Let $f(n)$ be the number of ordered pairs $(k, \ell)$ of positive integers such that $n = (2\ell-1)\cdot 2^k - k$, and let $g(n)$ be the number of ordered pairs $(k, \ell)$ of positive integers such that $n = \ell \cdot 2^{k+1}-k$. Compute $\sum_{i=1}^{\infty}\frac{f(i) - g(i)}{2^i}$. .

On the table, there're $1000$ cards arranged on a circle. On each card, a positive integer was written so that all $1000$ numbers are distinct. First, Vasya selects one of the card, remove it from the circle, and do the following operation: If on the last card taken out was written positive integer $k$, count the $k^{th}$ clockwise card not removed, from that position, then remove it and repeat the operation. This continues until only one card left on the table. Is it possible that, initially, there's a card $A$ such that, no matter what other card Vasya selects as first card, the one that left is always card $A$?

For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that: $y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ . Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$ (High School Affiliated to Nanjing Normal University )

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

If $ a > 1$, then the sum of the real solutions of \[\sqrt{a \minus{} \sqrt{a \plus{} x}} \equal{} x\] is equal to $ \textbf{(A)}\ \sqrt{a} \minus{} 1\qquad \textbf{(B)}\ \frac{\sqrt{a} \minus{} 1}{2}\qquad \textbf{(C)}\ \sqrt{a \minus{} 1}\qquad \textbf{(D)}\ \frac{\sqrt{a \minus{} 1}}{2}\qquad \textbf{(E)}\ \frac{\sqrt{4a \minus{} 3} \minus{} 1}{2}$

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

Suppose $P (x)$ is a polynomial with real coefficients such that $P (t) = P (1)t^2 + P (P (1))t + P (P (P (1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(P(1))))$.

Are there natural numbers $n$ and $N$ such that $n > 10^{10}$, $$n^n < 2^{2^{\frac{8N}{\omega (N)}}}$$ and $n$ is divisible by $p^{2022(v_p(N)-1)}(p-1)$ for every prime divisor $p$ of $N$? (For a natural number $N$, we denote by $\omega (N)$ the number of its different prime divisors and with $v_p(N)$ the power of the prime number $p$ in its canonical representation.)

You come across an ancient mathematical manuscript. It reads, "To find out whether a number is divisible by seventeen, take the number formed by the last two digits of the number, subtract the number formed by the third- and fourth-to-last digits of the number, add the number formed by the fifth- and sixth-to-last digits of the number and so on. The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen." What is the sum of the five smallest bases the ancient culture might have been using? (Note that "seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)

If $n$ is a positive integer which can be expressed in the form $n=a^{2}+b^{2}+c^{2}$, where $a, b, c$ are positive integers, prove that for each positive integer $k$, $n^{2k}$ can be expressed in the form $A^2 +B^2 +C^2$, where $A, B, C$ are positive integers.

Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

Let $f:\mathbb{R} \to \mathbb{R}$ a continuos function and $\alpha$ a real number such that $$\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.$$ Prove that for any $r > 0,$ there exists $x,y \in \mathbb{R}$ such that $y-x = r$ and $f(x) = f(y).$

Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.

Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

Find the number of permutations of the numbers $1,1,2,2,3,3,4,4$ such that no two consecutive numbers are equal. [i]Team #5[/i]

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.