**Grades 5 and 6
Vilnius University, Faculty of Mathematics and Informatics
September the 12th 2009**

1. Every year Geppetto gives pocket-money his beloved patient and modest Pinocchio on his birthday. The sum of that support is established according to the unshakable rules and, counting in humble cents, is always equal to the product of the ages of Pinocchio and his noblest patron himself. This year Pinocchio got 7.81 Euro. What amount of money did Pinocchio get in the previous year?

2. On their infrequent leisure time the immortal Bremen four – Donkey, Dog, Cat and Rooster divided the usual chess board into four equal parts and started examining one of these parts containing 16 fields (8 white and 8 black fields, colored in the usual chess order), i.e. a 4 x 4 square.

The zigzag-form path consisting of 4 white fields, one from each row, such that any two neighbouring fields share only a common corner was called by them a Bremen path. The musicians immediately started furious discussions about how many Bremen paths there are in that small 4 x 4 square. Patron of the Bremen city Roland gave evidence that they sat late and couldn‘t come to the common conclusion how many Bremen paths there are in that small 4 x 4 square.

Could you explain in an understandable way to that immortal Bremen four how many Bremen paths could be detected in that (rather small) 4 x 4 square?

3. Nowadays only very few persons remember that before gaining the unbelievable popularity the members of the immortal Bremen four earned their bread working as unqualified city guards. The patron of the City Roland witnessed strongly that it was also their duty every night before the day-break to creep along the wall around the whole Bremen city checking whether everything is still running well. The very same Roland claims that he established for sure that the Donkey made twice as many walks around the city as the Cat did, three times as many as the Dog did and even four times as many as the Rooster did. Altogether they all made exactly 400 walks around the Bremen city. How many walks around the Bremen city did the Donkey make?

4. Yesterday the Cup of Nations (Bremenland soccer tournament) in which each team played exactly once against every other team came to the end. The matches were played according to the rules which generated a healthy hazard: a team was awarded even 3 points for a win, 1 point for a tie and no points for a loss. After all matches were over it was mentioned that all teams together were rewarded the total of 21 points. The troubadour of the Cup maestro Rooster spent the whole 3 days in the deepest confidence that knowing only what was told right now it is still impossible neither to conclude

(A) how many teams participated in that Cup of Nations,

nor to establish

(B) how many points each team was awarded (according to its final classification).

Was the troubadour of that Cup Rooster right in his belief?

(A) How many teams participated in that Cup of Nations?

(B) How many points was each team awarded?

5. On another honorable occasion the members of the immortal four during the camping near their beloved city Bremen saw 7 regions formed by 3 overlapping circles. They came to an idea to use 7 numbers 1, 2, 3, 4, 5, 6, 7 to rule these seven regions – one number for each region. They decided at once that the central region – the one belonging to all the 3 circles – should always be ruled by number 6 as the number having most divisors. They also expected that it is possible to nominate the numbers for ruling the regions in a democratic way, that is in such a way that in any of these 3 circles the sum all integers ruling regions contained in that circle is the same for each circle (and equals some number T).

All this would have been very nice and challenging but they could in no way find an example which would demonstrate them that such democratic ruling of regions is possible to establish. Help them in:

(A) demonstrating that such democratic ruling of regions is possible to establish; present them the corresponding example and, if necessary, also tell them even the corresponding value of T, the same for all 3 circles;

(B) finding all possible values of that sum T, equal for all three circles – each one, naturally, with its own example of democratic ruling.

**Grades 7 and 8
Vilnius University, Faculty of Mathematics and Informatics
September the 12th 2009**

1. The immortal Bremen four – Donkey, Dog, Cat and Rooster – after they became the absolute classics of the hard beat music could no more perform all together. On very rare occasions they performed being three, and that was taken as a sign of an absolute respect, and such performances and only these were called the Bremen ecstasies. When in recent summer in Bremen the World Session of Beautiful Young Math Minds took place then, to honor it, the immortal four had given several exceptional Bremen ecstasies. The soul of the city Roland who naturally took part in all these Bremen ecstasies certified that maestro Rooster participated more times than any other of them, that is, 8 times, and maestro Donkey participated less than any other of them – 5 times. The soul of the city Roland without saying any word made them feel that any clever mind which is able to concentrate at least a bit could really understand how many Bremen ecstasies were provided by the members of immortal Bremen four.

Are you also able to explain how many Bremen ecstasies were given in Bremen at the World Session of the Beautiful Young Math Minds by the members of that immortal Bremen four?

2. To the top league of soccer in Bremenland only 5 teams are allowed. In the recent tournament each team played exactly once against each of the other four teams. Each team received 3 points for a match it won, one point for a match it drew and no points for a match it lost. At the end of the competition the points were: Prairie’s Lions 10 points, Desert’s Bison 9 points, Alpine Grandsons 4 points, Peaceful Bulldogs 3 points, and Windmills 1 point.

The voice of the league maestro Rooster claimed that even knowing only as much as that it is already possible to make many fundamental insights and not only

(A) to state precisely how many matches resulted in a draw,

but also even to state

(B) what were the results of Alpine Grandsons’ matches against the other four teams.

We believe for sure that you having some time are also able to understand it and explain it in a proper way.

3. On silent winter nights when the last lonely passengers disappear from the streets the eternal patron of the Bremen city Roland is climbing down from his monument and together with maestro Cat are arranging, as they call it, the silent Bremen-17 game. For it 7 cards with numbers 0, 1, 2, 3, 4, 5, 6 are necessary (with every number written on exactly one card). Roland and maestro Cat are taking in turn one card each; Roland usually starts first. The player who is able using his cards to present a number divisible by 17 earlier than his opponent is declared to be a winner. On the news portal ihaha.com there were furious quarrels concerning the fact whether any of them is able to take the cards in such a way that he is always able to win independently what his partner undertakes.

There appeared one wiseacre whose name was Rex who kept claiming that

(A) if any of them is really able to win independently of what his partner is ever able to undertake then that person is the person who starts first.

Is the wiseacre Rex right? Explain your answer.

(B) So how is that: are indeed any of them able to win independently of what his partner is ever able to undertake?

Explain your answer.

4. Maestro Cat eagerly intends to solve the cross-number. The solution to each clue of this cross-number is a two-digit number. Maestro Cat remembers too well that none of these numbers begins with zero. He is ready to complete the cross-number, stating the order in which he solved the clues and explaining why there is only one possibility at each stage. Can you help him?

Clues horizontally: 1. Multiple of 3. 3. Three times a prime

Clues vertically: 1. Multiple of 25. 2. Square number.

5. Starting with an equilateral triangle ABC with length of side 2 meters, the hidden geometer maestro Rex on the three its sides AB, BC and CA constructed three outward-pointed squares ABPQ, BCTU, CARS. Maestro Donkey who just arrived was kept repeating some irresponsible statements that it is not maestro Rex who could today establish the area of the hexagon PQRSTU. Maestro Rooster who heard this tried eagerly to help Rex with that area and acting together they both in two hours were able to get the right answer.

What is the area of the hexagon PQRSTU?