Found problems: 85335
Let $a$, $b$, $c$ be positive reals. Prove that \[ \sqrt{\frac{a^2(bc+a^2)}{b^2+c^2}}+\sqrt{\frac{b^2(ca+b^2)}{c^2+a^2}}+\sqrt{\frac{c^2(ab+c^2)}{a^2+b^2}}\ge a+b+c. \][i]Proposed by Robin Park[/i]
Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
In how many ways can $10$ distinct books be placed onto $3$-shelf bookcase in such a way that no shelf is empty?
$
\textbf{(A)}\ 36\cdot 10!
\qquad\textbf{(B)}\ 50 \cdot 10!
\qquad\textbf{(C)}\ 55 \cdot 10!
\qquad\textbf{(D)}\ 81 \cdot 10!
\qquad\textbf{(E)}\ \text{None of the above}
$
Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.
For any number $n\in\mathbb{N},n\ge 2$, denote by $P(n)$ the number of pairs $(a,b)$ whose elements are of positive integers such that
\[\frac{n}{a}\in (0,1),\quad \frac{a}{b}\in (1,2)\quad \text{and}\quad \frac{b}{n}\in (2,3). \]
$a)$ Calculate $P(3)$.
$b)$ Find $n$ such that $P(n)=2002$.
Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.
Determine all pairs of prime numbers $(p; q)$ such that $p^2 + 5pq + 4q^2$ is the square of an integer.
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Given a triangle $ABC$.
Let $M$ and $N$ be the points where the angle bisectors of the angles $ABC$ and $BCA$ intersect the sides $CA$ and $AB$, respectively.
Let $D$ be the point where the ray $MN$ intersects the circumcircle of triangle $ABC$.
Prove that $\frac{1}{BD}=\frac{1}{AD}+\frac{1}{CD}$.
Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.
For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, b, c)$ be the minimum number of stamps in such a collection. Find the sum of the three least values of $c$ such that $f(a, b, c) = 97$ for some choice of $a$ and $b$.
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\]
Let $f$ be a function from nonnegative integers to nonnegative integers such that $f(0)=0$ and $$f(m)=f\left(\left\lfloor \frac{m}{2}\right\rfloor\right)+\left\lceil\frac{m}{2}\right\rceil^2$$ for all positive integers $m.$ Compute $$\frac{f(1)}{1\cdot2}+\frac{f(2)}{2\cdot3}+\frac{f(3)}{3\cdot4}+\cdots+\frac{f(31)}{31\cdot32}.$$(Here, $\lfloor z \rfloor$ is the greatest integer less than or equal to $z,$ and $\lceil z \rceil$ is the least positive integer greater than or equal to $z.$)
For a positive integer $n$, let
\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]
Answer the following questions:
(1) Show the following inequality.
\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]
(2) Express $T_n-2S_n$ in terms of $n$.
(3) Find the limit $\lim_{n\to\infty} T_n.$
Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andy Xu[/i]
Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$.
[i]Proposed by Angelo Di Pasquale, Australia[/i]
Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?
An acute triangle $ABC$ is given, $AC \not= BC$. The altitudes drawn from $A$ and $B$ meet at $H$ and intersect the external bisector of the angle $C$ at $Y$ and $X$ respectively. The external bisector of the angle $AHB$ meets the segments $AX$ and $BY$ at $P$ and $Q$ respectively. If $PX = QY$, prove that $AP + BQ \ge 2CH$.
Show that [list=a] [*] if $n>49$, then there are positive integers $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}<1$. [*] if $n>4$, then there are $a>1$ and $b>1$ such that $a+b=n$ and $\frac{\phi(a)}{a}+\frac{\phi(b)}{b}>1$.[/list]
In triangle $[ABC]$, the bissector of the angle $\angle{BAC}$ intersects the side $[BC]$ at $D$. Suppose that $\overline{AD}=\overline{CD}$. Find the lengths $\overline{BC}$, $\overline{AC}$ and $\overline{AB}$ that minimize the perimeter of $[ABC]$, given that all the sides of the triangles $[ABC]$ and $[ADC]$ have integer lengths.
Given a graph with $n$ ($n\ge 4$) vertices . It is known that for any two vertices $A$ and $B$ there exists a vertex which is connected by edges both with $A$ and $B$. Find the smallest possible numbers of edges in the graph.
E. Barabanov
Let $a<b<c<d$ be positive integers which satisfy $ad=bc.$ Prove that $2a+\sqrt{a}+\sqrt{d}<b+c+1.$
[i]Marius Mînea[/i]
Given are $a_0,a_1, ... , a_n$, satisfying $a_0=a_n = 0$, and $a_{k-1} - 2a_k+a_{k+1}\ge 0$ for $k=0, 1, ... , n-1$. Prove that all the numbers are negative or zero.