (b)\) Show that if \(\displaystyle p\equiv3\pmod4\), then \(\displaystyle \dfrac{S(p)}{\sqrt{p}}\) is an odd integer.

(5 points)

**B. 4870.** The diagonals of a convex quadrilateral \(\displaystyle ABCD\) intersect at point \(\displaystyle E\). Prove that the centres of the Feuerbach circles of the triangles \(\displaystyle ABE\), \(\displaystyle BCE\), \(\displaystyle CDE\) and \(\displaystyle DAE\) either form a parallelogram or are collinear.

(Proposed by *Sz. Miklós,* Herceghalom)

(5 points)

**B. 4872.** Let \(\displaystyle p\), \(\displaystyle q\) and \(\displaystyle r\) denote three distinct primes, and let \(\displaystyle n=pq^2r^3\). Show that

\(\displaystyle
(n,1)+(n,2)+\dots+(n,n) =qr^2(2p-1)(3q-2)(4r-3),
\)

where \(\displaystyle (a,b)\) denotes the greatest common divisor of \(\displaystyle a\) and \(\displaystyle b\).

(Proposed by *J. Szoldatics,* Budapest)

(6 points)

**C. 1419.** In an acute-angled triangle \(\displaystyle ABC\), consider the two circular sectors, each bounded by an arc \(\displaystyle AB\), with one arc passing through the foot of the altitude drawn from \(\displaystyle A\), and the other arc passing through the orthocentre of the triangle. Prove that if \(\displaystyle \angle ACB =45^\circ\) then the areas of the two circular sectors are equal.

(5 points)

This problem is for grade 11 - 12 students only.