In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so. Determine the largest possible value of $m$.

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

Let $ABC$ be an acute triangle with $| AB | > | AC |$. Let $D$ be the foot of the altitude from $A$ to $BC$, let $K$ be the intersection of $AD$ with the internal bisector of angle $B$, Let $M$ be the foot of the perpendicular from $B$ to $CK$ (it could be in the extension of segment $CK$) and$ N$ the intersection of $BM$ and $AK$ (it could be in the extension of the segments). Let $T$ be the intersection of$ AC$ with the line that passes through $N$ and parallel to $DM$. Prove that $BM$ is the internal bisector of the angle $\angle TBC$

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that: $$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$

Let $a_1, a_2, a_3, \ldots$ a sequence of positive integers that satisfy the following conditions: $$a_1=1, a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor}, \forall n\ge 1$$Prove that for every positive integer $k$ there exists a term $a_i$ that is divisible by $k$

Let $n \ge 2$. Ana and Beto play the following game: Ana chooses $2n$ non-negative real numbers $x_1, x_2,\ldots , x_{2n}$ (not necessarily different) whose total sum is $1$, and shows them to Beto. Then Beto arranges these numbers in a circle in the way she sees fit, calculates the product of each pair of adjacent numbers, and writes the maximum value of these products. Ana wants to maximize the number written by Beto, while Beto wants to minimize it. What number will be written if both play optimally?