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).$

Determine all functions $f: \mathbb {N} \to \mathbb {N}$ such that $n!\hspace{1mm} +\hspace{1mm} f(m)!\hspace{1mm} |\hspace{1mm} f(n)!\hspace{1mm} +\hspace{1mm} f(m!)$, for all $m$, $n$ $\in$ $\mathbb{N}$.

If $a, b,c,p,q, r $are real numbers such that, for every real number $x,$ $$ax^2 - 2bx + c \ge 0 \ \ and \ \ px^2 + 2qx + r \ge 0;$$ prove that then $$apx^2 + bqx + cr \ge 0$$ for all real $x$.

Given is a trihedral angle with edges $ SA $, $ SB $, $ SC $, all plane angles of which are acute, and the dihedral angle at edge $ SA $ is right. Prove that the section of this triangle with any plane perpendicular to any edge, at a point different from the vertex $ S $, is a right triangle.

Find the smallest real $\alpha$, such that for any convex polygon $P$ with area $1$, there exist a point $M$ in the plane, such that the area of convex hull of $P\cup Q$ is at most $\alpha$, where $Q$ denotes the image of $P$ under central symmetry with respect to $M$.

Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.

Simplify $\frac{1}{\sqrt[3]{81} + \sqrt[3]{72} + \sqrt[3]{64}}$

Find the largest constant $K\in \mathbb{R}$ with the following property: if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]

Let $ C$ denote the set of functions $ f(x)$, integrable (according to either Riemann or Lebesgue) on $ (a,b)$, with $ 0\le f(x)\le1$. An element $ \phi(x)\in C$ is said to be an "extreme point" of $ C$ if it can not be represented as the arithmetical mean of two different elements of $ C$. Find the extreme points of $ C$ and the functions $ f(x)\in C$ which can be obtained as "weak limits" of extreme points $ \phi_n(x)$ of $ C$. (The latter means that $ \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx$ holds for every integrable function $ h(x)$.)

Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$ [i]Proposed by A. Golovanov[/i]

In snack bar Pita Goras, the owner checks his accounts. He writes on every line either a positive amount in case of an income or a negative amount in case of an expense. He says to his accountant, “If I change the amounts of random $5$ adding consecutive lines, I always get a strictly positive result.” "Indeed," the accountant answers him, “but if you put the sums of $7$ consecutive lines add up, you always get a strictly negative result.” How many lines are there maximum on his sheet?

A triangle has vertices $ (0,0)$, $ (1,1)$, and $ (6m,0)$, and the line $ y \equal{} mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $ m$? $ \textbf{(A)}\minus{} \!\frac {1}{3} \qquad \textbf{(B)} \minus{} \!\frac {1}{6} \qquad \textbf{(C)}\ \frac {1}{6} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$

Determine all $n \geq 3$ for which there are $n$ positive integers $a_1, \cdots , a_n$ any two of which have a common divisor greater than $1$, but any three of which are coprime. Assuming that, moreover, the numbers $a_i$ are less than $5000$, find the greatest possible $n$.

Find the total area of the shaded area in the figure if all circles have an equal radius $R$ and the centers of the outer circles divide into six equal parts of the middle circle. [img]http://3.bp.blogspot.com/-Ax0QJ38poYU/XovXkdaM-3I/AAAAAAAALvM/DAZGVV7TQjEnSf2y1mbnse8lL6YIg-BQgCK4BGAYYCw/s400/estonia%2B2000%2Bo.j.1.5.png[/img]

Let $n$ cards are placed in a circle. Each card has a white side and a black side. On each move, you pick one card with black side up, flip it over, and also flip over the two neighboring cards. Suppose initially, there are only one black-side-up card. (a)If $n=2015$ , can you make all cards white-side-up through a finite number of moves? (b)If $n=2016$ , can you make all cards white-side-up through a finite number of moves?

Outside of the trapezoid $ABCD$ with the smaller base $AB$ are constructed the squares $ADEF$ and $BCGH$. Prove that the perpendicular bisector of $AB$ passes through the midpoint of $FH$.

$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$. $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$. Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$, and $r$ meets $A_{1}B_{1}$ at $C_{2}$. Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$, and that $X$ lies on the circumcircle of $ABC$.

A number $n$ is said to be [i]nice[/i] if it exists an integer $r>0$ such that the expression of $n$ in base $r$ has all its digits equal. For example, 62 and 15 are $\emph{nice}$ because 62 is 222 in base 5, and 15 is 33 in base 4. Show that 1993 is not [i]nice[/i], but 1994 is.

Each cell of a beehive is constructed from a right regular 6-angled prism, open at the bottom and closed on the top by a regular 3-sided pyramidical mantle. The edges of this pyramid are connected to three of the rising edges of the prism and its apex $T$ is on the perpendicular line through the center $O$ of the base of the prism (see figure). Let $s$ denote the side of the base, $h$ the height of the cell and $\theta$ the angle between the line $TO$ and $TV$. (a) Prove that the surface of the cell consists of 6 congruent trapezoids and 3 congruent rhombi. (b) the total surface area of the cell is given by the formula $6sh - \dfrac{9s^2}{2\tan\theta} + \dfrac{s^2 3\sqrt{3}}{2\sin\theta}$ [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=286[/img]

Let $\{a_i\}$ be a strictly increasing sequence of positive integers. Define the sequence $\{s_k\}$ as $$s_k = \sum_{i=1}^{k}\frac{1}{[a_i,a_{i+1}]},$$ where $[a_i,a_{i+1}]$ is the least commun multiple of $a_i$ and $a_{i+1}$. Show that the sequence $\{s_k\}$ is convergent.

Find all prime numbers $p$ and $q$ such that $$1 + \frac{p^q - q^p}{p + q}$$ is a prime number. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Let $\{ a_n\}$ be a series consisting of positive integers such that $n^2 \mid \sum_{i=1}^{n}{a_i}$ and $a_n\leq (n+2016)^2$ for all $n\geq 2016$. Define $b_n=a_{n+1}-a_n$. Prove that the series $\{ b_n\}$ is eventually constant.

Let $ABCD$ be a trapezoid such that $AB$ is parallel to $CD$, and let $E$ be the midpoint of its side $BC$. Suppose we can inscribe a circle into the quadrilateral $ABED$, and that we can inscribe a circle into the quadrilateral $AECD$. Denote $|AB|=a$, $|BC|=b$, $|CD|=c$, $|DA|=d$. Prove that \[a+c=\frac{b}{3}+d;\] \[\frac{1}{a}+\frac{1}{c}=\frac{3}{b}\]

We define the sequence $a_n=(2n)^2+1$ for each natural number $n$. We will call one number [i]bad[/i], if there don’t exist natural numbers $a>1$ and $b>1$ such that $a_n=a^2+b^2$. Prove that the natural number $n$ is [i]bad[/i], if and only if $a_n$ is prime.

How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side lengths and perimeter less than 20?