The natural number $a_n$ is obtained by writing together and ordered, in decimal notation , all natural numbers between $1$ and $n$. So we have for example that $a_1 = 1$,$a_2 = 12$, $a_3 = 123$, $. . .$ , $a_{11} = 1234567891011$, $...$ . Find all values of $n$ for which $a_n$ is not divisible by $3$.

Let $P (x) = x^{3}-3x+1.$ Find the polynomial $Q$ whose roots are the fifth powers of the roots of $P$.

Three positive integers $a,b,c$ with $a>c$ satisfy the following equations : $$ac + b+c = bc + a + 66, \; \; \; \; a+b+c=32$$ Find the value of $a$.

A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length $ L$ and width $ W$ to enclose the greatest area. Find $ L\plus{}W$.

$a$ and $b$ are positive integers. When written in binary, $a$ has $2004$ $1$'s, and $b$ has $2005$ $1$'s (not necessarily consecutive). What is the smallest number of $1$'s $a + b$ could possibly have?

Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.

Let $g:[2013,2014]\to\mathbb{R}$ a function that satisfy the following two conditions: i) $g(2013)=g(2014) = 0,$ ii) for any $a,b \in [2013,2014]$ it hold that $g\left(\frac{a+b}{2}\right) \leq g(a) + g(b).$ Prove that $g$ has zeros in any open subinterval $(c,d) \subset[2013,2014].$

The quadrangle $ ABCD $ contains two circles of radii $ R_1 $ and $ R_2 $ tangent externally. The first circle touches the sides of $ DA $,$ AB $ and $ BC $, moreover, the sides of $ AB $ at the point $ E $. The second circle touches sides $ BC $, $ CD $ and $ DA $, and sides $ CD $ at $ F $. Diagonals of the quadrangle intersect at $ O $. Prove that $ OE + OF \leq 2 (R_1 + R_2) $. (F. Bakharev, S. Berlov)

[color=darkred]Let $f,g\colon [0,1]\to [0,1]$ be two functions such that $g$ is monotonic, surjective and $|f(x)-f(y)|\le |g(x)-g(y)|$ , for any $x,y\in [0,1]$ . [list] [b]a)[/b] Prove that $f$ is continuous and that there exists some $x_0\in [0,1]$ with $f(x_0)=g(x_0)$ . [b]b)[/b] Prove that the set $\{x\in [0,1]\, |\, f(x)=g(x)\}$ is a closed interval. [/list][/color]

Geetha wants to cut a cube of size $4 \times 4\times 4$ into $64$ unit cubes (of size $1\times 1\times 1$). Every cut must be straight, and parallel to a face of the big cube. What is the minimum number of cuts that Geetha needs? Note: After every cut, she can rearrange the pieces before cutting again. At every cut, she can cut more than one pieces as long as the pieces are on a straight line.

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$ where both sums are taken over the positive divisors of $n$. [i] (Vlad Robu) [/i]

Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*} & f(g(h(x))) = h(g(f(x))) = g(x) \\ & g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\ & h(f(g(x))) = g(f(h(x))) = f(x), \\ \end{align*} for all $x\in \{1,2,3,4,5\}$.

Let $a, b$ and $c$ be the distinct solutions to the equation $x^3-2x^2+3x-4=0$. Find the value of $$\frac{1}{a(b^2+c^2-a^2)}+\frac{1}{b(c^2+a^2-b^2)}+\frac{1}{c(a^2+b^2-c^2)}.$$

A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$. [i]Proposed by Mahdi Etesamifard[/i]

A non-negative integer $n$ is called [I]redundant[/I] if the sum of all his proper divisors is bigger than $n$. Prove that for each non-negative integer $N$ there are $N$ consecutive redundant non-negative integers. [I]Proposed by V. Bragin[/I]

For which $n\ge 2$ is it possible to find $n$ pairwise non-similar triangles $A_1, A_2,\ldots , A_n$ such that each of them can be divided into $n$ pairwise non-similar triangles, each of them similar to one of $A_1,A_2 ,\ldots ,A_n$?

Let $ n > 1, n \in \mathbb{Z}$ and $ B \equal{}\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?

Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$ Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$

The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$ (i) $ F(4n) \equal{} F(2n) \plus{} F(n),$ (ii) $ F(4n \plus{} 2) \equal{} F(4n) \plus{} 1,$ (iii) $ F(2n \plus{} 1) \equal{} F(2n) \plus{} 1.$ Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) \equal{} F(3n)$ is $ F(2^{m \plus{} 1}).$

$ R$ varies directly as $ S$ and inverse as $ T$. When $ R \equal{} \frac43$ and $ T \equal{} \frac {9}{14}$, $ S \equal{} \frac37.$ Find $ S$ when $ R \equal{} \sqrt {48}$ and $ T \equal{} \sqrt {75}.$ $ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 42 \qquad \textbf{(E)}\ 60$

$a,b,c,d$ are positive integers, and $\log_{a}b=\frac{3}{2},\log_{c}d=\frac{5}{4}$. If $a-c=9$, then $b-d=$________.

We call a graph with n vertices $k-flowing-chromatic$ if: 1. we can place a chess on each vertex and any two neighboring (connected by an edge) chesses have different colors. 2. we can choose a hamilton cycle $v_1,v_2,\cdots , v_n$, and move the chess on $v_i$ to $v_{i+1}$ with $i=1,2,\cdots ,n$ and $v_{n+1}=v_1$, such that any two neighboring chess also have different colors. 3. after some action of step 2 we can make all the chess reach each of the n vertices. Let T(G) denote the least number k such that G is k-flowing-chromatic. If such k does not exist, denote T(G)=0. denote $\chi (G)$ the chromatic number of G. Find all the positive number m such that there is a graph G with $\chi (G)\le m$ and $T(G)\ge 2^m$ without a cycle of length small than 2017.

Let $S = \{1,..., 999\}$. Determine the smallest integer $m$. for which there exist $m$ two-sided cards $C_1$,..., $C_m$ with the following properties: $\bullet$ Every card $C_i$ has an integer from $S$ on one side and another integer from $S$ on the other side. $\bullet$ For all $x,y \in S$ with $x\ne y$, it is possible to select a card $C_i$ that shows $x$ on one of its sides and another card $C_j$ (with $i \ne j$) that shows $y$ on one of its sides.

Find all real polynomials $p(x) $ such that $p(x-1)p(x+1)= p(x^2-1)$.