[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other. [b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?

Prove that the entire space can be partitioned into “crosses” made of seven unit cubes as shown in the picture. [img]https://cdn.artofproblemsolving.com/attachments/2/b/77c4a4309170e8303af321daceccc4010da334.png[/img]

Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?

[b](i)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges). [b](ii)[/b] Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).

Each of the numbers $1, 2, 3,... 25$ is arranged in a $5$ by $5$ table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column. (Folklore)

The exact quantity of gas needed for a car to complete a single loop around a track is distributed among $n$ containers placed along the track. Prove that there exists a position starting at which the car, beginning with an empty tank of gas, can complete a loop around the track without running out of gas. The tank of gas is assumed to be large enough.

All sequences of $k$ elements $(a_1,a_2,...,a_k)$ are considered, where each $a_i$ belongs to the set $\{1,2,... ,2021\}$. What is the sum of the smallest elements of all these sequences?

Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$

Homer Grog has written the numbers $1, 3, 4, 5, 7, 9, 11, 13, 15,17$, one number on each note. He arranges the bills in a circle and tries to get the largest sum $S$ of the numbers of three consecutive bills to be the least possible. What is the smallest value $S$ can assume?

The interval $[0,1]$ is covered by a finite number of intervals. Show that one can choose a number of these intervals which are pairwise disjoint and have the total length at least $1/2$.

At a mathematical competition $n$ students work on $6$ problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is $0$ or $2$. Find the maximum possible value of $n$.

Let $n > 2$ be a natural number. A subset $A$ of $R$ is said $n$-[i]small [/i]if there exist $n$ real numbers $t_1 , t_2 , ..., t_n$ such that the sets $t_1 + A , t_2 + A ,... , t_n + A$ are different . Show that $R$ can not be represented as a union of $ n - 1$ $n$-[i]small [/i] sets . Notation : if $r \in R$ and $B \subset R$ , then $r + B = \{ r + b | b \in B\}$ .

In a qualification football round there are six teams and each two play one versus another exactly once. No two matches are played at the same time. At every moment the difference between the number of already played matches for any two teams is $0$ or $1$. A win is worth $3$ points, a draw is worth $1$ point and a loss is worth $0$ points. Determine the smallest positive integer $n$ for which it is possible that after the $n$-th match all teams have a different number of points and each team has a non-zero number of points.

Robert colors each square in an empty 3 by 3 grid either red or green. Find the number of colorings such that no row or column contains more than one green square.

Kamil wrote on a board an expression consisting of alternating addition and subtraction of natural numbers from $1$ to $100$: \[1-2+3-4+5-6+\ldots-98+99-100.\] Then, Kamil erased one of the plus or minus signs and replaced it with an equals sign, obtaining a true equality. Which number preceded the erased sign? Find all possibilities and justify your answer.

Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that $n\le(k-1)^2$. [i]Linus Tang[/i]

On a \( 10 \times 10 \) chessboard, there are 91 white pawns placed in different squares. Nico picks a white pawn, paints it black, and places it in an empty square, repeating the process until all pawns have been painted. Prove that at some point, there will be two pawns of different colors placed on squares that share a common edge.

[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Let $A$ be the set of all ordered sequences $(a_1,a_2,...,a_{11})$ of zeros and ones. The elements of $A$ are ordered as follows: The first element is $(0,0,...,0)$, and the $n + 1$−th is obtained from the $n$−th by changing the first component from the right such that the newly obtained sequence was not obtained before. Find the $1992$−th term of the ordered set $A$

Given a sequence of real numbers $a_1, a_2, ... , a_n$. Prove that it is possible to choose some of the numbers providing $3$ conditions: a) not a triple of successive members is chosen, b) at least one of every triple of successive members is chosen, c) the absolute value of chosen numbers sum is not less that one sixth part of the initial numbers' absolute values sum.

[u]Set 3[/u] [b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw? [b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standing at $(0, 0)$ in the coordinate plane. She takes $T$ steps, each of which moves her by $1$ unit in either the $+x$, $-x$, $+y$, or $-y$ direction (e.g. her first step takes her to $(1, 0)$, $(1, 0)$, $(0, 1)$ or $(0, -1)$). After the T steps, how many possibilities are there for Charlotte’s location? [b]3.3[/b] Let $T$ be the answer from the previous problem, and let $S$ be the sum of the digits of $T$. Francesca has an unfair coin with an unknown probability $p$ of landing heads on a given flip. If she flips the coin $S$ times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability $p$. PS. You should use hide for answers.

The set of quadruples $(a,b,c,d)$ where each of $a,b,c,d$ is either $0$ or $1$ is [i]called vertices of the four dimensional unit cube[/i] or [i]4-cube[/i] for short. Two vertices are called [i]adjacent[/i], if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from $(0,0,0,0)$ and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to $(0,0,0,0)$ after $4042$ turns? Note that it is [u]NOT[/u] forbidden for the robot to pass through $(0,0,0,0)$ before the $4042$-nd turn.

Let $n$ be a positive integer and let $a_1$, $a_2$,..., $a_n$ be positive integers from set $\{1, 2,..., n\}$ such that every number from this set occurs exactly once. Is it possible that numbers $a_1$, $a_1 + a_2 ,..., a_1 + a_2 + ... + a_n$ all have different remainders upon division by $n$, if: $a)$ $n=7$ $b)$ $n=8$

For a set $A$, we define $A + A = \{a + b: a, b \in A \}$. Determine whether there exists a set $A$ of positive integers such that $$\sum_{a \in A} \frac{1}{a} = +\infty \quad \text{and} \quad \lim_{n \rightarrow +\infty} \frac{|(A+A) \cap \{1,2,\cdots,n \}|}{n}=0.$$ [hide=Note]Google translated from [url=http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales]http://ciim.uan.edu.co/ciim-2020-pruebas-virtuales/pruebas-virtuales[/url][/hide]