Found problems: 85335
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$
\[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
There are several white and black points. Every white point is connected with every black point by a segment. Each segment is equipped with a positive integer. For any closed circuit the product of the integers on the segments passed in the direction from white to black point is equal to the product of the integers on the segments passed in the opposite direction. Can one always place the integer at each point so that the integer on each segment is the product of the integers at its ends?
How many numbers between $ 1$ and $ 2005$ are integer multiples of $ 3$ or $ 4$ but not $ 12$?
$ \textbf{(A)}\ 501\qquad
\textbf{(B)}\ 668\qquad
\textbf{(C)}\ 835\qquad
\textbf{(D)}\ 1002\qquad
\textbf{(E)}\ 1169$
Let $x,y,z$ be positive real numbers that satisfy at least one of the inequalities, $2xy>1$, $yz>1$. Find the least possible value of
$$xy^3z^2+\frac{4z}{x}-8yz-\frac{4}{yz}$$ .
Joao had blue, white and red pearls and with them he made a necklace with $20$ pearls that has as many blue as white pearls. João noticed that, regardless of how he cut the necklace into two parts, both with an even number of pearls, one of the parts would always have more blue pearls than white ones. How many red pearls are in Joao's necklace?
Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.
a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares.
b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.
Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$.
Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).
(a) What is the smallest and biggest representable number?
(b) Construct 45 representable numbers.
Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.
Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.
There are $n$ airports which form a regular $n$-gon. In the beginnig, there is exactly one plane at only $k$ airports. Each of the planes flies to one of the nearest airport each day. For which of the following ordered pairs $(n,k)$, it is impossible to gather all planes at a airport on one day however the planes are arranged initially?
$
\textbf{(A)}\ (10,6)
\qquad\textbf{(B)}\ (10,4)
\qquad\textbf{(C)}\ (11,3)
\qquad\textbf{(D)}\ (11,5)
\qquad\textbf{(E)}\ (13,8)
$
Let $a$ and $b$ be positive real numbers satisfying
$$\frac{a}{b} \left( \frac{a}{b}+ 2 \right) + \frac{b}{a} \left( \frac{b}{a}+ 2 \right)= 2022.$$
Find the positive integer $n$ such that $$\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}=\sqrt{n}.$$
An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the [i]least[/i] possible value of the $ 50$th term and let $ G$ be the [i]greatest[/i] possible value of the $ 50$th term. What is the value of $ G \minus{} L$?
You are given a row made by $2018$ squares, numbered consecutively from $0$ to $2017$. Initially, there is a coin in the square $0$. Two players $A$ and $B$ play alternatively, starting with $A$, on the following way: In his turn, each player can either make his coin advance $53$ squares or make the coin go back $2$ squares. On each move the coin can never go to a number less than $0$ or greater than $2017$. The player who puts the coin on the square $2017$ wins. ¿Who is the one with the wining strategy and how should he play to win?
Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$
Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.
The integer 287 is exactly divisible by
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 6$
let $ABC$ be a triangle and $I$ be its incentre. let the incircle of $ABC$ touch $BC$ at $D$.
let incircle of triangle $ABD$ touch $AB$ at $E$ and incircle of triangle $ACD$ touch $AC$ at $F$.
prove that $B,E,I,F$ are concyclic.
For a positive integer $n$, let $\phi(n)$ denote the number of positive integers between $1$ and $n$, inclusive, which are relatively prime to $n$. We say that a positive integer $k$ is total if $k=\frac n{\phi(n)}$, for some positive integer $n$. Find all total numbers.
On sides $AC$ and $BC$ of acute-angled triangle $ABC$ rectangles with equal areas $ACPQ$ and $BKLC$ were built exterior. Prove that midpoint of $PL$, point $C$ and center of circumcircle are collinear.
On a typical morning Aiden gets out of bed, goes through his morning preparation, rides the bus, and walks from the bus stop to work arriving at work 120 minutes after getting out of bed. One morning Aiden got out of bed late, so he rushed through his morning preparation getting onto the bus in half the usual time, the bus ride took 25 percent longer than usual, and he ran from the bus stop to work in half the usual time it takes him to walk arriving at work 96 minutes after he got out of bed. The next morning Aiden got out of bed extra early, leisurely went through his morning preparation taking 25 percent longer than usual to get onto the bus, his bus ride took 25 percent less time than usual, and he walked slowly from the bus stop to work taking 25 percent longer than usual. How many minutes after Aiden got out of bed did he arrive at work that day?
Given points $A_1 (x_1, y_1, z_1), A_2 (x_2, y_2, z_2), .., A_n (x_n, y_n, z_n)$ let $P (x, y, z)$ be the point which minimizes $\Sigma ( |x - x_i| + |y -y_i| + |z -z_i| )$. Give an example (for each $n > 4$) of points $A_i $ for which the point $P$ lies outside the convex hull of the points $A_i$.
The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.
A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether?
$ \textbf{(A)}\ 9
\qquad\textbf{(B)}\ 11
\qquad\textbf{(C)}\ 13
\qquad\textbf{(D)}\ 14
\qquad\textbf{(E)}\ 15
$
Let $n \ge 2$ be an integer. Show that there exist $n+1$ numbers $x_1, x_2, \ldots, x_{n+1} \in \mathbb{Q} \setminus \mathbb{Z}$, so that
$\{ x_1^3 \} + \{ x_2^3 \} + \cdots + \{ x_n^3 \}=\{ x_{n+1}^3 \}$, where $\{ x \}$ is the fractionary part of $x$.