Found problems: 85335
Given $n \geq 2$ reals $x_1 , x_2 , \dots , x_n.$ Show that
$$\prod_{1\leq i < j \leq n} (x_i - x_j)^2 \leq \prod_{i=0}^{n-1} \left(\sum_{j=1}^{n} x_j^{2i}\right)$$
and find all the $(x_1 , x_2 , \dots , x_n)$ where the equality holds.
For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation:
\[ x^2 - xy + y^2 = n. \]
a) Determine $f(2022)$.
b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.
Denote by $m(a,b)$ the arithmetic mean of positive real numbers $a,b$. Given a positive real function $g$ having positive derivatives of the first and second order, define $\mu (a,b)$ the mean value of $a$ and $b$ with respect to $g$ by $2g(\mu (a,b)) = g(a)+g(b)$. Decide which of the two mean values $m$ and $\mu$ is larger.
The circumference of a circle is $100$ inches. The side of a square inscribed in this circle, expressed in inches, is:
$ \textbf{(A) }\frac{25\sqrt{2}}{\pi} \qquad\textbf{(B) }\frac{50\sqrt{2}}{\pi}\qquad\textbf{(C) }\frac{100}{\pi}\qquad\textbf{(D) }\frac{100\sqrt{2}}{\pi}\qquad\textbf{(E) }50\sqrt{2} $
Let $a_n$ be a sequence such that $a_1 = 1$ and $a_{n+1} = \lfloor a_n +\sqrt{a_n} +\frac12 \rfloor $, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. What are the last four digits of $a_{2012}$?
In the Crelle $[ABCD]$ tetrahedron, we note with $A',B',C',A'',B'',C''$ the tangent points of the hexatangent sphere $\varphi(J,\rho)$, associated with the tetrahedron, with the edges $|BC|,|CA|,|AB|,|DA|,|DB|,|DC|$. Show that these inequalities occur:
a)
$$2\sqrt3R\ge6\rho\ge A'A''+B'B''+C'C''\ge6\sqrt3r$$
b)
$$4R^2\ge12\rho^2\ge(A'A'')^2+(B'B'')^2+(C'C'')^2\ge36r^2$$
c)
$$\frac{8R^3}{3\sqrt3}\ge8\rho^3\ge A'A''\cdot B'B''\cdot C'C''\ge24\sqrt3r^3$$
where $r,R$ is the length of the radius of the sphere inscribed and respectively circumscribed to the tetrahedron.
[i]Proposed by Marius Olteanu[/i]
There are distinct quadratics $e(x)$, $p(x)$, $h(x)$, $r(x)$, $a(x)$, and $m(x)$ with leading coefficient $1$, such that their roots are $2$ distinct values from the set $\{3, 4, 5, 6\}$. James takes three of these quadratics, sums two, and subtracts the last. Given that this new quadratic has a root at $0$, find its other root.
Let $n$ be a positive integer such that there exist positive integers $x_1,x_2,\cdots ,x_n$ satisfying $$x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.$$ Find the greatest possible value of $n$.
Let $f$ be a function defined on $[0, 2022]$, such that $f(0) = f(2022) = 2022$, and $$|f(x) - f(y)| \le 2|x -y|,$$ for all $x, y$ in $[0, 2022]$. Prove that for each $x, y$ in $[0, 2022]$, the distance between $f(x)$ and $f(y)$ does not exceed $2022$.
The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.
Solve the system of equations:
$\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$
The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh $100\text{ kg}$ in total. Mr. Yamaimo realizes that $99\%$ of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only $95\%$ of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?
[b](a)[/b] For every positive integer $n$ prove that
\[1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2\]
[b](b)[/b] Let $X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1)$ and let $A_k$ be non-empty subsets of $X \ (k=1,2,3, \ldots , 2^n -1).$ If $a_k$ be the product of all elements of the set $A_k,$ prove that
\[\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1\]
A positive integer $n$ is [i]Mozart[/i] if the decimal representation of the sequence $1, 2, \ldots, n$ contains each digit an even number of times.
