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In a vase, there is a bouquet of 7 white and blue lilac branches. It is known that 1$)$ at least one branch is white, 2$)$ out of any two branches, at least one is blue. How many white branches and how many blue are there in the bouquet?

In a race between 6 athletes, Andrew falls behind Boris and two athletes finish between them. Vincent finished after Declan, but before George. Declan finished before Boris but after Eric. Which order did the athletes finish the race in?

This problem is from Ancient Rome.

$\\$ A rich senator died, leaving his wife pregnant. After the senator’s death it was found out that he left a property of 210 talents (an Ancient Roman currency) in his will as follows: “In the case of the birth of a son, give the boy two thirds of my property (i.e. 140 talents) and the other third (i.e. 70 talents) to the mother. In the case of the birth of a daughter, give the girl one third of my property (i.e. 70 talents) and the other two thirds (i.e. 140 talents) to the mother.”

$\\$ The senator’s widow gave birth to twins: one boy and one girl. This possibility was not foreseen by the late senator. How can the property be divided between three inheritors so that it is as close as possible to the instructions of the will?

We meet three people: Alex, Brian and Ben. One of them is an architect, the other is a baker and the third is an archeologist. One lives in Aberdeen, the other in Birmingham and the third in Brighton.$\\$

1) Ben is in Birmingham only for trips, and even then very rarely. However, all his relatives live in this city.$\\$

2) For two of these people the first letter of their name, the city they live in and their job is the same.$\\$

3) The wife of the architect is Ben’s younger sister.

Three friends – Peter, Ryan and Sarah – are university students, each studying a different subject from one of the following: mathematics, physics or chemistry. If Peter is the mathematician then Sarah isn’t the physicist. If Ryan isn’t the physicist then Peter is the mathematician. If Sarah isn’t the mathematician then Ryan is the chemist. Can you determine which subject each of the friends is studying?

In the numbers of MEXAILO and LOMONOSOV, each letter denotes a number $($different letters correspond to different numbers$)$. It is known that the products of the numbers of these two words are equal. Can both numbers be odd?

Two weighings. There are 7 coins which are identical on the surface, including 5 real ones $($all of the same weight$)$ and 2 counterfeit coins $($both of the same weight, but lighter than the real ones$)$. How can you find the 3 real coins with the help of two weighings on scales without weights?

Jessica, Nicole and Alex received 6 coins between them: 3 gold coins and 3 silver coins. Each of them received 2 coins. Jessica doesn’t know which coins the others received but only which coins she has. Think of a question which Jessica can answer with either “yes”, “no” or “I don’t know” such that from the answer you can know which coins Jessica has.

Author: I.S. Rubanov

On the table, there are 7 cards with numbers from 0 to 6. Two take turns in taking one card. The winner is the one is the first person who can, from his cards, make up a natural number that is divisible by 17. Who will win in a regular game the person who goes first or second?

There are scales and 100 coins, among which several (more than 0 but less than 99) are fake. All of the counterfeit coins weigh the same and all of the real ones also weigh the same, while the counterfeit coin is lighter than the real one. You can do weighings on the scales by paying with one of the coins (whether real or fake) before weighing. Prove that it is possible with a guarantee to find a real coin.

Find all of the solutions of the puzzle: ARKA + RKA + KA + A = 2014. (Different letters correspond to different numbers, and the same letters correspond to the same numbers.)

Author: E.V. Bakaev

After a hockey match Anthony said that he scored 3 goals, and Ilya only one. Ilya said that he scored 4 goals, and Serge scored 5 goals. Serge said that he scored 6 goals, and Anthony only two. Could it be that the three of them scored 10 goals, if it is known that each of them once told the truth, and once lied?

Author: E.V. Bakaev

From the beginning of the academic year, Andrew wrote down his marks for mathematics. When he received another evaluation (2, 3, 4 or 5), he called it unexpected, if before that time this mark was met less often than each of the other possible marks. (For example, if he had received the following marks: 3, 4, 2, 5, 5, 5, 2, 3, 4, 3 from the beginning of the year, the first five and the second four would have been unexpected). For the whole academic year, Andrew received 40 marks – 10 fives, fours, threes and twos (it is not known in which order). Is it possible to say exactly how many marks were unexpected?

Authors: B.R. Frenkin, T.V. Kazitcina

On the tree sat 100 parrots of three kinds: green, yellow, multi-coloured. A crow flew past and croaked: “Among you, there are more green parrots than multi-coloured ones!” – “Yes!” – agreed 50 parrots, and the others shouted “No!”. Glad to the dialogue, the crow again croaked: “Among you, there are more multi-coloured parrots than yellow ones!” Again, half of the parrots shouted “Yes!”, and the rest – “No!”. The green parrots both told the truth, the yellow ones lied both times, and each of the multi-coloured ones lied once, and once told the truth. Could there be more yellow than green parrots?

Author: N. Medved

Peter and Victoria are playing on a board measuring 7 × 7. They take turns putting the numbers from 1 to 7 in the board cells so that the same number does not appear in one line nor in one column. Peter goes first. The player who loses is the one who cannot make a move. Who of them can win, no matter how the opponent plays?

Author: A.V. Shapovalov

To the cabin of the cable car leading up to the mountain, four people arrived who weigh 50, 60, 70 and 90 kg. A supervisor does not exist, but the cable car travels back and forth in automatic mode only with a load from 100 to 250 kg (in particular, it does not go anywhere when the cable car is empty), provided that passengers can be seated on two benches so that the weights on the benches differ by no more than 25 kg. How can they all climb the mountain?

Ali Baba followed by 40 robbers lined up on the crossing across the Bosporus Strait. There is only one boat and in it there can be either two or three people (there cannot be one person in the boat). Among those in the boat there should not be people who are not friends with each other. Will all of them be able to cross, if every two people standing next to each other in the queue are friends, while Ali Baba is also friends with the robber standing behind the person next to him?

Author: M.A. Khachaturyan

Mum baked identical pies with the same appearance: 7 with cabbage, 7 with meat and one with cherries, and laid them out in a circle on a round dish in this order. Then she put the dish into a microwave and to warm up the pies. Olga knows how she originally arranged the pies, but she does not know the dish turned in the microwave. She wants to eat a pie with cherries, and she thinks that the rest are tasteless. How does Olga surely achieve this, after biting into no more than three tasteless pies?

Author: N.K. Agakhanov

On the board, the equation $xp^3$ + * x² + * x + * = 0 is written. Peter and Vlad take turns to replace the asterisks with rational numbers: first, Peter replaces any of the asterisks, then Vlad – any of the two remaining ones, and then Peter replaces the remaining asterisk. Is it true that for any of Vlad’s actions, Peter can get an equation in which the difference of some two roots is equal to 2014?

Author: A.V. Khachaturyan

Replace the letters of the word MATEMATIKA with numbers and signs of addition and subtraction so that a numeric expression equal to 2014 is obtained.

(The same letters denote the same numbers or signs, different letters denote different numbers or signs. Note that it is enough to give an example.)