Several crocodiles, dragons and snakes were left on an island. Animals were eating each other according to the following rules. Every day at the breakfast, each snake ate one dragon; at the lunch, each dragon ate one crocodile; and at the dinner, each crocodile ate one snake. On the Saturday after the dinner, only one crocodile and no snakes and dragons remained on the island. How many crocodiles, dragons and snakes were there on the Monday in the same week before the breakfast?

The eyes of a magician are blindfolded while a person $A$ from the audience arranges $n$ identical coins in a row, some are heads and the others are tails. The assistant of the magician asks $A$ to write an integer between $1$ and $n$ inclusive and to show it to the audience. Having seen the number, the assistant chooses a coin and turns it to the other side (so if it was heads it becomes tails and vice versa) and does not touch anything else. Afterwards, the bandages are removed from the magician, he sees the sequence and guesses the written number by $A$. For which $n$ is this possible? [hide=Spoiler hint] The original formulation asks: a) Show that if $n$ is possible, so is $2n$; b) Show that only powers of $2$ are possible; I have omitted this from the above formulation, for the reader's interest. [/hide]

There are 100 houses in the street, divided into 50 pairs. In each pair houses are right across the street one from another. On the right side of the street the houses have even numbers, while the houses on the left side have odd numbers. On both sides of the street the numbers increase from the beginning to the end of the street, but are not necessarily consecutive (some numbers may be omitted). For each house on the right side of the street, the difference between its number and the number of the opposite house was computed, and it turned out that all these values were different. Let $n$ be the greatest number of a house on this street. Find the smallest possible value of $n$. (8 points) Maxim Didin

In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims: (i) Less than $12$ employees have a more difficult work; (ii) At least $30$ employees take a higher salary. Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kopeks wins. Which player has a winning strategy? [i](5 points)[/i]

Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.

$n$ people sit in a circle. Each of them is either a liar (always lies) or a truthteller (always tells the truth). Every person knows exactly who speaks the truth and who lies. In their turn, each person says 'the person two seats to my left is a truthteller'. It is known that there's at least one liar and at least one truthteller in the circle. [list=a] [*] Is it possible that $n=2017?$ [*] Is it possible that $n=5778?$ [/list]

Imagine the unit square in the plane to be a [i]carrom board[/i]. Assume the [i]striker[/i] is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x,y)$ and its initial velocity $\overrightarrow{(p,q)}$. If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its [i]bounce number[/i] to be the number of edges it hits before returning to its initial state for the $1^{\text{st}}$ time. For example, the trajectory with initial state $[(.5,.5);\overrightarrow{(1,0)}]$ has bounce number $2$ and it returns to its initial state for the $1^{\text{st}}$ time in $2$ time units. And the trajectory with initial state $[(.25,.75);\overrightarrow{(1,1)}]$ has bounce number $4$. $\textbf{(a)}$ Suppose the striker has initial state $[(.5,.5);\overrightarrow{(p,q)}]$. If $p>q\geqslant 0$ then what is its velocity after it hits an edge for the $1^{\text{st}}$ time ? What if $q>p\geqslant 0$ ? $\textbf{(b)}$ Draw a trajectory with bounce number $5$ or justify why it is impossible. $\textbf{(c)}$ Consider the trajectory with initial state $[(x,y);\overrightarrow{(p,0)}]$ where $p$ is a positive integer. In how much time will the striker $1^{\text{st}}$ return to its initial state ? $\textbf{(d)}$ What is the bounce number for the initial state $[(x,y);\overrightarrow{(p,q)}]$ where $p,q$ are relatively prime positive integers, assuming the striker never hits a corner ?

Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.

Call a (non-trivial) lattice class a pseudo-variety if it is closed under taking a homomorphic image, a direct product, and a convex subset. Prove that the smallest distributive pseudo-variety cannot be defined by a first-order set of formulas.