Prove that:
1. All Mozart numbers are even.
2. There are infinitely many Mozart numbers.
A wire figure is held in different ways in a bundle of parallel light rays, so that different shadow figures are created in a plane perpendicular to the light rays. In this way one can form:
(a) an isosceles triangle;
(b) an isosceles triangle with altitude from the apex;
(c) a rectangle containing an isosceles triangle;
(d) a rhombus with one diagonal.
The wire figure consists of eight straight pieces of iron wire, with each piece connected to both ends are attached to at least one other piece. Determine a figure corresponding to the above description is satisfactory, and indicate the direction of the light rays at which the shadow figures (a) to (d) arise.
[hide=original wording]Men houdt een draadfiguur op verschillende manieren in een bundel evenwijdige lichtstralen, waardoor er in een vlak loodrecht op de lichtstralen verschillende schaduwfiguren ontstaan. Op deze wijze kan men vormen:
(a) een gelijkbenige driehoek;
(b) een gelijkbenige driehoek met hoogtelijn uit de top;
(c) een rechthoek met daarin een gelijkbenige driehoek;
(d) een ruit met één diagonaal.
De draadfiguur bestaat uit acht rechte stukjes ijzerdraad, waarbij ieder stukje aan beide
uiteinden aan tenminste één ander stukje vastzit. Bepaal een figuur die aan bovenstaande
beschrijving voldoet, en geef de richting van de lichtstralen aan waarbij de
schaduwfiguren (a) tot en met (d) ontstaan.[/hide]
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
What is the greatest multiple of $9$ that can be formed by using each of the digits in the set $\{1, 3,5, 7, 9\}$ at most once.
Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set
$A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$
$B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$
Prove that $2|A|\geq |B|$.
The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if
$A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$.
i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors
of $A$.
ii) Find the maximal number of distinct pairwise commuting involutions on $V$.
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for $24$ hours. If she is using it constantly, the battery will last for only $3$ hours. Since the last recharge, her phone has been on $9$ hours, and during that time she has used it for $60$ minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
$\textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 11\qquad
\textbf{(D)}\ 14\qquad
\textbf{(E)}\ 15$
Grisha wrote $n$ different natural numbers, the sum of which does not exceed $S$. The saboteur added to each of them a number from the half-interval $[0, 1)$. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than $1$. At what minimum $S$ can Grisha ensure that the sabotage will definitely not be succeeded?
[b]a)[/b] Let $ D_1,D_2,D_3 $ be pairwise skew lines. Through every point $ P_2\in D_2 $ there is an unique common secant of these three lines that intersect $ D_1 $ at $ P_1 $ and $ D_3 $ at $ P_3. $ Let coordinate systems be introduced on $ D_2 $ and $ D_3 $ having as origin $ O_2, $ respectively, $ O_3. $ Find a relation between the coordinates of $ P_2 $ and $ P_3. $
[b]b)[/b] Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants.
[b]c)[/b] Let $ F_1,F_2,F_3,F_4 $ be any four secants of $ D_1,D_2, D_3. $ Prove that $ F_1,F_2, F_3, F_4 $ have infinitely many common secants.
$777$ pairwise distinct complex numbers are written on a board. It turns out that there are exactly 760 ways to choose two numbers \(a\) and \(b\) from the board such that:
\[
a^2 + b^2 + 1 = 2ab
\]
Ways that differ by the order of selection are considered the same. Prove that there exist two numbers \(c\) and \(d\) from the board such that:
\[
c^2 + d^2 + 2025 = 2cd
\]
Given any positive integer $n$ , show that there are two positive rational numbers $a$ and $b$ , $a \not= b$, which are not integers and which are such that $a - b, a^2 - b^2 , \ldots a^n - b^n$ are all integers.
If $m$ is a positive integer, let $S_m$ be the set of rational numbers in reduced form with denominator at most $m$. Let $f(m)$ be the sum of the numerator and denominator of the element of $S_m$ closest to $e$ (Euler's constant). Given that $f(2007) = 3722$, find the remainder when $f(1000)$ is divided by $2008$.