Let $a,b,c$ be distinct complex numbers with $|a|=|b|=|c|=1.$ If $|a+b-c|^2+|b+c-a|^2+|c+a-b|^2=12,$ prove that the points of affixes $a,b,c$ are the vertices of an equilateral triangle.

How many solutions, depending on the value of the parameter $a$, has the equation $$\sqrt{x^2-4}+\sqrt{2x^2-7x+5}=a ?$$

For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?

Consider the sequence $(a_n)_{n\ge 1}$ such that $a_1=1$ and $a_{n+1}=\sqrt{a_n+n^2}$, $\forall n\ge 1$. $\textbf{(a)}$ Prove that there is exactly one rational number among the numbers $a_1,a_2,a_3,\dots$. $\textbf{(b)}$ Consider the sequence $(S_n)_{n\ge 1}$ such that $$S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}.$$ Prove that there exists an integer $N$ such that $S_n>0.9$, $\forall n>N$. [i] (Stefan Obadă)[/i]

The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.

Let $n$ be a positive integer. Prove that the inequality \[n \sum_{i=1}^n \sum_{j = 1}^n \sum_{k=1}^n \frac{3}{a_ja_k + a_ka_i + a_i a_j} \ge \left(\sum_{j=1}^n \sum_{k=1}^n \frac{2}{a_j + a_k}\right)^2 \] holds for any positive real numbers $a_1$, $a_2$, $\dots$, $a_n$. [i]Proposed by Li4 and Ming Hsiao.[/i]

p1. Among the numbers $\frac15$ and $\frac14$ there are infinitely many fractional numbers. Find $999$ decimal numbers between $\frac15$ and $\frac14$ so that the difference between the next fractional number with the previous fraction constant. (i.e. If $x_1, x_2, x_3, x_4,..., x_{999}$ is a fraction that meant, then $x_2 - x_1= x_3 - x_3= ...= x_n - x_{n-1}=...=x_{999}-x_{998}$) p2. The pattern in the image below is: "Next image obtained by adding an image of a black equilateral triangle connecting midpoints of the sides of each white triangle that is left in the previous image." The pattern is continuous to infinity. [img]https://cdn.artofproblemsolving.com/attachments/e/f/81a6b4d20607c7508169c00391541248b8f31e.png[/img] It is known that the area of ​​the triangle in Figure $ 1$ is $ 1$ unit area. Find the total area of ​​the area formed by the black triangles in figure $5$. Also find the total area of the area formed by the black triangles in the $20$th figure. p3. For each pair of natural numbers $a$ and $b$, we define $a*b = ab + a - b$. The natural number $x$ is said to be the [i]constituent [/i] of the natural number $n$ if there is a natural number $y$ that satisfies $x*y = n$. For example, $2$ is a constituent of $6$ because there is a natural number 4 so that $2*4 = 2\cdot 4 + 2 - 4 = 8 + 2 - 4 = 6$. Find all constituent of $2005$. p4. Three people want to eat at a restaurant. To find who pays them to make a game. Each tossing one coin at a time. If the result is all heads or all tails, then they toss again. If not, then "odd person" (i.e. the person whose coin appears different from the two other's coins) who pay. Determine the number of all possible outcomes, if the game ends in tossing: a. First. b. Second. c. Third. d. Tenth. p5. Given the equation $x^2 + 3y^2 = n$, where $x$ and $y$ are integers. If $n < 20$ what number is $n$, and which is the respective pair $(x,y)$ ? Show that it is impossible to solve $x^2 + 3y^2 = 8$ in integers.

An eighth degree polynomial funtion $ y \equal{} ax^8 \plus{} bx^7 \plus{} cx^6 \plus{} dx^5 \plus{} ex^4 \plus{} fx^3 \plus{} gx^2\plus{}hx\plus{}i\ (a\neq 0)$ touches the line $ y \equal{} px \plus{} q$ at $ x \equal{} \alpha ,\ \beta ,\ \gamma ,\ \delta \ (\alpha < \beta < \gamma <\delta).$ Find the area of the region bounded by these graphs in terms of $ a,\ \alpha ,\ \beta ,\gamma ,\ \delta .$

(a)Given any natural number N, prove that there exists a strictly increasing sequence of N positive integers in harmonic progression. (b)Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]

Determine all real numbers $a, b,c$ such that the polynomial $f(x) = ax^2 + bx + c$ satisfies simultaneously the folloving conditions $\begin{cases} |f(x)| \le 1 \text{ for } |x | \le 1 \\ f(x) \ge 7 \text{ for } x \ge 2 \end{cases} $

$a > 0$ and $b$, $c$ are integers such that $ac$ – $b^2$ is a square-free positive integer P. [hide="For example"] P could be $3*5$, but not $3^2*5$.[/hide] Let $f(n)$ be the number of pairs of integers $d, e$ such that $ad^2 + 2bde + ce^2= n$. Show that$f(n)$ is finite and that $f(n) = f(P^{k}n)$ for every positive integer $k$. [b]Original Statement:[/b] Let $a,b,c$ be given integers $a > 0,$ $ac-b^2 = P = P_1 \cdots P_n$ where $P_1 \cdots P_n$ are (distinct) prime numbers. Let $M(n)$ denote the number of pairs of integers $(x,y)$ for which \[ ax^2 + 2bxy + cy^2 = n. \] Prove that $M(n)$ is finite and $M(n) = M(P_k \cdot n)$ for every integer $k \geq 0.$ Note that the "$n$" in $P_N$ and the "$n$" in $M(n)$ do not have to be the same.

$(HUN 5)$ Let $r$ and $m (r \le m)$ be natural numbers and $Ak =\frac{2k-1}{2m}\pi$. Evaluate $\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$

The teacher wrote $5$ distinct real numbers on the board. After this, Petryk calculated the sums of each pair of these numbers and wrote them on the left part of the board, and Vasyl calculated the sums of each triple of these numbers and wrote them on the left part of the board (each of them wrote $10$ numbers). Could the multisets of numbers written by Petryk and Vasyl be identical?

Prove that with $ n\ge 1 $ distinct numbers we can form an arithmetic progression if and only if there are exactly $ n-1 $ distinct elements in the set of positive differences between any two of these numbers.

Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$, \[f(a+b)=f(a)g(b)+g(a)f(b)\\ g(a+b)=g(a)g(b)-f(a)f(b).\]

For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $?

Two different positive numbers $ a$ and $ b$ each differ from their reciprocals by 1. What is $ a \plus{} b$? \[ \textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } \sqrt {5} \qquad \textbf{(D) } \sqrt {6} \qquad \textbf{(E) } 3 \]

On the surface of the planet lives one inhabitant, that can move with the speed not greater than $u$. A spaceship approaches to the planet with its speed $v$. Prove that if $v/u > 10$ , the spaceship can find the inhabitant, even it is trying to hide.

Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number \[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\] is a perfect square.

Through the point $F(0,\frac{1}{4})$ of the coordinate plane two perpendicular lines pass, that intersect parabola $y=x^2$ at points $A,B,C,D$ ($A_x<B_x<C_x<D_x$) The difference of projections of segments $AD$ and $BC$ onto the $Ox$ line is $m$ Find the area of $ABCD$

Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds : $$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$ $$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$ $P$ is square-free. Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it. [i]Proposed by Sina Azizedin[/i]

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

The numbers $a, b, c, d$ and $e$ satisfy $$a + b < c + d < e + a < b + c < d + e .$$ Which of the numbers is the smallest, and which is the largest?