a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$. b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations \begin{align*} abcd &= 2007,\\ a &= \sqrt{55 + \sqrt{k+a}},\\ b &= \sqrt{55 - \sqrt{k+b}},\\ c &= \sqrt{55 + \sqrt{k-c}},\\ d &= \sqrt{55 - \sqrt{k-d}}. \end{align*}

Let $\left(a_k\right)_{k=1}^\infty$ be a sequence of real numbers such that \[a_{k-1}+a_{k+1}\ge2a_k\] for all $k>1.$ For $n\ge1$ denote \[A_n=\frac1n\left(a_1+\cdots+a_n\right).\] Show that also the inequality \[A_{n-1}+A_{n+1}\ge2A_n\] holds for every $n>1.$

For real numbers $x$ and $y$ we define $M(x, y)$ to be the maximum of the three numbers $xy$, $(x- 1)(y - 1)$, and $x + y - 2xy$. Determine the smallest possible value of $M(x, y)$ where $x$ and $y$ range over all real numbers satisfying $0 \le x, y \le 1$.

Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as [i]nice triple.[/i] Find all nice triples.

Let $f$ be a polynomial function such that, for all real $x$, \[f(x^2 + 1) = x^4 + 5x^2 + 3.\] For all real $x$, $f(x^2-1)$ is $ \textbf{(A)}\ x^4+5x^2+1\qquad\textbf{(B)}\ x^4+x^2-3\qquad\textbf{(C)}\ x^4-5x^2+1\qquad\textbf{(D)}\ x^4+x^2+3\qquad\textbf{(E)}\ \text{None of these} $

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

Points $A, B$ and $P$ lie on the circumference of a circle $\Omega_1$ such that $\angle APB$ is an obtuse angle. Let $Q$ be the foot of the perpendicular from $P$ on $AB$. A second circle $\Omega_2$ is drawn with centre $P$ and radius $PQ$. The tangents from $A$ and $B$ to $\Omega_2$ intersect $\Omega_1$ at $F$ and $H$ respectively. Prove that $FH$ is tangent to $\Omega_2$.

Part of an "$n$-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ are congruent. If the acute angle at $A_{1}$ is $10^{\circ}$ less than the acute angle at $B_{1}$, then $n = $ [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(90-2*36), B=dir(90-36), C=dir(90), D=dir(90+36), E=dir(90+2*36); pair F=2*dir(90-1.5*36), G=2*dir(90-0.5*36), H=2*dir(90+0.5*36), I=2*dir(90+1.5*36); draw(A--F--B--G--C--H--D--I--E); label("$B_2$", B, -0.3*dir(B)); label("$B_1$", C, -0.3*dir(C)); label("$B_n$", D, -0.3*dir(D)); label("$A_3$", F, dir(F)); label("$A_2$", G, dir(G)); label("$A_1$", H, dir(H)); label("$A_n$", I, dir(I)); [/asy] $ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 60 $

For every positive integer $m$ prove the inquality $|\{\sqrt{m}\} - \frac{1}{2}| \geq \frac{1}{8(\sqrt m+1)} $ (The integer part $[x]$ of the number $x$ is the largest integer not exceeding $x$. The fractional part of the number $x$ is a number $\{x\}$ such that $[x]+\{x\}=x$.) A. Golovanov

Let $x,y,z$ be real numbers such that $(x-z)(y-z)=x+y+z-3.$ Prove that $x^2+y^2+z^2\ge3.$

Let there be $131$ given distinct natural numbers, each having prime divisors not exceeding $42$. Prove that one can choose four of them whose product is a perfect square.

For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have \[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\] [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 2)[/i]

A point $O$ lies inside $\triangle ABC$ so that $\angle BOC=90-\angle BAC$. Let $BO, CO$ meet $AC, AB$ at $K, L$. Points $K_1, L_1$ lie on the segments $CL, BK$ so that $K_1B=K_1K$ and $L_1C=L_1L$. If $M$ is the midpoint of $BC$, then prove that $\angle K_1ML_1=90^{o}$. [i]Proposed by Anton Trygub[/i]

Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.

Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.

A pen costs $\mathrm{Rs.}\, 13$ and a note book costs $\mathrm{Rs.}\, 17$. A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ note books such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $\mathrm{Rs.}\, 10000$ and buys $y$ pens and $x$ note books. How much [b]more[/b] did the school pay?

There are 300 children in a camp. Everyone has no more than $k-1$ friends. What is the smallest $k{}$ for which it might be impossible to create some new friendships so that everyone has exactly $k{}$ friends?

For any set $S$, let $P(S)$ be its power set, the set of all of its subsets. Over all sets $A$ of $2015$ arbitrary finite sets, let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A), T \in P(P(A))$, $S \in T$, and $S \subseteq T$. (Note that by convention, a set may never contain itself.) Find the remainder when $N$ is divided by $1000.$ [i] Proposed by Ashwin Sah [/i]

Let $O$ be the circumcenter of triangle $ABC$. Suppose the perpendicular bisectors of $\overline{OB}$ and $\overline{OC}$ intersect lines $AB$ and $AC$ at $D\neq A$ and $E\neq A$, respectively. Determine the maximum possible number of distinct intersection points between line $BC$ and the circumcircle of $\triangle ADE$. [i]Andrew Wen[/i]

Find all polygons $P$ that can be covered completely by three (possibly overlapping) smaller dilated versions of itself. [i]Proposed by Evan Chang (squareman), USA[/i]

Given a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let $A$ be the number of pairs of adjacent red vertices and $B$ be the number of pairs of adjacent blue vertices. a) Find all possible values of pair $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying $B=14.$ 2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

A convex quadrilateral $ABCD$ inscribed in a circle $k$ has the property that the rays $DA$ and $CB$ meet at a point $E$ for which $CD^2=AD\cdot
ED.$ The perpendicular to $ED$ at $A$ intersects $k$ again at point $F.$ Prove that the segments $AD$ and $CF$ are congruent if and only if the circumcenter of $\triangle ABE$ lies on $ED.$

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

Find all function $f:\mathbb R^+ \rightarrow \mathbb R^+$ such that: \[f\left(\frac{f(x)}{x}+y\right)=1+f(y), \quad \forall x,y \in \mathbb R^+.\]