Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

Find all injective $f:\mathbb{Z}\ge0 \to \mathbb{Z}\ge0 $ that for every natural number $n$ and real numbers $a_0,a_1,...,a_n$ (not everyone equal to $0$), polynomial $\sum_{i=0}^{n}{a_i x^i}$ have real root if and only if $\sum_{i=0}^{n}{a_i x^{f(i)}}$ have real root. [i]Proposed by Hesam Rajabzadeh [/i]

Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds. \[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]

The ordered pair of four-digit numbers $(2025, 3136)$ has the property that each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number. Find, with proof, all ordered pairs of five-digit numbers and ordered pairs of six-digit numbers with the same property: each number in the pair is a perfect square and each digit of the second number is $1$ more than the corresponding digit of the first number.

A sequence $a_0,a_1,a_2,\cdots$ satisfies the conditions $a_0 = 0$ , $a_{n-1}^2 - a_{n-1} = a_n^2 + a_n$ 1) determine the two possible values of $a_1$ . then determine all possible values of $a_2$ . 2)for each $n$, prove that $a_{n+1}=a_n+1$ or $a_{n+1} = -a_n$ 3)Describe the possible values of $a_{1435}$ 4)Prove that the values that you got in (3) are correct

A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.

An $n\times n$ square is divided into $n^2$ unit cells. Is it possible to cover this square with some layers of 4-cell figures of the following shape [img]https://cdn.artofproblemsolving.com/attachments/5/7/d42a8011ec4c5c91c337296d8033d412fade5c.png[/img](i.e. each cell of the square must be covered with the same number of these figures) if a) $n=6$? b) $n=7$? (The sides of each figure must coincide with the sides of the cells; the figures may be rotated and turned over, but none of them can go beyond the bounds of the square.)

On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?

Find the values of $a\in [0,\infty)$ for which there exist continuous functions $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(f(x))=(x-a)^2,\ (\forall)x\in \mathbb{R}$.

Assume that $(a_n)_{n \ge 1}$ is an increasing sequence of positive real numbers such that $\lim \tfrac{a_n}{n}=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\cdots,n-1$?

a) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)-1$ divisors among $a_i$s? b) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$, $a_m$ has exactly $d(m)+1$ divisors among $a_i$s?

Let $a, b$, and $c$ be positive integers such that $gcd(a, b) = 1$. Sequence $\{u_k\}$, is given such that $u_0 = 0$, $u_1 = 1$, and u$_{k+2} = au_{k+1} + bu_k$ for all $k \ge 0$. Let $m$ be the least positive integer such that $c | u_m$ and $n$ be an arbitrary positive integer such that $c | u_n$. Show that $m | n$. [hide=PS.] There was a typo in the last line, as it didn't define what n does. Wording comes from [b]tst-2011-1.pdf[/b] from [url=https://sites.google.com/site/imoidn/idntst/2011tst]here[/url]. Correction was made according to #2[/hide]

A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

A [i]bypass[/i] operation on a set $S$ is a mapping $B: S\times S \rightarrow S$ with the property $B(B(w, x), B(y,z)) = B(w,z)$ for all $w, x, y, z \in S$. (a) Prove that $B(a,b)=c$ implies $B(c,c)=c$ when $B$ is a bypass. (b) Prove that $B(a,b)=c$ implies $B(a,x)=B(c,x)$ for all $x\in S$ when $B$ is a bypass. (c) Construct a bypass operation $B$ on a finite set S with the following three properties [list=i] [*] $B(x,x)=x$ for all $x\in S$. [*] There exist $d$ and $e$ in $S$ with $B(d,e)=d \ne e.$ [*] There exist $f$ and $g$ in $S$ with $B(f,g)\ne f.$ [/list]

In each of the cells of a $13 \times 13$ board is written an integer such that the integers in adjacent cells differ by $1$. If there are two $2$s and two $24$s on this board, how many $13$s can there be?

Let $ g : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $ x+g(x)$ is strictly monotone (increasing or decreasing), and let $ u : [0,\infty) \rightarrow \mathbb{R}$ be a bounded and continuous function such that \[ u(t)+ \int_{t-1}^tg(u(s))ds\] is constant on $ [1,\infty)$. Prove that the limit $ \lim_{t\rightarrow \infty} u(t)$ exists. [i]T. Krisztin[/i]

While Theophrastus was talking to Aristotle about the classification of plants, had a dog tied to a perfectly smooth cylindrical column of radius $r$, with a very fine rope that wrapped around the column and with a loop. The dog had the extreme free from the rope around his neck. In trying to reach Theophrastus, he put the rope tight and it broke. Find out how far from the column the knot was in the time to break the rope. [hide=original wording]Mientras Teofrasto hablaba con Arist´oteles sobre la clasificaci´on de las plantas, ten´ıa un perro atado a una columna cil´ındrica perfectamente lisa de radio r, con una cuerda muy fina que envolv´ıa la columna y con un lazo. El perro ten´ıa el extremo libre de la cuerda cogido a su cuello. Al intentar alcanzar a Teofrasto, puso la cuerda tirante y ´esta se rompi´o. Averiguar a qu´e distancia de la columna estaba el nudo en el momento de romperse la cuerda.[/hide]

Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have? [i]Mircea Becheanu[/i]

Solve the system of equations $\begin{cases}6x-y+z^2=3\\ x^2-y^2-2z=-1\quad\quad (x,y,z\in\mathbb{R}.)\\ 6x^2-3y^2-y-2z^2=0\end{cases}$.

Consider the set of triangles with side lengths $1 \le x \le y \le z$ such that $x$, $y$, and $z$ are the solutions to the equation $t^3-at^2+bt = 12$ for some real numbers $a$ and $b$. Compute the smallest real number $N$ such that $N > ab$ for any choice of $x$, $y$, and $z$.

Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

Two thieves stole a container of $8$ liters of wine. How can they divide it into two parts of $4$ liters each if all they have is a $3 $ liter container and a $5$ liter container? Consider the general case of dividing $m+n$ liters into two equal amounts, given a container of $m$ liters and a container of $n$ liters (where $m$ and $n$ are positive integers). Show that it is possible iff $m+n$ is even and $(m+n)/2$ is divisible by $gcd(m,n)$.

From the first 64 positive integers are chosen two subsets with 16 numbers in each. The first subset contains only odd numbers while the second one contains only even numbers. Total sums of both subsets are the same. Prove that among all the chosen numbers there are two whose sum equals 65. [i](3 points)[/i]