One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements: [list] [*] There are no more than three of us here. We are all liars. [*] There are no more than four of us here. Not all of us are liars. [*] There are five of us here. At least three of us are liars. [/list] How many people are in the room and how many of them are liars?

Tick, Trick and Track have 20, 23 and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Also, before a ride, if they want, they can redistribute tickets among themselves as many times as they want according to the following rule: If one has an even number of tickets, he can give half of his tickets to any of the other two. Can it happen that after any trip: (a) exactly one has no ticket left, (b) exactly two have no ticket left, (c) all tickets are given away?

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

A student on vacation for $d$ days observed that $(1)$ it rained $7$ times, morning or afternoon $(2)$ when it rained in the afternoon, it was clear in the morning $(3)$ there were five clear afternoons $(4)$ there were six clear mornings. Then $d$ equals: $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

[b][u]BdMO National 2015 Secondary Problem 1.[/u][/b] A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement: Witness [b]One[/b] said exactly one of the four witnesses is a liar. Witness [b]Two[/b] said exactly two of the four witnesses is a liar. Witness [b]Three[/b] said exactly three of the four witnesses is a liar. Witness [b]Four[/b] said exactly four of the four witnesses is a liar. Assume that each of the statements is either true or false.How many of the winesses are liars?

Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?

In the land of knights (who always tell the truth) and liars (who always lie), 10 people sit at a round table, each at a vertex of an inscribed regular 10-gon, at least one of them is a liar. A traveler can stand at any point outside the table and ask the people: ”What is the distance from me to the nearest liar at the table?” After that each person at the table gives him an answer. What is the minimal number of questions the traveler has to ask to determine which people at the table are liars? (Both the people at the table and the traveler should be considered as points, and everyone can compute the distance between any two points) (10 points) Maxim Didin

Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

If I have 100 cards with all the numbers 1 through 100 on them, how should I put them in order to create the largest possible number?

On Halloween 31 children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order. "Are you a truth-teller?" The principal gave a piece of candy to each of the 22 children who answered yes. "Are you an alternater?" The principal gave a piece of candy to each of the 15 children who answered yes. "Are you a liar?" The principal gave a piece of candy to each of the 9 children who answered yes. How many pieces of candy in all did the principal give to the children who always tell the truth? $\textbf{(A) }7\qquad\textbf{(B) }12\qquad\textbf{(C) }21\qquad\textbf{(D) }27\qquad\textbf{(E) }31$

Prove that $\forall n\in\mathbb{Z}^+_0:(\exists b\in\mathbb{Z}^+_0:(\forall m\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(x+m = b))\lor(\exists s\in\mathbb{Z}^+_0:(\exists p\in\mathbb{Z}^+_0:((\neg(\exists x\in\mathbb{Z}^+_0:(p+x = 1)))\land(\neg(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:(p = (x+2) \cdot (y+2)))))\land(\exists x\in\mathbb{Z}^+_0:(p = m+x+1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + m) + y))))))\land(\forall u\in\mathbb{Z}^+_0:((\exists x\in\mathbb{Z}^+_0:(u = p+x))\lor(u = 0)\lor(u = n+1)\lor(\neg(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + u) + y)))))))\lor(\exists v\in\mathbb{Z}^+_0:(\exists k\in\mathbb{Z}^+_0:((\neg(v = 0))\land((u = v \cdot (k+2))\lor(u = v \cdot (k+2) + 1))\land(\exists r\in\mathbb{Z}^+_0:((\forall x\in\mathbb{Z}^+_0:(\forall y\in\mathbb{Z}^+_0:((\neg(x \cdot y = r))\lor(x = 1)\lor(\exists z\in\mathbb{Z}^+_0:(x = z \cdot p)))))\land(\exists x\in\mathbb{Z}^+_0:(\exists y\in\mathbb{Z}^+_0:((\exists z\in\mathbb{Z}^+_0:(r = y+z+1))\land(s = r \cdot (p \cdot x + v) + y)))))))))))))))))$. Proposed by [i]Warren Bei[/i]