Determine all triples $(x,y,z)$ of real numbers that satisfy all of the following three equations: $$\begin{cases} \lfloor x \rfloor + \{y\} =z \\ \lfloor y \rfloor + \{z\} =x \\ \lfloor z \rfloor + \{x\} =y \end{cases}$$

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? $\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$

1) Cells of $8 \times 8$ table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exists integer written in the same row or in the same column such that it is not relatively prime with $a$. Find maximum possible number of prime integers in the table. 2) Cells of $2n \times 2n$ table, $n \ge 2,$ contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer $a$ from the table there exist integers written in the same row and in the same column such that they are not relatively prime with $a$. Find maximum possible number of prime integers in the table.

Consider the sequence $(a_n)_{n\ge 1}$ defined by $a_n=n$ for $n\in \{1,2,3.4,5,6\}$, and for $n \ge 7$: $$a_n={\lfloor}\frac{a_1+a_2+...+a_{n-1}}{2}{\rfloor}$$ where ${\lfloor}x{\rfloor}$ is the greatest integer less than or equal to $x$. For example : ${\lfloor}2.4{\rfloor} = 2, {\lfloor}3{\rfloor} = 3$ and ${\lfloor}\pi {\rfloor}= 3$. For all integers $n \ge 2$, let $S_n = \{a_1,a_1,...,a_n\}- \{r_n\}$ where $r_n$ is the remainder when $a_1 + a_2 + ... + a_n$ is divided by $3$. The minus $-$ denotes the ''[i]remove it if it is there[/i]'' notation. For example : $S_4 = {2,3,4}$ because $r_4= 1$ so $1$ is removed from $\{1,2,3,4\}$. However $S_5= \{1,2,3,4,5\}$ betawe $r_5 = 0$ and $0$ is not in the set $\{1,2,3,4,5\}$. 1. Determine $S_7,S_8,S_9$ and $S_{10}$. 2. We say that a set $S_n$ for $n\ge 6$ is well-balanced if it can be partitioned into three pairwise disjoint subsets with equal sum. For example : $S_6 = \{1,2,3,4,5,6\} =\{1,6\}\cup \{2,5\}\cup \{3,4\}$ and $1 +6 = 2 + 5 = 3 + 4$. Prove that $S_7,S_8,S_9$ and $S_{10}$ are well-balanced . 3. Is the set $S_{2019}$ well-balanced? Justify your answer.

Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a fleet to which no new ship can be added.

Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. [i]Proposed by Kevin Sun[/i]

Prove that for any positive integer $n$, \[\left\lfloor \frac{n+1}{2}\right\rfloor+\left\lfloor \frac{n+2}{4}\right\rfloor+\left\lfloor \frac{n+4}{8}\right\rfloor+\left\lfloor \frac{n+8}{16}\right\rfloor+\cdots = n.\]

Let $S$ be a $k$-element set. [i](a)[/i] Find the number of mappings $f : S \to S$ such that \[\text{(i) } f(x) \neq x \text{ for } x \in S, \quad \text{(ii) } f(f(x)) = x \text{ for }x \in S.\] [i](b)[/i] The same with the condition $\text{(i)}$ left out.

The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which \[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\] is a rational number.

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

Let $x$ and $y$ be two real numbers. Prove that the equations \[\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor\] Holds if and only if at least one of $x$ or $y$ be integer.

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5$, $2 \leq a \leq 2005$, and $2 \leq b \leq 2005$.

Let $a\geq 2$ be a natural number. Prove that $\sum_{n=0}^\infty\frac1{a^{n^{2}}}$ is irrational.

What is the sum of the squares of the roots of the equation $x^2 -7 \lfloor x\rfloor +5=0$ ?

Consider a $2n \times 2n$ board. From the $i$th line we remove the central $2(i-1)$ unit squares. What is the maximal number of rectangles $2 \times 1$ and $1 \times 2$ that can be placed on the obtained figure without overlapping or getting outside the board?

Find \[\lim_{n\to\infty}\int _0^{2n} e^{-2x}\left|x-2\lfloor\frac{x+1}{2}\rfloor\right|\ dx.\] [i]1985 Tohoku University entrance exam/Mathematics, Physics, Chemistry, Biology[/i]

Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$.

Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which the equality holds.

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$ $0.a_1a_2...$ rational or irrational?

Let $x$ be a solution to the equation $\lfloor x \lfloor x + 2\rfloor + 2\rfloor = 10$. Compute the smallest $C$ such that for any solution $x$, $x < C$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor -4.25\rfloor = -5$.