**B. 4698.** Give an example for sets \(\displaystyle H_1,H_2,\ldots\subset\mathbb{N}\) for which the following conditions hold:

\(\displaystyle a)\) \(\displaystyle |H_n|=n\) for all positive integers \(\displaystyle n\).

\(\displaystyle b)\) For all positive integers \(\displaystyle n\) and \(\displaystyle k\), \(\displaystyle H_n \cap H_k = H_{(n,k)}\), where \(\displaystyle (n,k)\) is the greatest common divisor of \(\displaystyle n\) and \(\displaystyle k\).

(5 points)

**B. 4701.** Let \(\displaystyle A_1 B_1 C_1 D_1\) be a quadrilateral. For any set of four points \(\displaystyle A_n B_n C_n
D_n\) already defined for a positive integer \(\displaystyle n\), let \(\displaystyle A_{n+1}\) be the centroid of the triangle \(\displaystyle B_{n}C_{n}D_{n}\). \(\displaystyle B_{n+1}\), \(\displaystyle C_{n+1}\) and \(\displaystyle D_{n+1}\) are defined analogously, with a cyclic permutation of the points. Show that for any starting quadrilateral, the point sequence \(\displaystyle A_n\) only has a finite number of points lying outside the unit circle drawn about the centre of mass of the quadrilateral \(\displaystyle A_1 B_1
C_1 D_1\).

Suggested by *E. Gáspár Merse,* Budapest

(4 points)

**B. 4703.** Given that the absolute values of the numbers \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\), \(\displaystyle x_4\), \(\displaystyle x_5\), \(\displaystyle x_6\) are at most 1, and their sum is 0, prove that

\(\displaystyle
3\sum_{i=1}^{5} {\sqrt{1-x_i^2}} \le \sum_{i=1}^{5} {\sqrt{9-{(x_i+x_{i+1})}^2}}\,.
\)

Suggested by *K. Williams,* Szeged

(6 points)

**C. 1283.** The longer base, \(\displaystyle AB\), of trapezium \(\displaystyle ABCD\) is not greater than three times the base \(\displaystyle CD\). Lines \(\displaystyle e\) and \(\displaystyle f\) are parallel to the legs \(\displaystyle BC\) and \(\displaystyle DA\), respectively, and each of them halves the area of the trapezium. Let \(\displaystyle P\) and \(\displaystyle Q\) denote the intersections of \(\displaystyle AB\) with \(\displaystyle e\) and \(\displaystyle f\), respectively, and let \(\displaystyle P'\) and \(\displaystyle Q'\) denote their intersections with \(\displaystyle DC\).

\(\displaystyle a)\) Prove that the intersection \(\displaystyle M\) of lines \(\displaystyle e\) and \(\displaystyle f\) lies on the midline of the trapezium.

\(\displaystyle b)\) Given that the quadrilateral \(\displaystyle PQ'P'Q\) is a parallelogram, find the ratio of the area of triangle \(\displaystyle MPQ\) to the area of the trapezium \(\displaystyle ABCD\).

(5 points)

**C. 1286.** Solve the following simultaneous equations:

\(\displaystyle y^2 =x^3-3x^2+2x,\)

\(\displaystyle x^2 =y^3-3y^2+2y.\)

(5 points)

This problem is for grade 11 - 12 students only.

**K. 458.** Johnny wanted to buy some nails. In one shop, where 100 grams of nails cost 180 forints (HUF, Hungarian currency), he could not buy the quantity needed since he was 1430 forints short. So he went to another shop where 100 grams only cost 120 forints. He bought the quantity he needed, and 490 forints still remained in his pocket. How many kilos of nails did he need?

(6 points)

This problem is for grade 9 students only.

**K. 459.** Uncle Charlie had a ladder of length 2 metres and 60 cm. In order to replace a light bulb in a lamp on the wall, he leaned the ladder against the wall, with the bottom end at a distance of 156 cm from the wall. It turned out that he would need to climb 32 cm higher in order to reach the bulb, so he pushed the bottom of the ladder closer to the wall. By how many centimetres did he push it closer?

(6 points)

This problem is for grade 9 students only.

**K. 460.** A circle of radius 10 units is centred at point \(\displaystyle O\). \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) are points on the circle such that \(\displaystyle O\) lies in the interior of triangle \(\displaystyle ABC\). Given that the length of line segment \(\displaystyle AB\) is 12 units and the measure of angle \(\displaystyle ABC\) is \(\displaystyle 60^\circ\), find

\(\displaystyle a)\) the distance of point \(\displaystyle O\) from line segment \(\displaystyle AB\),

\(\displaystyle b)\) the length of line segment \(\displaystyle AC\).

(6 points)

This problem is for grade 9 students only.

**K. 462.** \(\displaystyle a)\) \(\displaystyle f\) is a function defined on the set of real numbers. Given that \(\displaystyle f(a)-f(b)=f(a\cdot b)\) for all \(\displaystyle a\) and \(\displaystyle b\), find the value of \(\displaystyle f(2015)\).

\(\displaystyle b)\) Is there a function \(\displaystyle g\) defined on the set of real numbers such that \(\displaystyle g(a) - g(b) =
2\cdot g(a\cdot b) - 2\) for all \(\displaystyle a\) and \(\displaystyle b\)?

(6 points)

This problem is for grade 9 students only.