Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.

[Help me] Determine the smallest value of the sum M =xy-yz-zx where x; y; z are real numbers satisfying the following condition $x^2+2y^2+5z^2 = 22$.

It is known that $ \angle BAC$ is the smallest angle in the triangle $ ABC$. The points $ B$ and $ C$ divide the circumcircle of the triangle into two arcs. Let $ U$ be an interior point of the arc between $ B$ and $ C$ which does not contain $ A$. The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AU$ at $ V$ and $ W$, respectively. The lines $ BV$ and $ CW$ meet at $ T$. Show that $ AU \equal{} TB \plus{} TC$. [i]Alternative formulation:[/i] Four different points $ A,B,C,D$ are chosen on a circle $ \Gamma$ such that the triangle $ BCD$ is not right-angled. Prove that: (a) The perpendicular bisectors of $ AB$ and $ AC$ meet the line $ AD$ at certain points $ W$ and $ V,$ respectively, and that the lines $ CV$ and $ BW$ meet at a certain point $ T.$ (b) The length of one of the line segments $ AD, BT,$ and $ CT$ is the sum of the lengths of the other two.

At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the board. Find the greatest persistent number.

The $1$st edition of OLCOMA was organized in $1989$, so in $2022$ the $34$th edition will be celebrated. Suppose that the Olympics will continue to be held annually without interruption. We say that a year $N$ is [i]good [/i] if the OLCOMA edition number of that year divides the product $N(N +1)$. For example, the year $2022$ is good because $34$ divides $2022 \cdot 2023$. Determine the last year $N$ in the $21$st century, $2000\le N \le 2099$, which is good.

For which positive integers $n$ does there exist a positive integer $m$ such that among the numbers $m + n, 2m + (n - 1), \dots, nm + 1$, there are no two that share a common factor greater than $1$?

Prove that there is a function $ F:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ (F\circ F) (n) =n^2, $ for all $ n\in\mathbb{N} . $

Find the smallest integer $n$ such that an $n\times n$ square can be partitioned into $40\times 40$ and $49\times 49$ squares, with both types of squares present in the partition, if a) $40|n$; b) $49|n$; c) $n\in \mathbb N$.

A collection of short stories written by Papadiamantis contains $70$ short stories, one of $1$ page, one of $2$ pages, ... one of $70$ pages . and not nessecarily in that order. Every short story starts on a new page and numbering of pages of the book starts from the first page . What is the maximum number of short stories that start on page with odd number?

[b]a)[/b] Calculate $ \int_{0}^1 x\sin\left( \pi x^2\right) dx. $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \frac{1}{n}\sum_{k=0}^{n-1} k\int_{\frac{k}{n}}^{\frac{k+1}{n}} \sin\left(\pi x^2\right) dx. $ [i]Florin Stănescu[/i]

Prove that : $\cos36-\sin18=\frac{1}{2}$

If $ \sqrt{x \minus{} 1} \minus{} \sqrt{x \plus{} 1} \plus{} 1 \equal{} 0$, then $ 4x$ equals: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4\sqrt{\minus{}1}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 1\frac{1}{4}\qquad \textbf{(E)}\ \text{no real value}$

Chords $ AB$ and $ CD$ of a circle intersect at a point $ E$ inside the circle. Let $ M$ be an interior point of the segment $ EB$. The tangent line at $ E$ to the circle through $ D$, $ E$, and $ M$ intersects the lines $ BC$ and $ AC$ at $ F$ and $ G$, respectively. If \[ \frac {AM}{AB} \equal{} t, \] find $\frac {EG}{EF}$ in terms of $ t$.

Let $A$ be the sequence of zeroes and ones (binary sequence). The sequence can be modified by the following operation: we may pick a block or a contiguous subsequence where there are an unequal number of zeroes and ones, and then flip their order within the block (so block $a_1, a_2, \ldots, a_r$ becomes $a_r, a_{r-1}, \ldots, a_1$). As an example, let $A$ be the sequence $1,1,0,0,1$. We can pick block $1,0,0$ and flip it, so the sequence $1,\boxed{1,0,0},1$ becomes $1,\boxed{0,0,1},1$. However, we cannot pick block $1,1,0,0$ and flip their order since they contain the same number of $1$s and $0$s. Two sequences $A$ and $B$ are called [i]related[/i] if $A$ can be transformed into $B$ using a finite number the operation mentioned above. Determine the largest natural number $n$ for which there exists $n$ different sequences $A_1, A_2, \ldots, A_n$ where each sequence consists of 2022 digits, and for every index $i \neq j$, the sequence $A_i$ is not related to $A_j$.

