**B. 4860.** Assume that \(\displaystyle a<b<c<d\) and \(\displaystyle a+d\ne b+c\). Show that the equation

\(\displaystyle
\frac1{a-x}-\frac1{b-x}-\frac{1}{c-x}+\frac1{d-x}=0
\)

has exactly two distinct roots, such that one of them lies in the interval \(\displaystyle (b,c)\), and the other one lies outside the interval \(\displaystyle (a,d)\).

(3 points)

**B. 4861.** Let \(\displaystyle r\) and \(\displaystyle R\), respectively, denote the radii of the inscribed and circumscribed circles of a right-angled triangle \(\displaystyle ABC\). Let \(\displaystyle CD\) be the altitude drawn to the hypotenuse \(\displaystyle AB\). Draw the square \(\displaystyle CEFG\) of side length \(\displaystyle CD\) that has vertex \(\displaystyle E\) on side \(\displaystyle AC\) and vertex \(\displaystyle G\) on side \(\displaystyle BC\). Let \(\displaystyle T\) and \(\displaystyle t\), respectively, denote the parts of the area of the square \(\displaystyle CEFG\) which lie inside and outside the triangle \(\displaystyle ABC\). Prove that

\(\displaystyle
\frac{t}{T}=\frac{r}{2R}.
\)

(*B. Bíró,* Eger)

(4 points)

**B. 4862.** \(\displaystyle M\), \(\displaystyle N\), \(\displaystyle P\), \(\displaystyle Q\), and \(\displaystyle R\), respectively, are the midpoints of the sides \(\displaystyle AB\), \(\displaystyle BC\), \(\displaystyle CD\), \(\displaystyle DE\) and \(\displaystyle EA\) of a convex pentagon \(\displaystyle ABCDE\). Show that if the line segments \(\displaystyle AP\), \(\displaystyle BQ\), \(\displaystyle CR\) and \(\displaystyle DM\) are concurrent then the common point also lies on line segment \(\displaystyle EN\).

(*S. Róka,* Nyíregyháza)

(5 points)

**B. 4866.** Xavier and Yvette take turns in choosing

\(\displaystyle a)\) real numbers;

\(\displaystyle b)\) complex numbers.

Xavier starts, and the game ends after the 100th number. Let the numbers be \(\displaystyle a_1,\dots,a_{100}\). Yvette's goal is to make the sum \(\displaystyle a_1a_2+a_1a_3+\ldots
+a_{99}a_{100}\) of the products of the \(\displaystyle \lbinom{100}2\) pairs of the numbers equal to 0, while Xavier is trying to avoid that. Who has got a winning strategy?

(6 points)

**C. 1407.** In a parallelogram \(\displaystyle ABCD\), \(\displaystyle M\) and \(\displaystyle N\) are points on sides \(\displaystyle AD\) and \(\displaystyle DC\), respectively, such that \(\displaystyle \frac{AM}{MD}=\frac{DN}{NC}=\frac{7}{11}\). Let \(\displaystyle P\) denote the intersection of lines \(\displaystyle BM\) and \(\displaystyle AN\).

Prove that the areas of triangle \(\displaystyle APB\) and quadrilateral \(\displaystyle DMPN\) are equal.

(Based on the idea of *L. Longáver,* Nagybánya)

(5 points)

This problem is for grade 1 - 10 students only.

**K. 542.** A shop sells a certain souvenir for 1100 forints (HUF, Hungarian currency). A tourist from an exotic country wants to buy one by paying for it with the currency of his own country. In his country, there are three kinds of coins: round, triangular and square. 11 round coins are worth exactly 1500 forints, 11 square coins are worth 1600 forints, and 11 triangular coins are worth 1700 forints. How many of each type of coin should the tourist hand over in order to pay the exact price of 1100 forints? Find all possible answers.

(6 points)

This problem is for grade 9 students only.

**K. 543.** In a food store, 27 cubical boxes are arranged to form a large cube. In each box, there is a large piece of cheese. A mouse crawls into one of the boxes lying on the floor and eats the cheese he finds there. Then he moves to another nonempty box that has a face in common with the first box, and eats the cheese again. He continues in this way. Is it possible for the mouse to organize his moves so that he should finish

\(\displaystyle a)\) in the box at the centre of a face of the large cube?

\(\displaystyle b)\) in the box in the interior of the large cube?

(6 points)

This problem is for grade 9 students only.

**K. 544.** Triangular numbers are defined as the numbers that can be represented as the sum of the first few consecutive natural numbers (\(\displaystyle 1, 3, 6, 10, \ldots\)). Hexagonal numbers are defined by the (infinite) sequence of *diagrams* represented below. Find all hexagonal numbers that are also triangular numbers.

(6 points)

This problem is for grade 9 students only.

**K. 546.** Barbara chooses two numbers out of the set \(\displaystyle A = \{8, 9, 10\}\) at random, and adds them together. Matthew chooses two numbers out of the set \(\displaystyle M = \{3, 5, 6\}\) at random and multiplies them together. What is the probability that Barbara's result is larger?

(6 points)

This problem is for grade 9 students only.