Found problems: 85335
A fair coin is flipped $6$ times. The probability that the coin lands on the same side $3$ flips in a row at some point can be expressed as a common fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.
Let $a, b, c$ be positive real numbers in the interval $[0, 1]$ with $a+b, b+c, c+a \ge 1$. Prove that
\[
1 \le (1-a)^2 + (1-b)^2 + (1-c)^2 +
\frac{2\sqrt{2} abc}{\sqrt{a^2+b^2+c^2}}.
\]
Let $a,b>1$ be two real numbers. Prove that $a>b$ if and only if there exists a function $f: (0,\infty)\to\mathbb{R}$ such that
i) the function $g:\mathbb{R}\to\mathbb{R}$, $g(x)=f(a^x)-x$ is increasing;
ii) the function $h:\mathbb{R}\to\mathbb{R}$, $h(x)=f(b^x)-x$ is decreasing.
A corner with arm $n$ is a figure made of $2n-1$ unit squares, such that 2 rectangles $1$ x $(n-1)$ are connected to two adjacent sides of a square $1$ x $1$, so that their unit sides coincide.
The squares or a chessboard $100$ x $100$ are colored in 15 colors. We say that a corner with arm 8 is [i]“multicolored”[/i], if it contains each of the colors on the board. What’s the greatest number of corners with arm 8 which could be [i]“mutlticolored”[/i]?
Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?
Given posetive real numbers $a_1,a_2,\dots,a_n$ such that $a_1^2+2a_2^3+\dots+na_n^{n+1} <1.$ Prove that $2a_1+3a_2^2+\dots+(n+1)a_{n}^n <3.$
Let ABCDEF be an equilateral hexagon in which all pairs of opposite sides are parallel. The triangle whose sides are the extensions of AB, CD and EF has side lengths 200, 240 and 300 respectively. Find the side length of the hexagon.
Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$.
Show that there exists $j<k$ and $l<m$ such that
\[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]
Let $(A_i)_{i\ge 1}$ be sequence of sets of two integer numbers, such that no integer is contained in more than one $A_i$ and for every $A_i$ the sum of its elements is $i$. Prove that there are infinitely many values of $k$ for which one of the elements of $A_k$ is greater than $13k/7$.
Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}$$
Find all real numbers$ x$ and $y$ such that $$x^2 + y^2 = 2$$
$$\frac{x^2}{2 - y}+\frac{y^2}{2 - x}= 2.$$
Show that there doesn't exist two infinite and separate sets $A,B$ of points such that
[b](i)[/b] There are no three collinear points in $A \cup B$,
[b](ii)[/b] The distance between every two points in $A \cup B$ is at least $1$, and
[b](iii)[/b] There exists at least one point belonging to set $B$ in interior of each triangle which all of its vertices are chosen from the set $A$, and there exists at least one point belonging to set $A$ in interior of each triangle which all of its vertices are chosen from the set $B$.
Circles with radii $1, 2$, and $3$ are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
$ \textbf{(A)}\ \frac{3}{5} \qquad
\textbf{(B)}\ \frac{4}{5} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{6}{5} \qquad
\textbf{(E)}\ \frac{4}{3}
$
Let $f(x)$ be the distance from $x$ to the nearest perfect square. For example, $f(\pi) = 4 - \pi$. Let $\alpha = \frac{3 + \sqrt{5}}{2}$ and let $m$ be an integer such that the sequence $a_n = f(m \; \alpha^n)$ is bounded. Prove that either $m=k^2$ or $m = 5k^2$ for some integer $k$.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil[/i].
Determine all rings $(A,+,\cdot)$ such that $x^3\in\{0,1\}$ for any $x\in A.$
[i]Mihai Opincariu[/i]
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]
Let $n\geq 2$ and $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $$\{\text{rank}(A^k)\mid k\geq 1\}=\{\text{rank}(B^k)\mid k\geq 1\}.$$ Prove that $\text{rank}(A^k)=\text{rank}(B^k)$ for all $k\geq 1$.
[i]Cristi Săvescu[/i]
Let $$N = \sum^{512}_{i=0}i {512 \choose i}.$$ What is the greatest integer $a$ such that $2^a$ is a divisor of $N$?
Let $f$ be a function that satisfies :
\[ \displaystyle f(x)+2f\left(\frac{x+\frac{2001}2}{x-1}\right) = 4014-x. \]
Find $f(2004)$.
For $0<x<1,$ express $$\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}$$ as a rational function of $x.$
The numbers 1 through 9 can be arranged in the triangles labeled $a$ through $i$ illustrated below so that the numbers in each of the $2\times2$ triangles sum to the value $n$; that is \[a+b+c+d=b+e+f+g=d+g+h+i=n.\] For each possible sum $n$, show an arrangement, labeled with the sum as shown below. Prove that there are no possible arrangements for any other values of $n$.
[asy]
size(150);
defaultpen(linewidth(0.7)+fontsize(12)); picture p = new picture;
draw(p,(-3,-3^.5)/2--(3,-3^.5)/2^^(-1,0)--(1,0)^^(-1,3^.5)/2--(1,3^.5)/2); add(p); add(rotate(120)*p); add(rotate(240)*p);
string[] hexlbl = {'d','c','b','f','g','h'}, trilbl = {'a','e','i'};
for(int i = 0; i < hexlbl.length; ++i) label('$'+hexlbl[i]+'$',dir(30+60*i)/3^.5);
for(int i = 0; i < trilbl.length; ++i) label('$'+trilbl[i]+'$',dir(90+120*i)*2/3^.5);[/asy]
The real numbers $x_1, x_2, ... , x_n$ belong to the interval $(0,1)$ and satisfy $x_1 + x_2 + ... + x_n = m + r$, where $m$ is an integer and $r \in [0,1)$. Show that $x_1 ^2 + x_2 ^2 + ... + x_n ^2 \leq m + r^2$.
Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.
Let $X$ be an arbitrary set and $f$ a bijection from $X$ to $X$. Show that there exist bijections $g,\ g':X\to X$ s.t. $f=g\circ g',\ g\circ g=g'\circ g'=1_X$.
The sequence Pn (x), n ∈ N of polynomials is defined as follows:
P0 (x) = x, P1 (x) = 4x³ + 3x
Pn+1 (x) = (4x² + 2)Pn (x) − Pn−1 (x), for all n ≥ 1
For every positive integer m, we consider the set A(m) = { Pn (m) | n ∈ N }. Show that the sets A(m) and A(m+4) have no common elements.