Found problems: 85335
Is the infinite series
$$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$
convergent?
Given a positive integer $n$, an operation consists of replacing $n$ with either $2n-1$, $3n-2$ or $5n-4$. A number $b$ is said to be a [i]follower[/i] of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$.
Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle.
i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle.
ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle.
[i](Dan Schwarz)[/i]
Jacek has $n$ cards numbered consecutively with the numbers $1,. . . , n$, which he places in a row on the table, in any order he chooses. Jacek will remove cards from the table in the sequence consistent with the numbering of cards: first they will remove the card number $1$, then the card number $2$, and so on. Before Jacek starts taking the cards, Pie will color each one of cards in red, blue or yellow. Prove that Pie can color the cards in such a way that when Jacek takes them off, it will be fulfilled at every moment the following condition: between any two cards of the same suit
there is at least one card of a different color.
Calculate the sum $\frac{1}{2.7.12} + \frac{1}{7.12.17} + ... + \frac{1}{1997.2002.2007}$.
In a particular game, each of $4$ players rolls a standard $6{ }$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game?
$(\textbf{A})\: \frac{61}{216}\qquad(\textbf{B}) \: \frac{367}{1296}\qquad(\textbf{C}) \: \frac{41}{144}\qquad(\textbf{D}) \: \frac{185}{648}\qquad(\textbf{E}) \: \frac{11}{36}$
There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation:
The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him.
Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.
$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$.
Let $n$ be a natural number. Two numbers are called "unsociable" if their greatest common divisor is $1$. The numbers $\{1,2,...,2n\}$ are partitioned into $n$ pairs. What is the minimum number of "unsociable" pairs that are formed?
Prove that if $a,b$ are positive numbers, then
$$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$
The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.
Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.
We call $\overline{a_n\ldots a_2}$ the Fibonacci representation of a positive integer $k$ if \[k = \sum_{i=2}^n a_i F_i,\] where $a_i\in\{0,1\}$ for all $i$, $a_n=1$, and $F_i$ denotes the $i^{\text{th}}$ Fibonacci number ($F_0=0$, $F_1=1$, and $F_i=F_{i-1}+F_{i-2}$ for all $i\ge2$). This representation is said to be $\textit{minimal}$ if it has fewer 1’s than any other Fibonacci representation of $k$. Find the smallest positive integer that has eight ones in its minimal Fibonacci representation.
In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?
Let $ \pi_n(x)$ be a polynomial of degree not exceeding $ n$ with real coefficients such that \[ |\pi_n(x)| \leq \sqrt{1\minus{}x^2}
\;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ .\] Then \[ |\pi'_n(x)| \leq 2(n\minus{}1).\]
[i]P. Turan[/i]
For each positive integer $n$, determine the digits of units and hundreds of the decimal representation of the number $$\frac{1 + 5^{2n+1}}{6}$$
For a convex polygon $P$ in the plane let $P'$ denote the convex polygon with vertices at the midpoints of the sides of $P.$ Given an integer $n \geq 3,$ determine sharp bounds for the ratio
\[ \frac{\text{area } P'}{\text{area } P}, \] over all convex $n$-gons $P.$
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying $$xf(y)+yf(x) \leq 2$$
Given an arbitrary prime $p>2011$. Prove that there exist positive integers $a, b, c$ not all divisible by $p$ such that for all positive integers $n$ that $p\mid n^4- 2n^2+ 9$, we have $p\mid 24an^2 + 5bn + 2011c$.
Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.
Let $n\geq1$ be an integer and let $t_1<t_2<\dots<t_n$ be positive integers. In a group of $t_n+1$ people, some games of chess are played. Two people can play each other at most once. Prove that it is possible for the following two conditions to hold at the same time:
(i) The number of games played by each person is one of $t_1,t_2,\dots,t_n$.
(ii) For every $i$ with $1\leq i\leq n$, there is someone who has played exactly $t_i$ games of chess.
Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point.
[i]Proposed by Aaron Thomas, UK[/i]
How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?
Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that
$a^4 - 2019a = b^4 - 2019b = c$.
Prove that $- \sqrt{c} < ab < 0$.
Given an $8\times 8$ chess board, in how many ways can we select $56$ squares on the board while satisfying both of the following requirements:
(a) All black squares are selected.
(b) Exactly seven squares are selected in each column and in each row.