Compute the sum of all positive real numbers $x \le 5$ satisfying $$x =\frac{ \lceil x^2 \rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{ \lceil x\rceil + \lfloor x \rfloor}$$

Find all $a,b \in \mathbb{N}$ such that $$2049^ba^{2048}-2048^ab^{2049}=1.$$ [i]Proposed by fattypiggy123 and 61plus[/i]

Does there exist a polynomial \( P(x,y) \) in two variables with real coefficients, such that the following two conditions hold: 1) \( P(x,y) = P(x, x-y) = P(y-x, y) \) for any real numbers \( x \) and \( y \); 2) There does not exist a polynomial \( Q(z) \) in one variable with real coefficients such that \( P(x,y) = Q(x^2 - xy + y^2) \) for any real numbers \( x \) and \( y \)?

Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

Consider systems of three linear equations with unknowns $x,$ $y,$ and $z,$ \begin{align*} a_1 x + b_1 y + c_1 z = 0 \\ a_2 x + b_2 y + c_2 z = 0 \\ a_3 x + b_3 y + c_3 z = 0 \end{align*} where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x = y = z = 0.$ For example, one such system is $\{1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0\}$ with a nonzero solution of $\{x, y, z\} = \{1, -1, 1\}.$ How many such systems are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.) $\textbf{(A) } 302 \qquad \textbf{(B) } 338 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 343 \qquad \textbf{(E) } 344$

Let $x, y, z \in R$, find all triples $(x, y, z)$ that satisfy the following system of equations: $2x^2 - 3xy + 2y^2 = 1$ $y^2 - 3yz + 4z^2 = 2$ $z^2 + 3zx - x^2 = 3$

Let $\lambda$ the positive root of the equation $t^2-1998t-1=0$. It is defined the sequence $x_0,x_1,x_2,\ldots,x_n,\ldots$ by $x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots$ Find the remainder of the division of $x_{1998}$ by $1998$. Note: $\lfloor{x}\rfloor$ is the greatest integer less than or equal to $x$.

Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ [i]naughty[/i] if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is [i]not[/i] in $T$. Prove that 1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements; 2) If $S$ is a set of 2006 arbitrary positive integers, there exists a naughty subset of $S$ which has 669 elements.

One considers for $n > 2$ the polynomial: $$(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n$$ Show that the degree of this polynomial is $2n - 2$. The polynomial is written in the form $$a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}$$ Prove that $a_2+a_3+...+a_{2n-2}=0$

$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations $$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$ Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.

What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?

Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

Positive rational number $a$ and $b$ satisfy the equality \[a^3 + 4a^2b = 4a^2 + b^4.\] Prove that the number $\sqrt{a}-1$ is a square of a rational number.

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

A sequence of positive real numbers $a_1, a_2, \dots$ is called $\textit{phine}$ if it satisfies $$a_{n+2}=\frac{a_{n+1}+a_{n-1}}{a_n},$$ for all $n\geq2$. Is there a $\textit{phine}$ sequence such that, for every real number $r$, there is some $n$ for which $a_n>r$?

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

If $a, b, c$ are real numbers such that a$b + bc + ca = 0$, prove the inequality $$2(a^2 + b^2 + c^2)(a^2b^2 + b^2c^2 + c^2a^2) \ge 27a^2b^2c^2$$ When does the equality hold ? Leonard Giugiuc

Does there exist an infinite sequence of integers \(a_0\), \(a_1\), \(a_2\), \(\ldots\) such that \(a_0\ne0\) and, for any integer \(n\ge0\), the polynomial \[P_n(x)=\sum_{k=0}^na_kx^k\] has \(n\) distinct real roots? [i]Proposed by Amol Rama and Espen Slettnes[/i]

Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that: $$W(x,y,z) = U(x,y,z)+V(x,y,z),$$ $$U(x,y,z) = U(y,x,z),$$ $$V(x,y,z) = -V(x,z,y).$$

Let ${(a_n)}_{n=0}^{\infty}$ and ${(b_n)}_{n=0}^{\infty}$ be real squences such that $a_0=40$, $b_0=41$ and for all $n\geq 0$ the given equalities hold. $$a_{n+1}=a_n+\frac{1}{b_n} \hspace{0.5 cm} \text{and} \hspace{0.5 cm} b_{n+1}=b_n+\frac{1}{a_n}$$ Find the least possible positive integer value of $k$ such that the value of $a_k$ is strictly bigger than $80$.

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

Find the number of sequences of $2005$ terms with the following properties: (i) No three consecutive terms of the sequence are equal, (ii) Every term equals either $1$ or $-1$, (iii) The sum of all terms of the sequence is at least $666$.