For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions.

On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each row. This process continues through June 12 with a different number of students per row each day. However, on June 13, they cannot find a new way of organizing the students. What is the smallest possible number of students in the group? $ \textbf{(A) } 21 \qquad \textbf{(B) } 30 \qquad \textbf{(C) } 60 \qquad \textbf{(D) } 90 \qquad \textbf{(E) } 1080 $

Nine people sit in three rows of three chairs each. The probability that two of them, Celery and Drum, sit next to each other in the same row is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Tang[/i]

Let $ABCD$ be a convex quadrilateral. Lines $AB$ and $CD$ meet at $P$, and lines $AD$ and $BC$ meet at $Q$. Let $O$ be a point in the interior of $ABCD$ such that $\angle BOP = \angle DOQ$. Prove that $\angle AOB +\angle COD = 180$.

Let $p$ be an odd prime congruent to 2 modulo 3. Prove that at most $p-1$ members of the set $\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}$ are divisible by $p$.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $x_1$, $x_2$, $\cdots$, $x_n$ be positive real numbers, and let \[ S = x_1 + x_2 + \cdots + x_n. \] Prove that \[ (1 + x_1)(1 + x_2) \cdots (1 + x_n) \leq 1 + S + \frac{S^2}{2!} + \frac{S^3}{3!} + \cdots + \frac{S^n}{n!} \]

Given a sequence $\{ a_n\}_{n\in \mathbb{Z}^+}$ defined by $a_1=1$ and $a_{2k}=a_{2k-1}+a_k,a_{2k+1}=a_{2k}$ for all positive integer $k$. Prove that, for any positive integer $n$, $a_{2^n}>2^{\frac{n^2}{4}}$.

Let $n\ge 2$ be a positive integer, and consider an $n\times n$ grid of $n^2$ equilateral triangles. Two triangles are adjacent if they share at least one vertex. Each triangle is colored red or blue, splitting the grid into regions. Find, with proof, the minimum number of triangles in the largest region. [i]Rohan Bodke[/i]

In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.

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 $C'K = A'K$.