**A. 619.** There are given four rays, \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\) and \(\displaystyle d\) in space, starting from the same point, laying in a plane \(\displaystyle \varPi\). For an arbitrary acute angle \(\displaystyle \varphi\), rotate \(\displaystyle \varPi\) by angle \(\displaystyle \varphi\) in positive direction around each of the four rays; denote the rotated planes by \(\displaystyle A_\varphi\), \(\displaystyle B_\varphi\), \(\displaystyle \varGamma_\varphi\) and \(\displaystyle \varDelta_\varphi\), respectively. Let \(\displaystyle \varSigma_\varphi\) be the plane through the intersection line of \(\displaystyle A_\varphi\) and \(\displaystyle B_\varphi\), and the intersection line of \(\displaystyle \varGamma_\varphi\) and \(\displaystyle \varDelta_\varphi\). Show that the planes \(\displaystyle \varSigma_\varphi\) share a common line.

(5 points)

**B. 4638.** Let \(\displaystyle x_{1}, x_{2}, \ldots, x_{n}\) denote arbitrary real numbers. Prove that \(\displaystyle \sqrt{\left(\sum_{k=1}^{n}
\frac{x_{k}^{4}+k^{2}}{x_{k}^{2}}\right)^{2}-n^{2} (n+1)^{2}}\ge
\sum_{k=1}^{n} \frac{x_{k}^{4}-k^{2}}{x_{k}^{2}}\).

Suggested by *Z. Paulovics,* Zalaegerszeg

(5 points)

**B. 4639.** Let \(\displaystyle P\) be an exterior point of the ellipse \(\displaystyle \mathcal E\) with foci \(\displaystyle F_1\) and \(\displaystyle F_2\). It does not lie on the line of the major axis. Let \(\displaystyle M_1\) be the intersection of the line segment \(\displaystyle PF_1\) with \(\displaystyle \mathcal E\), let \(\displaystyle M_2\) be the intersection of the line segment \(\displaystyle PF_2\) with \(\displaystyle \mathcal E\), and let \(\displaystyle R\) be the intersection of the lines \(\displaystyle M_1F_2\) and \(\displaystyle M_2F_1\). Prove that the quadrilateral \(\displaystyle PM_1RM_2\) has an inscribed circle.

Suggested by *G. Holló,* Budapest

(5 points)

**C. 1232.** In a triangle, the median drawn to side \(\displaystyle b\) is twice as long as the median drawn to side \(\displaystyle c\), and the two medians are perpendicular. Given that the length of the median drawn to side \(\displaystyle a\) is 60 cm, find the perimeter of the triangle.

(5 points)

This problem is for grade 1 - 10 students only.

**C. 1235.** In Flora's flower garden, tulips are grown for mothers' day. One of the flowerbeds has 53 rows with 38 flowers in each row. When Flora inspected all flowers from the first one to the last one, row by row, she observed that every other tulip had coloured streaks in it, every 19th had a broken petal, and every 53rd was not fully open yet. She also discovered that the sum of the numbers of the positions of the perfect tulips (with no coloured streaks or broken petals, fully open) was equal to nineteen times her income in forints (HUF, Hungarian currency). For how many forints did she sell a dozen of perfect tulips?

Suggested by *Á. Meszlényi,* Budapest

(5 points)