Found problems: 85335
Denote by $E(n)$ the number of $1$'s in the binary representation of a positive integer $n$. Call $n$ [i]interesting[/i] if $E(n)$ divides $n$. Prove that
(a) there cannot be five consecutive interesting numbers, and
(b) there are infinitely many positive integers $n$ such that $n$, $n+1$ and $n+2$ are each interesting.
Let $a,b,$ and $c$ be real numbers satisfying the system of equations
\begin{align*}
a\sqrt{1+b^2}+b\sqrt{1+a^2}&=\tfrac{3}{4},\\
b\sqrt{1+c^2}+c\sqrt{1+b^2}&=\tfrac{5}{12}, \ \text{and} \\
c\sqrt{1+a^2}+a\sqrt{1+c^2}&=\tfrac{21}{20}.
\end{align*}
Compute $a.$
What is the largest amount of elements that can be taken from the set $\{1, 2, ... , 2012, 2013\}$, such that within them there are no distinct three, say $a$, $b$,and $c$, such that $a$ is a divisor or multiple of $b-c$?
Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.
Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive numbers such that \[a_{n+1}^{2}= a_{n}+1, \;\; n \in \mathbb{N}.\] Show that the sequence contains an irrational number.
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.
Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.
A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$ $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdots,20\}$ for $i=1,\cdots,5$ ?
For a natural number $n \ge 2$, consider all representations of $n$ as a sum of its distinct divisors, $n = t_1 + t_2 + ... + t_k, t_i| n$. Two such representations differing only in order of the summands are considered the same (for example, $20 = 10+5+4+1$ and $20 = 5+1+10+4$). Let $a(n)$ be the number of different representations of $n$ in this form. Prove or disprove: There exists M such that $a(n) \le M$ for all $n \ge 2$.
Find all pairs of polynomials $p(x),q(x)\in R[x]$ satisfying the equality $p(x^2)=p(x)q(1-x)+p(1-x)q(x)$ for all real $x$.
I.Voronovich
A convex polygon on the plane is called [i]wide[/i] if the projection of the polygon onto any line in the same plane is a segment with length at least 1. Prove that a circle of radius $\tfrac{1}{3}$ can be placed completely inside any wide polygon.
$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that:
$\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$
Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$.
[i]Proposed by Justin Hsieh[/i]
Consider an inscriptible polygon $ABCDE$. Let $H_1,H_2,H_3,H_4,H_5$ be the orthocenters of the triangles $ABC,BCD,CDE,DEA,EAB$ and let $M_1,M_2,M_3,M_4,M_5$ be the midpoints of $DE,EA,AB,BC$ and $CD$, respectively. Prove that the lines $H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5$ have a common point.
[i]Dinu Serbanescu[/i]
let $f:R\to R$ be a continuous function tf(t)>0 for $t\neq 0$. Prove that there exists a non-zero differentiable function $y:[0,\infty)\to R$ such that $y'(t)=f(y(t-1))\,\forall t>1$ and the roots of y are bounded.
Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.
After rationalizing the numerator of $ \frac {\sqrt{3}\minus{}\sqrt{2}}{\sqrt{3}}$, the denominator in simplest form is:
$\textbf{(A)}\ \sqrt{3}(\sqrt{3}+\sqrt{2}) \qquad
\textbf{(B)}\ \sqrt{3}(\sqrt{3}-\sqrt{2}) \qquad
\textbf{(C)}\ 3-\sqrt{3}\sqrt{2} \qquad\\
\textbf{(D)}\ 3+\sqrt6 \qquad
\textbf{(E)}\ \text{None of these answers}$
There are $ 6n \plus{} 4$ mathematicians participating in a conference which includes $ 2n \plus{} 1$ meetings. Each meeting has one round table that suits for $ 4$ people and $ n$ round tables that each table suits for $ 6$ people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time.
1. Determine whether or not there is the case $ n \equal{} 1$.
2. Determine whether or not there is the case $ n > 1$.
For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$. Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$. $BF$ and $DE$ meets at $G$, and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$. $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$. $XY$ and $ML$ meets at $P$, $YZ$ and $MN$ meets at $Q$. Prove that $BP$ and $DQ$ are parallel.
Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $f'(x) > f(x)>0$ for all real numbers $x$. Show that $f(8) > 2022f(0)$.
[i]Proposed by Ethan Tan[/i]
For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.
Solve the equation $\cos^n{x}-\sin^n{x}=1$ where $n$ is a natural number.
The $2011$th prime number is $17483$ and the next prime is $17489$.
Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?