Found problems: 85335
Show that for $a,b,c > 0$ the following inequality holds:
$\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.
For how many real values of $ x$ is $ \sqrt {120 \minus{} \sqrt {x}}$ an integer?
$ \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \textbf{(E) } 11$
Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression
\[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]
Let $(G,\cdot)$ be a group, and let $H_1,H_2$ be proper subgroups s.t. $H_1\cap H_2=\{e\}$, where $e$ is the identity element of $G$. They also have the following properties:
[b]i)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_1\setminus\{e\}\Rightarrow xy\in H_2$
[b]ii)[/b] $x\in G\setminus(H_1\cup H_2),y\in H_2\setminus\{e\}\Rightarrow xy\in H_1$
Prove that:
[b]a)[/b] $|H_1|=|H_2|$
[b]b)[/b] $|G|=|H_1|\cdot |H_2|$
Let $S$ be the set of fractions of the form $\frac{{\text {lcm}}(A, B)}{A+B}$, where $A$ and $B$ are positive integers and ${\text{lcm}}(A, B)$ is the least common multiple of $A$ and $B$. What is the smallest number exceeding 3 in $S$?
Evaluate $\dbinom{6}{0}+\dbinom{6}{1}+\dbinom{6}{4}+\dbinom{6}{3}+\dbinom{6}{4}+\dbinom{6}{5}+\dbinom{6}{6}$
[i]Proposed by Jonathan Liu[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{64}$
We have that $\dbinom{6}{4}=\dbinom{6}{2}$, so $\displaystyle\sum_{n=0}^{6} \dbinom{6}{n}=2^6=\boxed{64}.$
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Let $a,b,c$ be positive real numbers such that
\[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \]
Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?
How many positive integers of $n$ digits exist such that each digit is $1, 2$, or $3$? How many of these contain all three of the digits $1, 2$, and $3$ at least once?
Find the limit \[\lim_{n\to \infty}\frac{\sqrt[n+1]{(2n+3)(2n+4)\ldots (3n+3)}}{n+1}\]
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?
Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers
\[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\]
to be products of two primes (not necessarily distinct).
Let $N$ be the set of positive integers.
Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers
Let $f(x) = x^3 + ax^2 + bx + c$ have solutions that are distinct negative integers. If $a+b+c = 2014$, find $c$.
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
Evaluate
\[\left\lfloor\frac{1\times 5}{7}\right\rfloor + \left\lfloor\frac{2\times 5}{7}\right\rfloor + \left\lfloor\frac{3\times 5}{7}\right\rfloor+\cdots+\left\lfloor\frac{100\times 5}{7}\right\rfloor.\]
[i]Proposed by Nathan Xiong[/i]
Let $(m,r,s,t)$ be positive integers such that $m\geq s+1$ and $r\geq t$. Consider $m$ sets $A_1, A_2, \dots, A_m$ with $r$ elements each one. Suppose that, for each $1\leq i\leq m$, there exist at least $t$ elements of $A_i$, such that each one(element) belongs to (at least) $s$ sets $A_j$ where $j\neq i$. Determine the greatest quantity of elements in the following set $A_1 \cup A_2 \cup A_3 \dots \cup A_m$.
Let $ABCD$ be a cyclic quadrilateral. Let $E$ be the intersection of lines parallel to $AC$ and $BD$ passing through points $B$ and $A$, respectively. The lines $EC$ and $ED$ intersect the circumcircle of $AEB$ again at $F$ and $G$, respectively. Prove that points $C$, $D$, $F$, and $G$ lie on a circle.
In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?
Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.
A board of size $2015 \times 2015$ is covered with sub-boards of size $2 \times 2$, each of which is painted like chessboard. Each sub-board covers exactly $4$ squares of the board and each square of the board is covered with at least one square of a sub-board (the painted of the sub-boards can be of any shape). Prove that there is a way to cover the board in such a way that there are exactly $2015$ black squares visible. What is the maximum number of visible black squares?
Aidan, Boyan, Calvin, Derek, Ephram, and Fanalex are all chamber musicians at a concert. In each performance, any combination of musicians of them can perform for all the others to watch. What is the minimum number of performances necessary to ensure that each musician watches every other musician play?
Seven cyclists follow one another, in a line, on a narrow way. By the end of the training, each cyclist complains about the style of driving of the one in front of him. They agree to rearrange themselves the next day, so that no cyclist would follow the same cyclist he follows the first day. How many such rearrangements are possible?
Find all nonnegative integers $n$ for which $n^8-n^2$ is not divisible by $72$.
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that
\[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\]
then either all these numbers are nonnegative or all these numbers are nonpositive.
Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.