Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

The number $d_1d_2\cdots d_9$ has nine (not necessarily distinct) decimal digits. The number $e_1e_2\cdots e_9$ is such that each of the nine $9$-digit numbers formed by replacing just one of the digits $d_i$ in $d_1d_2\cdots d_9$ by the corresponding digit $e_i \;\;(1 \le i \le 9)$ is divisible by $7$. The number $f_1f_2\cdots f_9$ is related to $e_1e_2\cdots e_9$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_i$ by the corresponding $f_i$ is divisible by $7$. Show that, for each $i$, $d_i-f_i$ is divisible by $7$. [For example, if $d_1d_2\cdots d_9 = 199501996$, then $e_6$ may be $2$ or $9$, since $199502996$ and $199509996$ are multiples of $7$.]

Let $p$ be an odd prime number. Find all natural numbers $k$ such that $$\sqrt{k^2 - pk}$$ is a positive integer.

$ABC$ is a triangle and $D$ is an internal point such that $\angle DAB=\angle DBC =\angle DCA$. $O_a$ is the circumcenter of $DBC$. $O_b$ is the circumcenter of $DAC$. $O_c$ is the circumcenter of $DAB$. Show that if the area of $ABC$ and $O_aO_bO_c$ are equal then $ABC$ is equilateral.

Find all real polynomials $f$ and $g$, such that: \[(x^2+x+1)\cdot f(x^2-x+1)=(x^2-x+1)\cdot g(x^2+x+1), \] for all $x\in\mathbb{R}$.

For all $n>1$. Find all polynomials with complex coefficient and degree more than one such that $(p(x)-x)^2$ divides $p^n(x)-x$. ($p^0(x)=x , p^i(x)=p(p^{i-1}(x))$) [i]Proposed by Navid Safaie[/i]

Prove that each of the numbers $1, 2, 3, ..., 2^n$ can be written in one of two colors (red and blue) such that no non-constant $2n$-term arithmetic sequence chosen from these numbers is monochromatic .

An integer consists of 7 different digits, and is a multiple of each of its digits. What digits are in this nubmer?

Let $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size $25$. For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$. Define $n=\textstyle{\sum_{S \in T}} P(S)$. If you divide $n$ by 17, what is the remainder?

Let \[ f(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=x_1x_2x_4+x_2x_3x_5+x_3x_4x_6+x_4x_5x_7+x_5x_6x_1+x_6x_7x_2+x_7x_1x_3 \] be defined for non-negative real numbers $x_1,x_2,\dots,x_7$ with sum $1$. Prove that $f(x_1,x_2,\dots,x_7)$ has a maximum value and find that value.

Find all pairs of integers $ (x,y)$ satisfying $ 1\plus{}1996x\plus{}1998y\equal{}xy.$

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

The circle inscribed in triangle $ABC$ touches $AC$ at point $F$. The perpendicular from point $F$ on $BC$ intersects the bisector of angle $C$ at point $N$. Prove that segment $FN$ is equal to the radius of the circle inscribed in triangle $ABC$. (Oleksii Karliuchenko)

Let $P$ be a polynomial with integer coefficients. We say $P$ is [i]good [/i] if there exist infinitely many prime numbers $q$ such that the set $$X=\left\{P(n) \mod q : \quad n\in \mathbb N\right\}$$ has at least $\frac{q+1}{2}$ members. Prove that the polynomial $x^3+x$ is good.