K. 439. Each side of the figure in the diagram is 10 cm long, and each interior angle is \(\displaystyle 30^\circ\), \(\displaystyle 60^\circ\), \(\displaystyle 150^\circ\), or \(\displaystyle 300^\circ\). Find the area of the figure.

K. 440. Ten regular dice were rolled simultaneously. The product of the numbers rolled is 7776. Given that the largest number only occurred once, what may be the sum of the numbers?

K. 441. If the sum of the digits of a two-digit number is added to the product of the digits, the result is the number itself. Find all such two-digit numbers.

K. 442. Consider the set of the numbers 1, 3, 120. The product of any two of them is one smaller than a perfect square. Which positive integer less than 120 may be added to the set so that the four numbers should still have this property?

K. 443. The equality \(\displaystyle \overline{ab}_{7} +\overline{cd}_{7} =100_{7}\) holds in base seven notation. What may \(\displaystyle \overline{ab}_{10} +\overline{cd}_{10}\) be in decimal notation?

C. 1254. In a triangle \(\displaystyle ABC\), \(\displaystyle T\) is the foot of the altitude drawn from \(\displaystyle C\), and \(\displaystyle AT=3BT\). Let \(\displaystyle F\) denote the midpoint of \(\displaystyle AB\), and let \(\displaystyle D\) denote the point of altitude \(\displaystyle CT\) where \(\displaystyle AB\) subtends a right angle. Prove that if the orthocentre of triangle \(\displaystyle ABC\) coincides with the centroid of triangle \(\displaystyle FBD\) then \(\displaystyle AD\) bisects the angle \(\displaystyle BAC\).

C. 1259. I have three at least two-digit integers in mind. The number greater by one than my first number, the number greater by four than the double of my second number, and the number greater by nine than three times my third number are all equal. What is the minimum possible product of my three numbers?

C. 1260. The midpoints of sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DA\) of a unit square \(\displaystyle ABCD\) are \(\displaystyle E\), \(\displaystyle F\), \(\displaystyle I\), \(\displaystyle H\), respectively. Let \(\displaystyle M\) denote the intersection of the lines \(\displaystyle ED\) and \(\displaystyle HI\), and let \(\displaystyle G\) denote the intersection of the lines \(\displaystyle EC\) and \(\displaystyle FI\). Find the area of the quadrilateral \(\displaystyle MEGI\).

C. 1261. How many sets of three positive integers are there in which the sum of the numbers is 30, and the sum of any two numbers is greater than the third number?

C. 1264. The interior angle bisector drawn from vertex \(\displaystyle A\) of triangle \(\displaystyle ABC\) intersects the opposite side at point \(\displaystyle P\), and the perpendicular bisector of line segment \(\displaystyle AP\) intersects side \(\displaystyle AC\) at point \(\displaystyle Q\). Given the line segments \(\displaystyle AB\), \(\displaystyle AQ\), and the angle \(\displaystyle BAQ\), express the area of quadrilateral \(\displaystyle ABPQ\).

B. 4669. It is a known fact that 777-headed dragons have either 9 or 13 heads on their necks. Two dragons are considered identical if they have the same number of 9-headed necks. How many different 777-headed dragons are possible?

B. 4670. Let \(\displaystyle A_1\), \(\displaystyle B_1\) and \(\displaystyle C_1\) be the midpoints of the sides of a triangle \(\displaystyle ABC\). Drop a perpendicular from \(\displaystyle A_1\) to the angle bisector drawn from vertex \(\displaystyle A\), from \(\displaystyle B_1\) to the angle bisector drawn from vertex \(\displaystyle B\), and from \(\displaystyle C_1\) to the angle bisector drawn from vertex \(\displaystyle C\). Let \(\displaystyle A_2\) denote the intersection of the perpendiculars from \(\displaystyle B_1\) and from \(\displaystyle C_1\). The points \(\displaystyle B_2\) and \(\displaystyle C_2\) are obtained in a similar way. Show that the lines \(\displaystyle A_1A_2\), \(\displaystyle B_1B_2\) and \(\displaystyle C_1C_2\) are concurrent.

