**A. 701.** An airline operates flights between any two capital cities in the European Union. Each flight has a fixed price which is the same in both directions. Furthermore, the flight prices from any given city are pairwise distinct. Anna and Bella wish to visit each city exactly once, not necessarily starting from the same city. While Anna always takes the cheapest flight from her current city to some city she hasn't visited yet, Bella always continues her tour with the most expensive flight available. Is it true that Bella's tour will surely cost at least as much as Anna's tour?

(*Based on a Soviet problem*)

(5 points)

**A. 702.** Fix a triangle \(\displaystyle ABC\). We say that triangle \(\displaystyle XYZ\) is elegant if \(\displaystyle X\) lies on segment \(\displaystyle BC\), \(\displaystyle Y\) lies on segment \(\displaystyle CA\), \(\displaystyle Z\) lies on segment \(\displaystyle AB\), and \(\displaystyle XYZ\) is similar to \(\displaystyle ABC\) (i.e., \(\displaystyle \angle A=\angle X\), \(\displaystyle \angle B=\angle Y\), \(\displaystyle \angle C=\angle Z\)). Of all the elegant triangles, which one has the smallest perimeter?

(5 points)

**A. 703.** Let \(\displaystyle n\ge 2\) be an integer. We call an ordered \(\displaystyle n\)-tuple of integers primitive if the greatest common divisor of its components is \(\displaystyle 1\). Prove that for every finite set \(\displaystyle H\) of primitive \(\displaystyle n\)-tuples, there exists a non-constant homogenous polynomial \(\displaystyle f(x_1,x_2,\dots,x_n)\) with integer coefficients whose value is \(\displaystyle 1\) at every \(\displaystyle n\)-tuple in \(\displaystyle H\).

(Based on the sixth problem of the *58th IMO,* Brazil)

(5 points)

**B. 4885.** Let \(\displaystyle k\) and \(\displaystyle m\) be two distinct 14-digit positive integers, each containing two of each digit 1, 2, 3, 4, 5, 6 and 7 (like 22133456456717, for example). Prove that \(\displaystyle \frac km\) cannot be an integer.

(*M&IQ*)

(4 points)

**B. 4887.** Prove that there are infinitely many number pairs \(\displaystyle (a,b)\), such that \(\displaystyle a+\frac{1}{b}=b+\frac{1}{a}\), where \(\displaystyle a\ne b\). Find the possible values of \(\displaystyle ab\).

(Proposed by *J. Szoldatics,* Budapest)

(3 points)

**B. 4891.** The circles \(\displaystyle S_1\), \(\displaystyle S_2\), \(\displaystyle S_3\) pairwise touch each other on the outside. Let \(\displaystyle A\), \(\displaystyle B\) and \(\displaystyle C\) denote the common points of the circles \(\displaystyle S_1\) and \(\displaystyle S_2\), \(\displaystyle S_1\) and \(\displaystyle S_3\), \(\displaystyle S_2\) and \(\displaystyle S_3\), respectively. Line \(\displaystyle AB\) intersects the circles \(\displaystyle S_2\) and \(\displaystyle S_3\) again at points \(\displaystyle D\) and \(\displaystyle E\), respectively. Line \(\displaystyle DC\) intersects circle \(\displaystyle S_3\) again at \(\displaystyle F\). Prove that triangle \(\displaystyle DEF\) is right-angled.

(*Kvant*)

(5 points)

**B. 4892.** Two players, First and Second, play the following game: they place 2017 pebbles on the table. First starts by removing 1 pebble. Then Second may choose to remove either 1 or 2. Then First may remove 1, 2, 3 or 4. Then Second may remove any number from 1 to 8. And so on, the player in the \(\displaystyle i\)th step needs to remove at least 1 and at most \(\displaystyle 2^{i-1}\) pebbles. The player removing the last pebble from the table wins the game. Who has a winning strategy?

(6 points)

**C. 1431.** The lengths of the shorter base of a trapezium, then one leg, then the other leg and finally the longer base, in this order, form an arithmetic progression. Given that the length of the shortest side is 3 cm, and one of the angles lying on the longer base is 60 degrees, what is the common difference of the arithmetic progression?

(5 points)

**K. 548.** We have four boxes numbered 1 to 4, and four cards with the numbers 1, 2, 3, 4 on them. We place one card in each box, according to the following rule: every card shows the number of the box that contains the card corresponding to the number of the box containing it. In how many different ways is it possible to place the cards in the boxes?

(6 points)

This problem is for grade 9 students only.

**K. 549.** Three cars are travelling along the same road, in the same direction but at different uniform speeds. In principle, there are six possible orders for the three cars behind each other. Is it possible that all six orders actually occur during their journey?

(Proposed by *L. Loránt,* Budapest)

(6 points)

This problem is for grade 9 students only.

**K. 550.** An unusual telegraph company charges for the various words by the letters they contain. Consonants are free, but each vowel costs a certain amount. We do not know these prices, but we do know the charges for a few words we have sent before: TÉGLALAP, PARALELOGRAMMA, NÉGYZET, HÁROMSZÖG, NÉGYSZÖG, ROMBUSZ, TRAPÉZ, DELTOID. (These are all mathematical terms in Hungarian. Y is not a vowel, and vowels with accents on them count as different vowels.) Show a possible method to determine the charge for the word GEOMETRIA.

(6 points)

This problem is for grade 9 students only.