Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with. [i]Proposed by Ray Li[/i]

Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

Calculate $ \lim_{n\to\infty } \frac{f(1)+(f(2))^2+\cdots +(f(n))^n}{(f(n))^n} , $ where $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ is an unbounded and nondecreasing function. [i]Dan Popescu[/i]

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

Prove that the real and imaginary part of the number $ \prod_{j=1}^n (j^3+\sqrt{-1}) $ is positive, for any natural numbers $ n. $ [i]Nicolae Mușuroia[/i]

Let $ABC$ be a triangle with integer sides in which $AB<AC$. Let the tangent to the circumcircle of triangle $ABC$ at $A$ intersect the line $BC$ at $D$. Suppose $AD$ is also an integer. Prove that $\gcd(AB,AC)>1$.

We have connected four metal pieces to each other such that they have formed a tetragon in space and also the angle between two connected metal pieces can vary. In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane. [i]proposed by Erfan Salavati[/i]

Inside a convex $1000$-gon, $500$ points are selected so that no three of the $1500$ points — the ones selected and the vertices of the polygon — lie on the same straight line. This $1000$-gon is then divided into triangles so that all $1500$ points are vertices of the triangles, and so that these triangles have no other vertices. How many triangles will there be?

Yasha writes in the cells of the table $99 \times 99$ all positive integers from 1 to $99^2$ (each number once). Grisha looks at the table and selects several cells, among which there are no two cells sharing a common side, and then sums up the numbers in all selected cells. Find the largest sum Grisha can guarantee to achieve.

Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2\plus{}ab\plus{}b^2)|ab(a\plus{}b)$. Prove that: $ |a\minus{}b|>\sqrt [3] {ab}$.

Let $\alpha$ and $\beta$ be irrational numbers with the property that $$\frac{1}{\alpha} +\frac{1}{\beta}=1$$ Let$\{a_n\}$ and $\{b_n\}$ be the sequences given by $a_n= \lfloor n\alpha \rfloor$ and $b_n= \lfloor n\beta \rfloor$ respectively. Prove that the sequences $\{ a_n\}$ and $\{ b_n \} $ has no term in common and cover all the natural numbers. I know this theorem from long ago, but forgot the proof of it. Can anybody help me with this?

[b]p1.[/b] Find all solutions $a, b, c, d, e, f, g$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccccc} & & & a & b & c & d \\ x & & & & & a & b \\ \hline & & c & d & b & d & b \\ + & c & e & b & f & b & \\ \hline & c & g & a & e & g & b \\ \end{tabular}$ [b]p2.[/b] $5$ numbers are placed on the circle. It is known that the sum of any two neighboring numbers is not divisible by $3$ and the sum of any three consecutive numbers is not divisible by $3$. How many numbers on the circle are divisible by $3$? [b]p3.[/b] $n$ teams played in a volleyball tournament. Each team played precisely one game with all other teams. If $x_j$ is the number of victories and $y_j$ is the number of losses of the $j$th team, show that $$\sum^n_{j=1}x^2_j=\sum^n_{j=1} y^2_j $$ [b]p4.[/b] Three cars participated in the car race: a Ford $[F]$, a Toyota $[T]$, and a Honda $[H]$. They began the race with $F$ first, then $T$, and $H$ last. During the race, $F$ was passed a total of $3$ times, $T$ was passed $5$ times, and $H$ was passed $8$ times. In what order did the cars finish? [b]p5.[/b] The side of the square is $4$ cm. Find the sum of the areas of the six half-disks shown on the picture. [img]https://cdn.artofproblemsolving.com/attachments/c/b/73be41b9435973d1c53a20ad2eb436b1384d69.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For positive integers $m,n\geq 3$, let $h(m,n)$ be the maximum (finite) number of intersection points between a simple $m$-gon and a simple $n$-gon. (Note: a polygon is simple if it does not intersect itself.) Evaluate \[\sum_{m=3}^6\sum_{n=3}^{12}h(m,n).\]

Let $a_0 = 1, \ a_1 = 2,$ and $a_2 = 10,$ and define $a_{k+2} = a_{k+1}^3+a_k^2+a_{k-1}$ for all positive integers $k.$ Is it possible for some $a_x$ to be divisible by $2021^{2021}?$ [i]Flavian Georgescu[/i]

The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.

There are 2023 points on the plane. Prove that there exists a circle that contains 2000 points inside it and leaves the remaining 23 outside. For example, if we had 5 points on the plane, we could find a circle that contains 4 of them inside and leaves 1 outside. Similarly, for 10 points, there exists a circle that contains 7 inside and leaves 3 outside. This reasoning extends to 2023 points, ensuring that such a division is always possible.

A positive integer is [i]brazilian[/i] if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is [i]superbrazilian[/i] if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, but $561=484+77$ is not superbrazilian, because $561$ is not brazilian. How many $4$-digit numbers are superbrazilians?

Consider function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=\frac{4^x}{4^x+2},$ for any $x\in \mathbb{R}$ a) Prove that $f(x)+f(1-x)=1,$ b) Claculate the sum $$f\left(\frac{1}{1986} \right)+f\left(\frac{2}{1986} \right)+\cdots f\left(\frac{1986}{1986} \right).$$

Let $a,b,$ and $c$ be pairwise distinct positive integers such that $\tfrac{1}{a}, \tfrac{1}{b}, \tfrac{1}{c}$ is an increasing arithmetic sequence in that order. Prove that $\gcd(a,b)>1.$

The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.) Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$

Four couples play chess together. For the match, they're paired as follows: ("man Clara" indicates the husband of Clara, etc.) \[Bea \Longleftrightarrow Eddy\] \[An \Longleftrightarrow man\ Clara\] \[Freddy \Longleftrightarrow woman\ Guy\] \[Debby \Longleftrightarrow man\ An\] \[Guy \Longleftrightarrow woman\ Eddy\] Who is $Hubert$ married to?

Rectangle $ABCD$ has sides $AB = 10$ and $AD = 7$. Point $G$ lies in the interior of $ABCD$ a distance $2$ from side $\overline{CD}$ and a distance $2$ from side $\overline{BC}$. Points $H, I, J$, and $K$ are located on sides $\overline{BC}, \overline{AB}, \overline{AD}$, and $\overline{CD}$, respectively, so that the path $GHIJKG$ is as short as possible. Then $AJ = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

Let $[n] = \{1, 2, 3, ... ,n\}$ and for any set $S$, let$ P(S)$ be the set of non-empty subsets of $S$. What is the last digit of $|P(P([2013]))|$?

$A$ writes, at his choice, $8$ ones and $8$ twos on a $4\times 4$ board. Then $B$ covers the board with $8$ dominoes and for each domino she finds the smaller of the two numbers that that domino covers. Finally, $A$ adds these $8$ numbers and the result is her score. What is the highest score $A$ can secure, no matter how $B$ plays? Clarification: A domino is a $1\times 2$ or $2\times 1$ rectangle that covers exactly two squares on the board.