Found problems: 85335
Let the angle bisectors of $\angle BAC,$ $\angle CBA,$ and $\angle ACB$ meets the circumcircle of $\triangle ABC$ at the points $M,N,$ and $K,$ respectively. Let the segments $AB$ and $MK$ intersects at the point $P$ and the segments $AC$ and $MN$ intersects at the point $Q.$ Prove that $PQ\parallel BC$
Prove that in the interior of an equilateral triangle with side $a$ you can put a finite number of equal circles that do not overlap, with radius $r = \frac{a}{2010}$, so that the sum of their areas is greater than $\frac{17\sqrt3}{80}$ a$^2$.
Find all triples $(x,y,z)\in \mathbb{R}^{3}$ such that $x,y,z>3$ and $\frac{(x+2)^2}{y+z-2}+\frac{(y+4)^2}{z+x-4}+\frac{(z+6)^2}{x+y-6}=36$
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$.
Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
\[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$
If $n=p_1^{a_1},p_2^{a_2}...p_s^{a_s}$ then $\phi (n)=n \left(1- \frac{1}{p_1}\right)\left(1 - \frac{1}{p_2}\right)...\left(1- \frac{1}{p_s}\right)$. Find the smallest positive integer $n$ such that $\phi (n)=\frac{2^5}{47}n.$
An integer $n \geq 3$ is given. We call an $n$-tuple of real numbers $(x_1, x_2, \dots, x_n)$ [i]Shiny[/i] if for each permutation $y_1, y_2, \dots, y_n$ of these numbers, we have
$$\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.$$
Find the largest constant $K = K(n)$ such that
$$\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K$$
holds for every Shiny $n$-tuple $(x_1, x_2, \dots, x_n)$.
Find all integers $n$ (not necessarily positive) such that $7n^3-3n^2-3n-1$ is a perfect cube.
Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.
Determine positive integers $n\geq 3$ such that there exists a set $M$ of $n$ complex numbers and a positive integer $m$ such that $(1+z_1z_2z_3)^m=1$ for all pairwise distinct $z_1,z_2,z_3\in M$.
[i]Vlad Matei[/i]
Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.
Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.
Working alone, the expert can paint a car in one day, the amateur can paint a car in two days, and the beginner can paint a car in three days. If the three painters work together at these speeds to paint three cars, it will take them $\frac{m}{n}$ days where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
There are unique integers $ a_2, a_3, a_4, a_5, a_6, a_7$ such that \[ \frac {5}{7} \equal{} \frac {a_2}{2!} \plus{} \frac {a_3}{3!} \plus{} \frac {a_4}{4!} \plus{} \frac {a_5}{5!} \plus{} \frac {a_6}{6!} \plus{} \frac {a_7}{7!},\] where $ 0 \le a_i < i$ for $ i \equal{} 2,3...,7$. Find $ a_2 \plus{} a_3 \plus{} a_4 \plus{} a_5 \plus{} a_6 \plus{} a_7$.
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 10 \qquad
\textbf{(D)}\ 11 \qquad
\textbf{(E)}\ 12$
A line $d$ is called [i]faithful[/i] to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one [i]faithful[/i] line to both of them, or there exist infinitely [i]faithful[/i] lines to them.
Bob plants two saplings. Each day, each sapling has a $1/3$ chance of instantly turning into a tree. Given that the expected number of days it takes both trees to grow is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Powell Zhang[/i]
Let $P$ be a point inside a triangle $ABC$, and $A_1B_2,B_1C_2,C_1A_2$ be segments passing through $P$ and parallel to $AB, BC, CA$ respectively, where points $A_1, A_2$ lie on $BC, B_1, B_2$ on $CA$, and $C_1,C_2$ on $AB$. Prove that
\[ \text{Area}(A_1A_2B_1B_2C_1C_2) \ge \frac{1}{2}\text{Area}(ABC)\]
Alice and Brian are playing a game on the real line. To start the game, Alice places a checker on a number $x$ where $0 < x < 1$. In each move, Brian chooses a positive number $d$. Alice must move the checker to either $x + d$ or $x - d$. If it lands on $0$ or $1$, Brian wins. Otherwise the game proceeds to the next move. For which values of $x$ does Brian have a strategy which allows him to win the game in a finite number of moves?
Let $ABC$ be a triangle with $AB=13$, $AC=14$, and $BC=15$. Let $G$ be the point on $AC$ such that the reflection of $BG$ over the angle bisector of $\angle B$ passes through the midpoint of $AC$. Let $Y$ be the midpoint of $GC$ and $X$ be a point on segment $AG$ such that $\frac{AX}{XG}=3$. Construct $F$ and $H$ on $AB$ and $BC$, respectively, such that $FX \parallel BG \parallel HY$. If $AH$ and $CF$ concur at $Z$ and $W$ is on $AC$ such that $WZ \parallel BG$, find $WZ$.
In an isosceles trapezoid, one of the bases is three times larger than the other. Angle at a greater basis is equal to $45^o$. Show how to cut this trapezium into three parts and make a square with them. Justify your answer.
Let $a, b$, and $c$ be integers that satisfy $2a + 3b = 52$, $3b + c = 41$, and $bc = 60$. Find $a + b + c$
Area of $\triangle ABC$ is $\frac{1}{4}$, circumradius of $\triangle ABC$ is $1$.
Let $s=\sqrt{a}+\sqrt{b}+\sqrt{c},t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then
$\text{(A)}s>t\qquad\text{(B)}s=t\qquad\text{(C)}s<t\qquad\text{(D)}s>t$
A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.
Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\]
$\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$