Each detail of the “Young Solderer” instructor is a bracket in the shape of the letter $\Pi$, consisting of three single segments. Is it possible from the parts of this constructor are soldered together, a complete wire frame of the cube $2 \times 2 \times 2$, divided into $1 \times 1 \times 1$ cubes? (The frame consists of 27 points, connected by single segments; any two adjacent points must be connected by exactly one piece of wire.) [hide]=original wording]Каждая деталько нструктора ''Юный паяльщик'' — это скобка в виде буквы П, остоящая из трех единичных отрезков. Можно ли издеталей этого конструктора спаятьполный роволочный каркас куба 2 × × 2 × 2, разбитого на кубики 1 × 1 × 1? (Каркас состоит из 27 точек,соединенных единичными отрезками; любые две соседние точки должны бытьсоединены ровно одним проволочным отрезком.)[/hide]

In a soccer league with $2020$ teams every two team have played exactly once and no game have lead to a draw. The participating teams are ordered first by their points (3 points for a win, 1 point for a draw, 0 points for a loss) then by their goal difference (goals scored minus goals against) in a normal soccer table. Is it possible for the goal difference in such table to be strictly increasing from the top to the bottom? [i]Proposed by Abolfazl Asadi[/i]

In a certain language there are only two letters, $A$ and $B$. The words of this language obey the following rules: (i) The only word of length $1$ is $A$; (ii) A sequence of letters $X_1X_2...X_{n+1}$, where $X_i\in \{A,B\}$ for each $i$, forms a word of length $n+1$ if and only if it contains at least one letter $A$ and is not of the form $WA$ for a word $W$ of length $n$. Show that the number of words consisting of $1998 A$’s and $1998 B$’s and not beginning with $AA$ equals $\binom{3995}{1997}-1$

A very well known family of mathematicians has three children called [i]Antonia, Bernhard[/i] and [i]Christian[/i]. Each evening one of the children has to do the dishes. One day, their dad decided to construct of plan that says which child has to do the dishes at which day for the following $55$ days. Let $x$ be the number of possible such plans in which Antonia has to do the dishes on three consecutive days at least once. Furthermore, let $y$ be the number of such plans in which there are three consecutive days in which Antonia does the dishes on the first, Bernhard on the second and Christian on the third day. Determine, whether $x$ and $y$ are different and if so, then decide which of those is larger.

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$.

On a plane several straight lines are drawn in such a way that each of them intersects exactly $15$ other lines. How many lines are drawn on the plane? Find all possibilities and justify your answer. Poland

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

James has a red jar, a blue jar and a pile of $100$ pebbles. Initially both jars are empty. A move consists of moving a pebble from the pile into one of the jars or returning a pebble from one of the jars to the pile. The numbers of pebbles in the red and blue jars determine the state of the game. The followwing conditions must be satisfied: (a) The red jar may never contain fewer pebbles than the blue jar; (b) The game may never be returned to a previous state. What is the maximum number of moves that James can make?

Alice has 8 coins. She knows for sure only that 7 of these coins are genuine and weigh the same, while the remaining one is counterfeit and is either heavier or lighter than any of the other 7. Bob has a balance scale. The scale shows which plate is heavier but does not show by how much. For each measurement, Alice pays Bob beforehand a fee of one coin. If a genuine coin has been paid, Bob tells Alice the correct weighing outcome, but if a counterfeit coin has been paid, he gives a random answer. Alice wants to identify 5 genuine coins and not to give any of these coins to Bob. Can Alice achieve this result for sure?

Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.

Julia and James pick a random integer between $1$ and $10$, inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

We are given $n$ intervals $[l_1,r_1],[l_2,r_2],[l_3,r_3],\dots, [l_n,r_n]$ in the number line. We can divide the intervals into two sets such that no two intervals in the same set have overlaps. Prove that there are at most $n-1$ pairs of overlapping intervals.

In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.

A convex polygon $P$ can be partitioned into $27$ parallelograms. Prove that it can also be partitioned into $21$ parallelograms.

How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?

We are given $1998$ normal coins, $1$ heavy coin and $1$ light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance $4$ times or less.

$30$ segments of lengths$$1,\quad \sqrt{3},\quad \sqrt{5},\quad \sqrt{7},\quad \sqrt{9},\quad \ldots ,\quad \sqrt{59} $$ have been drawn on a blackboard. In each step, two of the segments are deleted and a new segment of length equal to the hypotenuse of the right triangle with legs equal to the two deleted segments is drawn. After $29$ steps only one segment remains. Find the possible values of its length.