A bookshelf contains $n$ volumes, labelled $1$ to $n$, in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label $k$, takes it out, and inserts it in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(x+f(y^2)) + f(xy) = f(x) + yf(x+y)$$ for all $x, y \in \mathbb{R}$. [i]Proposed by Md. Fuad Al Alam[/i]

A [i]composite[/i] (positive integer) is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

A sequence $ \left( x_n\right)_{n\ge 0} $ of real numbers satisfies $ x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) , $ for each $ n\ge 1. $ Prove that if $ \left( x_n\right)_{n\ge 0} $ is periodic, then $ x_0 $ is a root of a quadratic equation. Study the converse.

$D$ and $E$ are points on the sides $BC$ and $AC$ of a triangle $ABC$ such that $AD$ and $BE$ are angle bisectors of the triangle $ABC$. If $DE$ bisects $\angle ADC$, find $\angle A$. (SI Tokarev)

A triangle has side lengths of $7$, $8$, and $9$. Find the radius of the largest possible semicircle inscribed in the triangle.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

We call an integer $ n > 1$ [i]good[/i] if, for any natural numbers $ 1 \le b_1, b_2, \ldots , b_{n\minus{}1} \le n \minus{} 1$ and any $ i \in \{0, 1, \ldots , n \minus{} 1\}$, there is a subset $ I$ of $ \{1, \ldots , n \minus{} 1\}$ such that $ \sum_{k\in I} b_k \equiv i \pmod n$. (The sum over the empty set is zero.) Find all good numbers.

A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]

In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$

Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.

The triangle $ABC$ has a right angle at $A$. The centre of the circumcircle is called $O$, and the base point of the normal from $O$ to $AC$ is called $D$. The point $E$ lies on $AO$ with $AE = AD$. The angle bisector of $\angle CAO$ meets $CE$ in $Q$. The lines $BE$ and $OQ$ intersect in $F$. Show that the lines $CF$ and $OE$ are parallel.

Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$

For all positive integers $m$ and $n$ prove the inequality $$ |n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1. $$

$n$ points are marked on a plane. Each pair of these points is connected with a segment. Each segment is painted one of four different colors. Find the largest possible value of $n$ such that one can paint the segments so that for any four points there are four segments (connecting these four points) of four different colors. (E. Barabanov)

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.