**A. 657.** Let \(\displaystyle \{x_n\}\) be the van der Korput sequence, that is, if the binary representation of the positive integer \(\displaystyle n\) is \(\displaystyle n = \sum_i a_i2^i\) (\(\displaystyle a_i\in\{0,1\}\)), then \(\displaystyle x_n =
\sum_i a_i2^{-i-1}\). Let \(\displaystyle V\) be the set of points \(\displaystyle (n,x_n)\) in the plane where \(\displaystyle n\) runs over the positive integers. Let \(\displaystyle G\) be the graph with vertex set \(\displaystyle V\) that is connecting any two distinct points \(\displaystyle p\) and \(\displaystyle q\) if and only if there is a rectangle \(\displaystyle R\) which lies in a parallel position to the axes and \(\displaystyle R\cap V = \{p,q\}\). Prove that the chromatic number of \(\displaystyle G\) is finite.

*Miklós Schweitzer competition, *2015

(5 points)

**B. 4754.** Lines \(\displaystyle AD\), \(\displaystyle BD\) and \(\displaystyle CD\) passing through an interior point \(\displaystyle D\) of a triangle \(\displaystyle ABC\) intersect the opposite sides at \(\displaystyle A_{1}\), \(\displaystyle B_{1}\) and \(\displaystyle C_{1}\), respectively. The midpoints of the segments \(\displaystyle A_1B_1\), \(\displaystyle B_1C_1\) and \(\displaystyle C_1A_1\) are \(\displaystyle C_2\), \(\displaystyle A_2\) and \(\displaystyle B_2\), respectively. Show that the lines \(\displaystyle AA_{2}\), \(\displaystyle BB_{2}\) and \(\displaystyle CC_{2}\) are concurrent.

Proposed by *Sz. Miklós,* Herceghalom

(5 points)

**B. 4755.** In a triangle \(\displaystyle ABC\), the escribed circles \(\displaystyle k_A\) and \(\displaystyle k_B\) drawn to sides \(\displaystyle CB\) and \(\displaystyle CA\) touch the appropriate sides at \(\displaystyle D\) and \(\displaystyle E\), respectively. Show that line \(\displaystyle DE\) cuts out equal chords from the circles \(\displaystyle k_A\) and \(\displaystyle k_B\).

Proposed by *K. Williams,* Szeged

(4 points)

**B. 4756.** In the interior of a unit cube, there are some spheres with a total surface area of 2015. Show that

\(\displaystyle a)\) there exists a line that intersects at least 500 spheres,

\(\displaystyle b)\) there exists a plane that intersects at least 600 spheres.

Hungarian Mathematics Competition of Transylvania

(6 points)

**C. 1324.** Agnes is making gingerbread hearts for Christmas. The pastry cutter has the shape of a 6 cm by 6 cm square with two semicircles attached to two adjacent sides. She always rolls the dough the same thickness, forming a square whose side is a whole number of decimetres. (If any dough remains, she gives it to her sister.) She starts cutting the hearts out of the pastry by placing the corner of the cutter to the corner of the pastry square, carefully aligning the sides. Then she continues by placing the cutter next to the cut-out squares with the same orientation, as close as possible. How many squares can Agnes make if she starts out with a 1 m\(\displaystyle {}^2\) pastry, and she always kneads together the pastry remaining after cutting out the hearts?

(5 points)

**K. 482.** In a bicycle factory, the bicycles produced are tested systematically. The brakes are tested on every fifth bike, the gears are tested on every fourth, and the shifter is tested on every seventh one. They manufacture 435 bicycles a day. How many bicycles are issued from the factory per day without anything tested on them?

(6 points)

This problem is for grade 9 students only.

**K. 483.** In how many different ways is it possible to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 on the circumference of a circle so that no sum of adjacent numbers is a multiple of 3, 5 or 7?

(6 points)

This problem is for grade 9 students only.

**K. 485.** Tom Thumb and the giant arrive at the castle of the dragon. Although the giant is 3.5 metres taller than Tom, he still cannot reach the top of the castle wall when he stands on the ground. So he lifts Tom Thumb on his palm over his head. Tom can just climb the wall, which is 6 metres and 20 centimetres high. The giant has long hands: he can reach 40% of his height above the top of his head, while Tom can only reach 20% of his height above the top of his head. How tall is the giant, and how tall is Tom Thumb?

(6 points)

This problem is for grade 9 students only.