Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which \[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\] where $ x_0, x_1, \ldots , x_n$ are real variables.

Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$. (Oleksii Masalitin)

For positive real numbers $a,b,c,d$, with $abcd = 1$, determine all values taken by the expression \[\frac {1+a+ab} {1+a+ab+abc} + \frac {1+b+bc} {1+b+bc+bcd} +\frac {1+c+cd} {1+c+cd+cda} +\frac {1+d+da} {1+d+da+dab}.\] (Dan Schwarz)

Let $ a $ be a non-negative real number distinct from $ 1. $ [b]a)[/b] For which positive values $ x $ the equation $$ \left\lfloor\log_a x\right\rfloor +\left\lfloor \frac{1}{3} +\log_a x\right\rfloor =\left\lfloor 2\cdot\log_a x\right\rfloor $$ is true? [b]b)[/b] Solve $ \left\lfloor\log_3 x\right\rfloor +\left\lfloor \frac{1}{3} +\log_3 x\right\rfloor =3. $

A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$, \[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\] Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$.

Which number is larger: $A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}$, or $B = \log_{2023} 91125$?

The harmonic table is a triangular array: $1$ $\frac 12 \qquad \frac 12$ $\frac 13 \qquad \frac 16 \qquad \frac 13$ $\frac 14 \qquad \frac 1{12} \qquad \frac 1{12} \qquad \frac 14$ Where $a_{n,1} = \frac 1n$ and $a_{n,k+1} = a_{n-1,k} - a_{n,k}$ for $1 \leq k \leq n-1.$ Find the harmonic mean of the $1985^{th}$ row.

Given the quadratic trinomial $ f (x) = x ^ 2 + ax + b $ with integer coefficients, satisfying the inequality $ f (x) \geq - {9 \over 10} $ for any $ x $. Prove that $ f (x) \geq - {1 \over 4} $ for any $ x $.

Polynomial $ P(t)$ is such that for all real $ x$, \[ P(\sin x) \plus{} P(\cos x) \equal{} 1. \] What can be the degree of this polynomial?

For what values of $ m$ does the equation $ x^2 \plus{} (2m \plus{} 6)x \plus{} 4m \plus{} 12 \equal{} 0$ has two real roots, both of them greater than $ \minus{}1$.

Determine the value of the expression $x^2 + y^2 + z^2$, if $x + y + z = 13$ , $xyz= 72$ and $\frac1x + \frac1y + \frac1z = \frac34$.

Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$: $xP(x-c) = (x - 2014)P(x)$

We define a sequence ${a_n}$: $$a_1=1,a_{n+1}=\sqrt{a_n+n^2},n=1,2,...$$ (1)Find $\lfloor a_{2019}\rfloor$ (2)Find $\lfloor a_{1}^2\rfloor+\lfloor a_{2}^2\rfloor+...+\lfloor a_{20}^2\rfloor$

Given are the real numbers $x_1,x_2,...,x_n$ and $t_1,t_2,...,t_n$ for which holds: $\sum_{i=1}^n x_i = 0$. Prove that $$\sum_{i=1}^n \left( \sum_{j=1}^n (t_i-t_j)^2x_ix_j \right)\le 0.$$

What is the sum of the reciprocals of the roots of the equation \[ \frac {2003}{2004}x \plus{} 1 \plus{} \frac {1}{x} \equal{} 0? \] $ \textbf{(A)}\ \minus{}\! \frac {2004}{2003} \qquad \textbf{(B)}\ \minus{} \!1 \qquad \textbf{(C)}\ \frac {2003}{2004} \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac {2004}{2003}$

Determine all triples $(x, y, z)$ of non-zero numbers such that \[ xy(x + y) = yz(y + z) = zx(z + x). \]

Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.

Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$. [i]Proposed by Bojan Bašić, Serbia[/i]

Let $ T$ denote the set of all ordered triples $ (p,q,r)$ of nonnegative integers. Find all functions $ f: T \rightarrow \mathbb{R}$ satisfying \[ f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \frac{1}{6}(f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)) & \text{otherwise} \end{cases} \] for all nonnegative integers $ p$, $ q$, $ r$.

Given is a graph $G$ of $n+1$ vertices, which is constructed as follows: initially there is only one vertex $v$, and one a move we can add a vertex and connect it to exactly one among the previous vertices. The vertices have non-negative real weights such that $v$ has weight $0$ and each other vertex has a weight not exceeding the avarage weight of its neighbors, increased by $1$. Prove that no weight can exceed $n^2$.

In a function $f (x) = (x^2 + ax + b )/ (x^2 + cx + d)$ , the quadratics $x^2 + ax + b$ and $x^2 + cx + d$ have no common roots. Prove that the next two statements are equivalent: (i) there is a numerical interval without any values of $f(x)$ , (ii) $f(x)$ can be represented in the form $f (x) = f_1 (f_2( ... f_{n-1} (f_n (x))... ))$ where each of the functions $f_j$ is o f one of the three forms $k_j x + b_j, 1/x, x^2$ . (A Kanel)

Real numbers $x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}$ differ from $1$ and are such that $x_ky_k=1$ for every $k=1,2,...,2004$. Calculate the sum $$S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}$$

Consider a collection of stones whose total weight is $65$ pounds and each of whose stones is at most $w$ pounds. Find the largest number $w$ for which any such collection of stones can be divided into two groups whose total weights differ by at most one pound. Note: The weights of the stones are not necessarily integers.

Let $x_1,x_2,x_3,x_4,x_5$ ve non-negative real numbers, so that $x_1\le4$ and $x_1+x_2\le13$ and $x_1+x_2+x_3\le29$ and $x_1+x_2+x_3+x_4\le54$ and $x_1+x_2+x_3+x_4+x_5\le90$. Prove that $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}+\sqrt{x_5}\le20$.

One day, Ivan was imprisoned by an evil king. The evil king said : "If you can correctly determine the polynomial that I'm thinking of, you'll be free. If after $2014$ tries, you can't guess it, you'll be executed." Ivan answered : "Are there any clues?" The evil king replied : "I can tell you that the polynomial has real coefficients and is monic. Furthermore, all roots are positive real numbers." That night, a kind wizard, told him the polynomial. The conversation was heard by the king who was visiting Ivan. He killed the wizard. The next day, Ivan forgot the polynomial, except that the coefficients of $x^{2013}$ is $2014$, and that the constant term is $1$. Can Ivan guarantee freedom? And if so, in how many tries? (Assume that Ivan is very unlucky so any random guess fails.)