B. 4671. Let \(\displaystyle AB_1B_2\dots B_6\) and \(\displaystyle AC_1C_2\dots C_6\) be regular heptagons with their vertices labelled in the same direction around the clock. Show that the lines \(\displaystyle B_1C_1, B_2C_2, \dots, B_6C_6\) are concurrent.

B. 4672. Determine all real functions \(\displaystyle f\) defined on the positive integers, such that for all positive integer \(\displaystyle n\), \(\displaystyle \frac{p}{f(1)+f(2)+\dots +f(n)} =\frac{p+1}{f(n)}
-\frac{p+1}{f(n+1)}\), where \(\displaystyle p\) is a fixed positive number.

B. 4673. \(\displaystyle E\) is the intersection of the diagonals of a cyclic quadrilateral \(\displaystyle ABCD\), and \(\displaystyle K\) is the centre of the circumscribed circle. The intersection of the lines of sides \(\displaystyle AB\) and \(\displaystyle CD\) is \(\displaystyle F\), and the intersection of the lines of sides \(\displaystyle BC\) and \(\displaystyle DA\) is \(\displaystyle G\). The second intersection of the circumscribed circles of triangles \(\displaystyle BFC\) and \(\displaystyle CGD\) is \(\displaystyle H\). Prove that the points \(\displaystyle K\), \(\displaystyle E\) and \(\displaystyle H\) are collinear.

B. 4674. On the circumscribed circle of triangle \(\displaystyle ABC\), a point \(\displaystyle X\) is moving along the arc \(\displaystyle AC\) not containing vertex \(\displaystyle B\). Let \(\displaystyle Y\) and \(\displaystyle Z\) denote the points on the extensions of side \(\displaystyle BA\) beyond \(\displaystyle A\) and side \(\displaystyle BC\) beyond \(\displaystyle C\), respectively, for which \(\displaystyle AY=AX\) and \(\displaystyle CZ=CX\). What is the locus of the midpoint of line segment \(\displaystyle YZ\)?

B. 4676. A flea is jumping along the number line. It starts at the origin, and the length of each jump is 1. The probability of any jump being in the same direction as the preceding jump is \(\displaystyle p\), and the probability of the opposite direction is \(\displaystyle 1-p\). What is the probability that the flea will get back to the origin?

B. 4677. Prove that if opposite edges of a tetrahedron \(\displaystyle ABCD\) are equal in length then the foot of the altitude drawn from \(\displaystyle D\) lies on the Euler line of triangle \(\displaystyle ABC\).

A. 629. An infinite number of points have been marked in the square grid in such a way that no circle contains more than 2014 marked points. Show that there is a circular disk of diameter 100 (in the plane of the grid) that does not contain any marked point in its interior.

Proposed by: Péter Ágoston, Zoltán Gyenes and Bálint Hujter

A. 630. The konvex quadrilateral \(\displaystyle ABCD\) has an inscribed circle with center \(\displaystyle I\). The rays \(\displaystyle AB\) and \(\displaystyle DC\) meet at point \(\displaystyle F\), the rays \(\displaystyle AD\) and \(\displaystyle BC\) meet at point \(\displaystyle G\). Let \(\displaystyle \mathcal{E}\) be the ellipse with foci \(\displaystyle F\) and \(\displaystyle G\) that passes through points \(\displaystyle B\) and \(\displaystyle D\), and let \(\displaystyle \mathcal{H}\) be the hyperbola branch with foci \(\displaystyle F\) and \(\displaystyle G\) that passes through points \(\displaystyle A\) and \(\displaystyle C\). Denote by \(\displaystyle P\) and \(\displaystyle Q\) the intersections of \(\displaystyle \mathcal{E}\) and \(\displaystyle \mathcal{H}\). Show that the points \(\displaystyle P\), \(\displaystyle Q\) and \(\displaystyle I\) are collinear.

A. 631. Let \(\displaystyle k\ge1\) and let \(\displaystyle I_1,\ldots,I_k\) be non-degenerate subintervals of the interval \(\displaystyle [0, 1]\). Prove \(\displaystyle \sum \frac1{|I_i\cup I_j|} \ge k^2\) where the summation is over all pairs \(\displaystyle (i,j)\) of indices such that \(\displaystyle I_i\) and \(\displaystyle I_j\) are not disjoint.