Found problems: 85335
For some fixed positive integer $n>2$, suppose $x_1$, $x_2$, $x_3$, $\ldots$ is a nonconstant sequence of real numbers such that $x_i=x_j$ if $i \equiv j \pmod{n}$. Let $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$. Given that $$f(1)=f(2)=f(3)=\cdots$$ find all possible values of the product $x_1 x_2 \ldots x_n$.
Prove that any permutation group of an order equal to a power of $ 2 $ contains a commutative subgroup whose order is the square of the exponent of the order of the group.
Determine all positive integers which cannot be represented as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a,b$ being positive integers.
Chip and Dale play the following game. Chip starts by splitting $1001$ nuts between three piles, so Dale can see it. In response, Dale chooses some number $N$ from $1$ to $1001$. Then Chip moves nuts from the piles he prepared to a new (fourth) pile until there will be exactly $N$ nuts in any one or more piles. When Chip accomplishes his task, Dale gets an exact amount of nuts that Chip moved. What is the maximal number of nuts that Dale can get for sure, no matter how Chip acts? (Naturally, Dale wants to get as many nuts as possible, while Chip wants to lose as little as possible).
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$.
[i](Proposed by Oleksii Masalitin)[/i]
Define $ x \heartsuit y$ to be $ |x\minus{}y|$ for all real numbers $ x$ and $ y$. Which of the following statements is [b]not[/b] true?
$\textbf{(A)}\ x \heartsuit y \equal{} y \heartsuit x \text{ for all } x \text{ and } y$
$\textbf{(B)}\ 2(x \heartsuit y) \equal{} (2x) \heartsuit (2y) \text{ for all } x \text{ and } y$
$\textbf{(C)}\ x \heartsuit 0 \equal{} x \text{ for all } x$
$\textbf{(D)}\ x \heartsuit x \equal{} 0 \text{ for all } x$
$\textbf{(E)}\ x \heartsuit y > 0 \text{ if } x \ne y$
An acute triangle is a triangle that has all angles less than $90^{\circ}$ ($90^{\circ}$ is a Right Angle). Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$ meeting at $H$. The circle passing through points $D$, $E$, and $F$ meets $AD$, $BE$, and $CF$ again at $X$, $Y$, and $Z$ respectively. Prove the following inequality: $$\frac{AH}{DX}+\frac{BH}{EY}+\frac{CH}{FZ} \geq 3.$$
$p(x) $ is a real polynomial of degree $3$.
Find necessary and sufficient conditions on its coefficients in order that $p(n)$ is integral for every integer $n$.
Let $ \alpha,\beta,\gamma$ be the angles of a triangle. Prove that
$csc^2\frac{\alpha}{2}+csc^2\frac{\beta}{2}+csc^2\frac{\gamma}{2} \ge 12$
and find the conditions for equality.
Of the students attending a school party, $ 60\%$ of the students are girls, and $ 40\%$ of the students like to dance. After these students are joined by $ 20$ more boy students, all of whom like to dance, the party is now $ 58\%$ girls. How many students now at the party like to dance?
$5.$Let x be a real number with $0<x<1$ and let $0.c_1c_2c_3...$ be the decimal expansion of x.Denote by $B(x)$ the set of all subsequences of $c_1c_2c_3$ that consist of 6 consecutive digits.
For instance , $B(\frac{1}{22})={045454,454545,545454}$
Find the minimum number of elements of $B(x)$ as $x$ varies among all irrational numbers with $0<x<1$
How many zeros are there at the end of $ \binom{200}{124} $?
There are $n \geq 1$ notebooks, numbered from $1$ to $n$, stacked in a pile. Zahar repeats the following operation: he randomly chooses a notebook whose number $k$ does not correspond to its location in this stack, counting from top to bottom, and returns it to the $k$th position, counting from the top, without changing the location of the other notebooks. If there is no such notebook, he stops.
Is it guaranteed that Zahar will arrange all the notebooks in ascending order of numbers in a finite number of operations?
[i]Proposed by Zahar Naumets[/i]
In a square $10^{2019} \times 10^{2019}, 10^{4038}$ points are marked. Prove that there is such a rectangle with sides parallel to the sides of a square whose area differs from the number of points located in it by at least $6$.
In a game of Among Us, there are $10$ players and $12$ colors. Each player has a "default" color that they will automatically get if nobody else has that color. Otherwise, they get a random color that is not selected. If $10$ random players with random default colors join a game one by one, the expected number of players to get their default color can be expressed as $\frac{m}{n}$. Compute $m+n$. Note that the default colors are not necessarily distinct.
[i]Proposed by Jeff Lin[/i]
Show that for all $x > 1$ there is a triangle with sides, $x^4 + x^3 + 2x^2 + x + 1, 2x^3 + x^2 + 2x + 1, x^4 - 1.$
In the figure, the area of square WXYZ is $25 \text{cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\Delta ABC$, $AB = AC$, and when $\Delta ABC$ is folded over side BC, point A coincides with O, the center of square WXYZ. What is the area of $\Delta ABC$, in square centimeters?
[asy]
defaultpen(fontsize(8));
size(225);
pair Z=origin, W=(0,10), X=(10,10), Y=(10,0), O=(5,5), B=(-4,8), C=(-4,2), A=(-13,5);
draw((-4,0)--Y--X--(-4,10)--cycle);
draw((0,-2)--(0,12)--(-2,12)--(-2,8)--B--A--C--(-2,2)--(-2,-2)--cycle);
dot(O);
label("$A$", A, NW);
label("$O$", O, NE);
label("$B$", B, SW);
label("$C$", C, NW);
label("$W$",W , NE);
label("$X$", X, N);
label("$Y$", Y, N);
label("$Z$", Z, SE);
[/asy]
$ \textbf{(A)}\ \frac{15}4\qquad\textbf{(B)}\ \frac{21}4\qquad\textbf{(C)}\ \frac{27}4\qquad\textbf{(D)}\ \frac{21}2\qquad\textbf{(E)}\ \frac{27}2$
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$
Let $n$ a positive integer, $n > 1$. The number $n$ is wonderful if the number is divisible by sum of the your prime factors.
For example; $90$ is wondeful, because $90 = 2 \times 3^2\times 5$ and $2 + 3 + 5 = 10, 10$ divides $90$.
Show that, exist a number "wonderful" with at least $10^{2002}$ distinct prime numbers.
Prove that there is a unique $1000$-digit number $N$ in base $2022$ with the following properties:
[list=1]
[*] All of the digits of $N$ (in base $2022$) are $1$’s or $2$’s, and [/*]
[*] $N$ is a multiple of the base-$10$ number $2^{1000}$. [/*]
[/list]
(Note that you must prove both that such a number exists and that there is not more than one such number. You do not have to write down the number! In fact, please don’t!)
If $p$ is a prime number greater than $2$ and $a, b, c$ integers not divisible by $p$, prove that the equation
\[ax^2 + by^2 = pz + c\]
has an integer solution.
Given an odd number $n >1$, let
\begin{align*} S =\{ k \mid 1 \le k < n , \gcd(k,n) =1 \} \end{align*}
and let \begin{align*} T = \{ k \mid k \in S , \gcd(k+1,n) =1 \} \end{align*}
For each $k \in S$, let $r_k$ be the remainder left by $\frac{k^{|S|}-1}{n}$ upon division by $n$. Prove
\begin{align*} \prod _{k \in T} \left( r_k - r_{n-k} \right) \equiv |S| ^{|T|} \pmod{n} \end{align*}
Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?