Found problems: 85335
Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ=735$ and $QR=112,$ find $RC.$
Three lines $m$, $n$, and $\ell$ lie in a plane such that no two are parallel. Lines $m$ and $n$ meet at an acute angle of $14^{\circ}$, and lines $m$ and $\ell$ meet at an acute angle of $20^{\circ}$. Find, in degrees, the sum of all possible acute angles formed by lines $n$ and $\ell$.
[i]Ray Li[/i]
Let $f(x)=\frac{\cos 2x-(a+2)\cos x+a+1}{\sin x}.$ For constant $a$ such that $\lim_{x\rightarrow 0} \frac{f(x)}{x}=\frac 12$, evaluate $\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{1}{f(x)}dx.$
Let $ABC$ be a triangle $(A=90^{\circ})$ and $D\in (AC)$ such that $BD$ is the bisector of $B$. Prove that $BC-BD=2AB$ if and only if
\[\frac{1}{BD}-\frac{1}{BC}=\frac{1}{2AB} \]
Let $PABC$ be a tetrahedron such that $\angle{APB}=\angle{APC}=\angle{BPC}=90^\circ, \angle{ABC}=30^\circ,$ and $AP^2$ equals the area of triangle $ABC.$ Compute $\tan\angle{ACB}.$
$W$ is a polygon which has a center of symmetry $S$ such that if $P$ belongs to $W$, then so does $P'$, where $S$ is the midpoint of $PP'$. Show that there is a parallelogram $V$ containing $W$ such that the midpoint of each side of $V$ lies on the border of $W$.
Find all natural numbers $n$ such that the number $n(n+1)(n+2)(n+3)$ has exactly three different prime divisors.
For each $q\in\mathbb Q$, let $\pi(q)$ denote the period of the repeating base-$16$ expansion of $q$, with the convention of $\pi(q)=0$ if $q$ has a terminating base-$16$ expansion. Find the maximum value among \[\pi\left(\frac11\right),~\pi\left(\frac12\right),~\dots,~\pi\left(\frac1{70}\right).\]
$f(x)=x^2+x$
$b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$
Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$
Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices.
[asy]
draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0));
draw((0,3/2)--(6,3/2));
draw((2,0)--(2,3/2));
draw((4,0)--(4,3/2));
draw((3,3/2)--(3,3));
[/asy]
An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$. Is it possible to order the real numbers so it becomes mean-free?
related: [url]https://www.youtube.com/watch?v=ppaXUxsEjMQ[/url]
If $x$, $y$, and $z$ are real numbers such that $x^2+2y^2+3z^2=96$, what is the maximum possible value of $x+2y+3z$?
Find all pairs of non-negative integers $(x,y)$ such that
\[\sqrt{x+y}-\sqrt{x}-\sqrt{y}+2=0.\]
If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$, then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$. Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$?
Let $\beta$ be the length of the bisector of angle $B$, and $a', c'$ be the lengths of the segments into which this bisector divides the side $AC$ of the triangle $ABC$. Prove the relation $\beta^2 = ac-a'c'$ and derive from this the formula $\beta^2=ac-\frac{b^2ac}{(a+c)^2}$.
Let $m,n,k$ be positive integers, $m> n> k$. An $1 \times m$ strip of paper is divided into the $1 \times 1$ cells. A teacher asks Bill and Pit to place numbers $0$ and $1$ in the cells of the strip so that the sum of the numbers in any $n$ consecutive cells is equal to $k$. After the task was performed it turned out that the sum $S(B)$ of all numbers on the strip of Bill was different from the sum $S(P)$ of Pit.
Find the largest possible value of $|S(B) - S(P) |$.
(I. Voronovich)
Given an isosceles trapezoid $ABCD$ with $AB = 6$, $CD = 12$, and area $36$, find $BC$.
Find all $a,b,c \in \mathbb{N}$ such that
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad c^2a|a^3+b^3+c^3.\]
[PS: The original problem was this:
Find all $a,b,c \in \mathbb{N}$ such that
\[a^2b|a^3+b^3+c^3,\qquad b^2c|a^3+b^3+c^3, \qquad \color{red}{c^2b}|a^3+b^3+c^3.\]
But I think the author meant $c^2a|a^3+b^3+c^3$, just because of symmetry]
Given triangle $ABC$. Point $B_1$ is marked on line $AC$ so that $AB = AB_1$, while $B_1$ and $C$ are on the same side of $A$. Through points $C$, $B_1$ and the foot of the bisector of angle $A$ of triangle $ABC$, a circle $\omega$ is drawn, intersecting for second time the circle circumscribed around triangle $ABC$, at point $Q$. Prove that the tangent drawn to $\omega$ at point $Q$ is parallel to $AC$.
Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$,
where $n$ is a positive integer.
The sequence $x_{n}$ is de fined by:
$x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$
Prove that all members of the sequence are perfect squares.
There is a village of pile-built dwellings on a lake, set on the gridpoints of an $m \times n$ rectangular grid. Each dwelling is connected by exactly $p$ bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values $m,n, p$ it is possible to place the bridges so that from any dwelling one can reach any other dwelling.
Given a triangle $ABC$ and a point $O$ on a plane. Let $\Gamma$ be the circumcircle of $ABC$. Suppose that $CO$ intersects with $AB$ at $D$, and $BO$ and $CA$ intersect at $E$. Moreover, suppose that $AO$ intersects with $\Gamma$ at $A,F$. Let $I$ be the other intersection of $\Gamma$ and the circumcircle of $ADE$, and $Y$ be the other intersection of $BE$ and the circumcircle of $CEI$, and $Z$ be the other intersection of $CD$ and the circumcircle of $BDI$. Let $T$ be the intersection of the two tangents of $\Gamma$ at $B,C$, respectively. Lastly, suppose that $TF$ intersects with $\Gamma$ again at $U$, and the reflection of $U$ w.r.t. $BC$ is $G$.
Show that $F,I,G,O,Y,Z$ are concyclic.
Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.
[img]https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png[/img]
by Morteza Saghafian
The points lie on three parallel lines with distances as indicated in the figure $A, B$ and $C$ such that square $ABCD$ is a square. Find the area of this square.
[img]https://1.bp.blogspot.com/-xeFvahqPVyM/XzcFfB0-NfI/AAAAAAAAMYA/SV2XU59uBpo_K99ZBY43KSSOKe-veOdFQCLcBGAsYHQ/s0/1998%2BMohr%2Bp3.png[/img]