C. 1231. Consider those \(\displaystyle n\)-digit natural numbers that have no digits different from 1 and 2. Prove that there is a number among them that is divisible by \(\displaystyle 2^n\).

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.

C. 1234. One angle of a kite is a right angle, the measure of the opposite angle is \(\displaystyle 30^\circ\), and the length of the shorter side is 10 cm. Find the length of the side of the square that has a vertex on each of three sides of the kite, and has a side parallel to the longer side of the kite.

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?

C. 1236. Determine the sum of the radii of the circles centred on the \(\displaystyle y\)-axis that are tangent to the circle of equation \(\displaystyle {(x-5)}^2+ {(y-5)}^2=25\) and to the line of equation \(\displaystyle y=\frac{4}{3}x+6\).

C. 1237. A 45-decagram loaf of bread is cut in three parts. The difference between the largest and smallest pieces is 5 decagrams. What may be the masses of the pieces if the standard deviation of the masses is \(\displaystyle \sqrt{\frac{14}{3}}\) decagrams?

B. 4632. The intersection of two given lines is off the page. Consider the line that passes through a given point on the page and through the intersection of the given lines. Of this line, construct the part on the page.

B. 4633. There are some points marked in the interior of a triangle, such that no three of them (including the vertices) are collinear. The points are connected to each other and to the vertices of the triangle so that the resulting line segments should not intersect, and the triangle should be divided into the largest possible number of smaller triangles. Prove that the number of small triangles formed is odd.

B. 4635. In an acute-angled triangle \(\displaystyle ABC\), \(\displaystyle AB<AC\). The centre of the circumscribed circle is \(\displaystyle O\), the orthocentre is \(\displaystyle M\). Construct the point \(\displaystyle P\) on the side \(\displaystyle BC\) for which \(\displaystyle AOP\sphericalangle = PMA\sphericalangle\).

B. 4637. Sir Bedevir will only enter in a tournament if he is certain that he will win with a probability of at least 1/2. In any combat of two knights, the probability of the victory of the parties are proportional to their fighting potentials. Bedevir's fighting potential is 1, and that of his \(\displaystyle n\)th opponent is \(\displaystyle \frac{1}{2^{n+1}-1}\). How many knights may have entered in the tournament if Bedevir, having carried out some careful calculations, also decided to enter?

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

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.

B. 4641. The plane \(\displaystyle S\) divides a regular octahedron into two parts. Determine the ratio of the volumes of the parts, given that the intersection of \(\displaystyle S\) with the octahedron is a regular hexagon.

A. 617. Let \(\displaystyle \mathcal{F}\) be a finite family of finite sets and let \(\displaystyle A\) be an arbitrary finite set. We say that \(\displaystyle \mathcal{F}\) shatters the set \(\displaystyle A\) if for every \(\displaystyle X\subseteq A\) there is a set \(\displaystyle F\in \mathcal{F}\) such that \(\displaystyle A\cap F=X\). Show that \(\displaystyle \mathcal{F}\) shatters at least \(\displaystyle |\mathcal{F}|\) sets.

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.