Let $p$ be a prime number. For integers $r, s$ such that $rs(r^2 - s^2)$ is not divisible by $p$, let $f(r, s)$ denote the number of integers $n \in \{1, 2, \ldots, p - 1\}$ such that $\{rn/p\}$ and $\{sn/p\}$ are either both less than $1/2$ or both greater than $1/2$. Prove that there exists $N > 0$ such that for $p \geq N$ and all $r, s$, \[ \left\lceil \frac{p-1}{3} \right\rceil \le f(r, s) \le \left\lfloor \frac{2(p-1)}{3} \right\rfloor. \]

Determine all integers $n \geq 3$ for which there exists a conguration of $n$ points in the plane, no three collinear, that can be labelled $1$ through $n$ in two different ways, so that the following condition be satisfied: For every triple $(i,j,k), 1 \leq i < j < k \leq n$, the triangle $ijk$ in one labelling has the same orientation as the triangle labelled $ijk$ in the other, except for $(i,j,k) = (1,2,3)$.

In the Cartesian plane, a line segment is called [i]tame[/i] if it lies parallel to one of the coordinate axes and its distance to this axis is an integer. Otherwise it is called [i]wild[/i]. Let $m$ and $n$ be odd positive integers. The rectangle with vertices $(0,0),(m,0),(m,n)$ and $(0,n)$ is partitioned into finitely many triangles. Let $M$ be the set of these triangles. Assume that (1) Each triangle from $M$ has at least one tame side. (2) For each tame side of a triangle from $M$, the corresponding altitude has length $1$. (3) Each wild side of a triangle from $M$ is a common side of exactly two triangles from $M$. Show that at least two triangles from $M$ have two tame sides each.

Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

Let the continuous functions $ f_n(x), \; n\equal{}1,2,3,...,$ be defined on the interval $ [a,b]$ such that every point of $ [a,b]$ is a root of $ f_n(x)\equal{}f_m(x)$ for some $ n \not\equal{} m$. Prove that there exists a subinterval of $ [a,b]$ on which two of the functions are equal.

Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?

Let $n$ be an even positive integer. On a board n real numbers are written. In a single move we can erase any two numbers from the board and replace each of them with their product. Prove that for every $n$ initial numbers one can in finite number of moves obtain $n$ equal numbers on the board.

Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. [asy] size(200); pen tpen = defaultpen + 1.337; draw((1,0)--(1,8)); draw((2,0)--(2,8)); draw((3,0)--(3,8)); draw((4,0)--(4,8)); draw((5,0)--(5,8)); draw((6,0)--(6,8)); draw((7,0)--(7,8)); draw((8,0)--(8,8)); draw((9,0)--(9,8)); draw((10,0)--(10,8)); draw((11,0)--(11,8)); draw((12,0)--(12,8)); draw((13,0)--(13,8)); draw((0,1)--(13.5,1)); draw((0,2)--(13.5,2)); draw((0,3)--(13.5,3)); draw((0,4)--(13.5,4)); draw((0,5)--(13.5,5)); draw((0,6)--(13.5,6)); draw((0,7)--(13.5,7)); draw((1,1)--(5,7), tpen); draw((1,1)--(13,1),tpen); draw((5,7)--(13,1),tpen); [/asy]

Let $n$ be a positive integer. If $x_1, x_2, \ldots, x_n \geq 0$, $x_1+x_2+\ldots+x_n=1$ and, assuming $x_{n+1}=x_1$, find the maximal value of $$\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.$$

On a certain math exam, $ 10 \%$ of the students got 70 points, $ 25 \%$ got 80 points, $ 20 \%$ got 85 points, $ 15 \%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 4\qquad \textbf{(E)}\ 5$

By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$). For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$). Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.