Peter has a deck of $1001$ cards, and with a blue pen he has written the numbers $1,2,\ldots,1001$ on the cards (one number on each card). He replaced cards in a circle so that blue numbers were on the bottom side of the card. Then, for each card $C$, he took $500$ consecutive cards following $C$ (clockwise order), and denoted by $f(C)$ the number of blue numbers written on those $500$ cards that are greater than the blue number written on $C$ itself. After all, he wrote this $f(C)$ number on the top side of the card $C$ with a red pen. Prove that Peter's friend Basil, who sees all the red numbers on these cards, can determine the blue number on each card.

We number a hundred blank cards on both sides with the numbers $1$ to $100$. The cards are then stacked in order, with the card with the number $1$ on top. The order of the cards is changed step by step as follows: at the $1$st step the top card is turned around, and is put back on top of the stack (nothing changes, of course), at the $2$nd step the topmost $2$ cards are turned around, and put back on top of the stack, up to the $100$th step, in which the entire stack of $100$ cards is turned around. At the $101$st step, again only the top card is turned around, at the $102$nd step, the top most $2$ cards are turned around, and so on. Show that after a finite number of steps, the cards return to their original positions.

Let $n$ be a fixed positive integer. Oriol has $n$ cards, each of them with a $0$ written on one side and $1$ on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of $n$ steps: 1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cards Let $s_0, s_1, s_2, \dots, s_n$ be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step, $\dots$ after the last step, respectively. a) Find the greatest integer $k$ such that, no matter the initial card configuration, there exists at least $k$ distinct numbers between $s_0, s_1, \dots, s_n$. b) Find all positive integers $m$ such that, for each initial card configuration, there exists an index $r$ such that $s_r = m$. [i]Proposed by Dorlir Ahmeti[/i]

Oriol has a finite collection of cards, each one with a positive integer written on it. We say the collection is $n$-[i]complete[/i] if for any integer $k$ from $1$ to $n$ (inclusive), he can choose some cards such that the sum of the numbers on them is exactly $k$. Suppose that Oriol's collection is $n$-complete, but it stops being $n$-complete if any card is removed from it. What is the maximum possible sum of the numbers on all the cards? [i]Proposed by Ariel García[/i]

Prove that there are exactly $2(2^{n-1}-1)$ ways of dealing $n$ cards to two persons. (The persons may receive unequal numbers of cards.)

There are $N$ red cards and $N$ blue cards. Each card has a positive integer between $1$ and $N$ (inclusive) written on it. Prove that we can choose a (non-empty) subset of the red cards and a (non-empty) subset of the blue cards, so that the sum of the numbers on the chosen red cards equals the sum of the numbers on the chosen blue cards.

We have $100$ cards with two sides, the [i]even[/i] and the [i]odd[/i]. In each side there are written two succesive integers, in the [i]odd[/i] side and odd integer and at the back in the [i]even[/i] side the even number that follows the odd number of the [i]odd[/i] side, such that all intgers from $1$ to $200$ are used. Student $A$ randomly choses $21$ cards and sums all the numbers of boths sides and announces as their sum the number $913$. Student $B$ randomly choses from the remaining cards $20$ cards and sums all the numbers of boths sides and announces as their sum the number $2400$. a) Explain why student $A$ has done an error in the addition. b) If the correct result for student $A$ is $903$, explain why also student $B$ has done an error in the addition.

Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, having got the pair of cards with $(a,b)$ and $(b,c)$ produces new card with $(a,c)$. All the automates return the initial cards also. Suppose there was $(5,19)$ card initially. Is it possible to obtain a) $(1,50)$? b) $(1,100)$? c) Suppose there was $(a,b)$ card initially $(a<b)$. We want to obtain $(1,n)$ card. For what $n$ is it possible?

The wizards $A, B, C, D$ know that the integers $1, 2, \ldots, 12$ are written on 12 cards, one integer on each card, and that each wizard will get three cards and will see only his own cards. Having received the cards, the wizards made several statements in the following order. [list=A] [*]“One of my cards contains the number 8”. [*]“All my numbers are prime”. [*]“All my numbers are composite and they all have a common prime divisor”. [*]“Now I know all the cards of each wizard”. [/list] What were the cards of $A{}$ if everyone was right? [i]Mikhail Evdokimov[/i]

Lamija and Faris are playing the following game. Cards, which are numerated from $1$ to $100$, are placed one next to other, starting from $1$ to $100$. Now Faris picks every $7$th card, and after that every card which contains number $7$. After that Lamija picks from remaining cards ones divisible with $5$, and after that cards which contain number $5$. Who will have more cards and how many ? How would game end, if Lamija started with "$5$ rule" and Faris continues with "$7$ rule"?

Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it away. This action repeats until when there is no remaining card for Ali. Amin wants to pick cards in a way that the sum of the number of his cards is maximized and Ali wants to choose cards in a way that the sum of the number of Amin's cards is minimized. Find the most value of $k$ such that Amin can play in a way that is sure the sum of the number of his cards will be at least equal to $k$.