Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $ [i]Ion Tecu[/i]

Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.

19) Each cell of a $2$ × $5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2$ × $2$ square is a single color. 20) Let $n$ be a three-digit integer with nonzero digits, not all of which are the same. Define $f(n)$ to be the greatest common divisor of the six integers formed by any permutation of $n$s digits. For example, $f(123) = 3$, because $gcd(123, 132, 213, 231, 312, 321) = 3$. Let the maximum possible value of $f(n)$ be $k$. Find the sum of all $n$ for which $f(n) = k$. 21) Consider a $2$ × $2$ grid of squares. Each of the squares will be colored with one of $10$ colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? 22) Find all the roots of the polynomial $x^5 - 5x^4 + 11x^3 -13x^2+9x-3$ 23) Compute the smallest positive integer $n$ for which $0 < \sqrt[4]{n} - \left \lfloor{\sqrt[4]{n}}\right \rfloor < \dfrac{1}{2015}$. 24) Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the same time. What is the probability that all three ants are at the same vertex when they stop? 25) Let $ABC$ be a triangle that satisfies $AB = 13$, $BC = 14$, $AC = 15$. Given a point $P$ in the plane, let $PA$, $PB$, $PC$ be the reflections of $A$, $B$, $C$ across $P$. Call $P$ [i]good[/i] if the circumcircle of $P_A P_B P_C$ intersects the circumcircle of $ABC$ at exactly 1 point. The locus of good points $P$ encloses a region $S$. Find the area of $S$. 26. Let $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ be a continuous function satisfying $f(xy) = f(x) + f(y) + 1$ for all positive reals ${x,y}$. If $f(2) = 0$, compute $f(2015)$. 27) Let $ABCD$ be a quadrilateral with $A = (3,4)$, $B=(9,-40)$, $C = (-5,-12)$, $D = (-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $\overline{AP} + \overline{BP} + \overline{CP} + \overline{DP}$.

For $n\geq 3$ , determine all real solutions of the system of n equations : $x_1+x_2+...+x_{n-1}=\frac{1}{x_n}$ ....................... $x_1+x_2+...+x_{i-1}+x_{i+1}+...+x_n=\frac{1}{x_i}$ ....................... $x_2+...+x_{n-1}+x_n=\frac{1}{x_1}$

If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Determine all geometric progressions such that the product of the three first terms is $64$ and the sum of them is $14$.

Each lattice point of the plane is labeled by a positive integer. Each of these numbers is the arithmetic mean of its four neighbors (above, below, left, right). Show that all the numbers are equal.

If $a,b,c$ are real numbers such that $a^2+2b=7$, $b^2+4c=-7$, and $c^2+6a=-14$, find $a^2+b^2+c^2$. $\textbf{(A) }14\qquad\textbf{(B) }21\qquad\textbf{(C) }28\qquad\textbf{(D) }35\qquad\textbf{(E) }49$

Suppose Bob begins walking at a constant speed from point $N$ to point $S$ along the path indicated by the following figure. [img]https://cdn.artofproblemsolving.com/attachments/6/2/f5819267020f2bd38e52c6e873a2cf91ce8c49.png[/img] After Bob has walked a distance of $x$, Alice begins walking at point $N$, heading towards point $S$ along the same path. Alice walks $1.28$ times as fast as Bob when they are on the same line segment and $1.06$ times as fast as Bob otherwise. For what value of $x$ do Alice and Bob meet at point $S$?

$x$ and $y$ are real numbers such that $6 -x$, $3 + y^2$, $11 + x$, $14 - y^2$ are greater than zero. Find the maximum of the function $$f(x,y) = \sqrt{(6 -x)(3 + y^2)} + \sqrt{(11 + x)(14 - y^2)}.$$

Suppose that $p>3$ is prime. Prove that the products of the primitive roots of $p$ between $1$ and $p-1$ is congruent to $1$ modulo $p$.

Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)

Let $(a_n)_{n=1}^{\infty}$ be a sequence with $a_n \in \{0,1\}$ for every $n$. Let $F:(-1,1) \to \mathbb{R}$ be defined by \[F(x)=\sum_{n=1}^{\infty} a_nx^n\] and assume that $F\left(\frac{1}{2}\right)$ is rational. Show that $F$ is the quotient of two polynomials with integer coefficients.

Prove that if $a, b, c$ are positive real numbers, then the least possible value of \[6a^3 + 9b^3 + 32c^3 + \frac{1}{4abc}\] is $6$. For which values of $a, b$ and $c$ is equality attained?

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

Show that if we take any six numbers from the following array, one from each row and column, then the product is always the same: 4 6 10 14 22 26 6 9 15 21 33 39 10 15 25 35 55 65 16 24 40 56 88 104 18 27 45 63 99 117 20 30 50 70 110 130

Let the sequence $\{a_n\}$ , $n \ge 0$ satisfy the recurrence relation $$a_{n + 2} =4a_{n + 1}-3a_n, \ \ (1) $$ Let us define the sequence $\{b_n\}$ , $n \ge 1$ by the relation $$b_n= \left[ \frac{a_{n+1}}{a_{n-1}} \right]$$ where we put $b_n =1$ for $a_{n-1}=0$. Prove that, starting from a certain term, the sequence also satisfies the recurrence relation (1). Note: $[x]$ indicates the whole part of the number $x$.

The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

Prove that $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x,y$ with $x\le 1980$. [i] Variant: [/i] Prove the equation $4x^3-3x+1=2y^2$ has infinitely many solutions in positive integers x,y.

For which positive integers $n$, one can find real numbers $x_1,x_2,\cdots ,x_n$ such that $$\dfrac{x_1^2+x_2^2+\cdots+x_n^2}{\left(x_1+2x_2+\cdots+nx_n\right)^2}=\dfrac{27}{4n(n+1)(2n+1)}$$ and $i\leq x_i\leq 2i$ for all $i=1,2,\cdots ,n$ ?

Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.

Given a positive integer $n$, determine all functions $f$ from the first $n$ positive integers to the positive integers, satisfying the following two conditions: [b](1)[/b] $\sum_{k=1}^{n}{f(k)}=2n$; and [b](2)[/b] $\sum_{k\in K}{f(k)}=n$ for no subset $K$ of the first $n$ positive integers.

Let $a_0,a_1,\dots,a_n \in \mathbb N$. Prove that there exist positive integers $b_0,b_1,\dots,b_n$ such that for $0 \leq i \leq n : a_i \leq b_i \leq 2a_i$ and polynomial \[P(x) = b_0 + b_1 x + \dots + b_n x^n\] is irreducible over $\mathbb Q[x]$. (10 points)

Over all real numbers $x$, let $k$ be the minimum possible value of the expression $$\sqrt{x^2 + 9} +\sqrt{x^2 - 6x + 45}.$$ Determine $k^2$.