Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$. Determine $f(2014)$.

Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$ $0.a_1a_2...$ rational or irrational?

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2)+12$ is a prime number that properly divides the positive number $a^2+b^2+c^2+abc-2017$?

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? $ \textbf{(A)}\ \frac{3\sqrt2}{\pi} \qquad \textbf{(B)}\ \frac{3\sqrt3}{\pi} \qquad \textbf{(C)}\ \sqrt3 \qquad \textbf{(D)}\ \frac{6}{\pi} \qquad \textbf{(E)}\ \sqrt3\pi$

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$. Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that: $$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2) $$

The quadrangles $AMBE, AHBT, BKXM$, and $CKXP$ are parallelograms. Prove that the quadrangle $ABTE$ is also parallelogram. (the vertices are mentioned counterclockwise)

Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$

Find all integer solutions to the equation \[ x^2 + y^2 + z^2 = 2xyz \]

Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

Let $f(z) = a_m z^n + a_{n-1} z^{n-1} + \cdots + a_1 z + a_0$ be a polynomial of degree $n \geq 3$ with real coefficients.Suppose all roots of $f(z) =0$ lie in the half plane ${\ z \in \mathbb{C} : Re(z) < 0 \}}$. Prove that $a_k a_{k+3} < a_{k+1}a_{k+2}$ for $k = 0,1,2,3,.... n-3$

In the $x$-$y$ plane, for $t>0$, denote by $S(t)$ the area of the part enclosed by the curve $y=e^{t^2x}$, the $x$-axis, $y$-axis and the line $x=\frac{1}{t}.$ Show that $S(t)>\frac 43.$ If necessary, you may use $e^3>20.$

Let's consider all possible natural seven-digit numbers, in the decimal notation of which the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$ are used once each. Let's number these numbers in ascending order. What number will be the $1995th$ ?

Let $ABCD$ be a convex quadrilateral. Two squares are constructed such that $AB{}$ and $CD{}$ are their diagonals. Show that if these squares have a common vertex inside $ABCD$, then the squares that have $BC{}$ and $AD{}$ as diagonals also have a common vertex inside $ABCD$.

$a,b,c$ are non-negative integers. Solve: $a!+5^b=7^c$ [i]Proposed by Serbia[/i]

Let $a, b$ be two reals such that $a+b<2ab$. Prove that $a+b>2$

Let \[P(m)=\frac{m}{2} + \frac{m^2}{4}+ \frac{m^4}{8} + \frac{m^8}{8}.\] How many of the values of $P(2022)$, $P(2023)$, $P(2024)$, and $P(2025)$ are integers? $ \textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad $

Let $\gamma,\Gamma$ be two concentric circles with radii $r,R$ with $r<R$. Let $ABCD$ be a cyclic quadrilateral inscribed in $\gamma$. If $\overrightarrow{AB}$ denotes the Ray starting from $A$ and extending indefinitely in $B's$ direction then Let $\overrightarrow{AB}, \overrightarrow{BC}, \overrightarrow{CD} , \overrightarrow{DA}$ meet $\Gamma$ at the points $C_1,D_1,A_1,B_1$ respectively. Prove that \[\frac{[A_1B_1C_1D_1]}{[ABCD]} \ge \frac{R^2}{r^2}\] where $[.]$ denotes area.

Let $AD$ and $BE$ be angle bisectors in triangle $ABC$. Let $x$, $y$ and $z$ be distances from point $M$, which lies on segment $DE$, from sides $BC$, $CA$ and $AB$, respectively. Prove that $z=x+y$

[list=a] [*]Determine the positive integer $x$ for which $\dfrac14-\dfrac{1}{x}=\dfrac16.$ [*]Determine all pairs of positive integers $(a,b)$ for which $ab-b+a-1=4.$ [*]Determine the number of pairs of positive integers $(y,z)$ for which $\dfrac{1}{y}-\dfrac{1}{z}=\dfrac{1}{12}.$ [*]Prove that, for every prime number $p$, there are at least two pairs $(r,s)$ of positive integers for which $\dfrac{1}{r}-\dfrac{1}{s}=\dfrac{1}{p^2}.$[/list]

[b]6.[/b] Prove or disprove the following two propositions: [b](i)[/b] If $a$ and $b$ are positive integers such that $a<b$, then in any set of $b$ consecutive integers there are two whose product is divisible by $ab$ [b](ii)[/b] If $a,b$ and $c$ are positive integers such that $a<b<c$, then in any set of $c$ consecutive integers there are three whose product is divisible by $abc$. [b](N.8)[/b]

In convex hexagon $ ABCDEF$ all sides have equal length and $ \angle A\plus{}\angle C\plus{}\angle E\equal{}\angle B\plus{}\angle D\plus{}\angle F$. Prove that the diagonals $ AD,BE,CF$ are concurrent.

We are given a cube with edges of length $n$ cm. At our disposal is a long piece of insulating tape of width $1$ cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that $n$ is an integer.) (A Spivak)

Let $a, b, c$ be reals and consider three lines $y=ax+b, y=bx+c, y=cx+a$. Two of these lines meet at a point with $x$-coordinate $1$. Show that the third one passes through a point with two integer coordinates.

A set of $n$ points in Euclidean 3-dimensional space, no four of which are coplanar, is partitioned into two subsets $\mathcal{A}$ and $\mathcal{B}$. An $\mathcal{AB}$-tree is a configuration of $n-1$ segments, each of which has an endpoint in $\mathcal{A}$ and an endpoint in $\mathcal{B}$, and such that no segments form a closed polyline. An $\mathcal{AB}$-tree is transformed into another as follows: choose three distinct segments $A_1B_1$, $B_1A_2$, and $A_2B_2$ in the $\mathcal{AB}$-tree such that $A_1$ is in $\mathcal{A}$ and $|A_1B_1|+|A_2B_2|>|A_1B_2|+|A_2B_1|$, and remove the segment $A_1B_1$ to replace it by the segment $A_1B_2$. Given any $\mathcal{AB}$-tree, prove that every sequence of successive transformations comes to an end (no further transformation is possible) after finitely many steps.