Found problems: 85335
Define the operator " $ * $ " on $ \mathbb{R} $ as $ x*y=x+y+xy. $
[b]a)[/b] Show that $ \mathbb{R}\setminus\{ -1\} , $ along with the operator above, is isomorphic with $ \mathbb{R}\setminus\{ 0\} , $ with the usual multiplication.
[b]b)[/b] Determine all finite semigroups of $ \mathbb{R} $ under " $ *. $ " Which of them are groups?
[b]c)[/b] Prove that if $ H\subset_{*}\mathbb{R} $ is a bounded semigroup, then $ H\subset [-2, 0]. $
Let \(x\) and \(y\) be positive real numbers satisfying the following system of equations:
\[
\begin{cases}
\sqrt{x}\left(2 + \dfrac{5}{x+y}\right) = 3 \\\\
\sqrt{y}\left(2 - \dfrac{5}{x+y}\right) = 2
\end{cases}
\]
Find the maximum value of \(x + y\).
Pieck the Frog hops on Pascal's Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Steven Yu[/i]
Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.
Let $CD$ be an altitude of right-angled triangle $ABC$ with $\angle C = 90$. Regular triangles$ AED$ and $CFD$ are such that $E$ lies on the same side from $AB$ as $C$, and $F$ lies on the same side from $CD$ as $B$. The line $EF$ meets $AC$ at $L$. Prove that $FL = CL + LD$
The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?
Prove that a binomial coefficient $\binom nk$ is odd if and only if all digits $1$ of $k$, when $k$ is written in binary, are on the same positions when $n$ is written in binary.
[i]I. Dimovski[/i]
Evaluate the following definite integral.
\[\int_{e^{2}}^{e^{3}}\frac{\ln x\cdot \ln (x\ln x)\cdot \ln \{x\ln (x\ln x)\}+\ln x+1}{\ln x\cdot \ln (x\ln x)}\ dx\]
The average of the numbers $ 1,2,3,...,98,99$, and $ x$ is $ 100x$. What is $ x$?
$ \textbf{(A)}\ \frac{49}{101} \qquad\textbf{(B)}\ \frac{50}{101} \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ \frac{51}{101} \qquad\textbf{(E)}\ \frac{50}{99}$
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13\qquad\textbf{(D)}\ 14\qquad\textbf{(E)}\ 15 $
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults?
$ \text{(A)}\ 26\qquad\text{(B)}\ 27\qquad\text{(C)}\ 28\qquad\text{(D)}\ 29\qquad\text{(E)}\ 30 $
The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$).
$M$ is infinite set of natural numbers. If $a,b, a\neq b$ are in $M$, then $a^b+2$ or $a^b-2$ ( or both) are in $M$. Prove that there is composite number in $M$
Compute
\[\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.\]
For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.
A cube of side length $n$ is divided into unit cubes by [i]partitions[/i] (each [i]partition[/i] separates a pair of adjacent unit cubes). What is the smallest number of [i]partitions[/i] that can be removed so that from each cube, one can reach the surface of the cube without passing through a partition ?
In the figure below ,$4$ of the $6$ disks are to be colored black and $2$ are to be colored white. Two colorings that can be obtained from one another by rotation or reflection of the entire figure are considered the same. [img]https://i.imgur.com/57nQwBI.jpg[/img]
There are only four such colorings for the given two colors, as shown.
In how many ways can we color the $6$ disks such that $2$ are colored black , $2$ are colored white, $2$ are colored blue with the given identification condition?
Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.
Find all injective functions $f: \mathbb R \rightarrow \mathbb R$ such that for every real number $x$ and every positive integer $n$,$$ \left|\sum_{i=1}^n i\left(f(x+i+1)-f(f(x+i))\right)\right|<2016$$
[i](Macedonia)[/i]
Let $m$ be an integer where $|m|\ge 2$. Let $a_1,a_2,\cdots$ be a sequence of integers such that $a_1,a_2$ are not both zero, and for any positive integer $n$, $a_{n+2}=a_{n+1}-ma_n$.
Prove that if positive integers $r>s\ge 2$ satisfy $a_r=a_s=a_1$, then $r-s\ge |m|$.
A necklace contains 2024 pearls, each one of them having one of the following colours: black, green and yellow. Each moment, we will switch each one of all pearls simultaneously to a new one following the following rules:
i) If its two neighbours are of the same colour, then it'll be switched to that same colour.
ii) If its two neighbours are of different colours, then it'll be switched to the third colour.
a) Does there exist any necklace that can be transformed into a necklace that consists of only yellow pearls if initially half of the pearls are black and the other half is green?
b) Does there exist a necklace that can be transformed into a necklace that consists of only yellow pearls if initially 998 pearls are black and the rest 1026 pearls are green?
Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)
For real numbers $a_1,a_2,\cdots,a_{100}$, $a_1=a_2=1,a_3=2$. For any positive integer $n$, $a_na_{n+1}a_{n+2}\neq1,a_na_{n+1}a_{n+2}a_{n+3}=a_n+a_{n+1}+a_{n+2}+a_{n+3}$, then $a_1+a_2+\cdots+a_{100}=$________.