Found problems: 85335
(a) Let $z$ be a solution of the quadratic equation
$$az^2 +bz+c=0$$
and let $n$ be a positive integer. Show that $z$ can be expressed as a rational function of $z^n , a,b,c.$
(b) Using (a) or by any other means, express $x$ as a rational function of $x^{3}$ and $x+\frac{1}{x}.$
Let $n\ge 2$ be a positive integer such that the set of $n$th roots of unity has less than $2^{\lfloor\sqrt n\rfloor}-1$ subsets with the sum $0$. Show that $n$ is a prime number.
[i]Cristi Săvescu[/i]
Let $k\geqslant 2$ be an integer. Consider the sequence $(x_n)_{n\geqslant 1}$ defined by $x_1=a>0$ and $x_{n+1}=x_n+\lfloor k/x_n\rfloor$ for $n\geqslant 1.$ Prove that the sequence is convergent and determine its limit.
Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \] What is the minimum degree of $P$?
[i]Proposed by Loke Zhi Kin[/i]
Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$?
A circle is tangent to sides $AD,\text{ }DC,\text{ }CB$ of a convex quadrilateral $ABCD$ at $\text{K},\text{ L},\text{ M}$ respectively. A line $l$, passing through $L$ and parallel to $AD$, meets $KM$ at $N$ and $KC$ at $P$. Prove that $PL=PN$.
Triangle $ABC$ is divided into six smaller triangles by lines that pass through the vertices and through a common point inside of the triangle. The areas of four of these triangles are indicated. Calculate the area of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/9/2/2013de890e438f5bf88af446692b495917b1ff.png[/img]
Two circles $\Gamma_1$ and $\Gamma_2$ meet at $A$ and $B$. A line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ repsectively. Another line through $B$ meets $\Gamma_1$ and $\Gamma_2$ again at $E$ and $F$ repsectively. Line $CF$ meets $\Gamma_1$ and $\Gamma_2$ again at $P$ and $Q$ respectively. $M$ and $N$ are midpoints of arc $PB$ and arc $QB$ repsectively. Show that if $CD = EF$, then $C,F,M,N$ are concyclic.
A reaction in which a carboxylic acid reacts with an alcohol to form an organic compound and water is called
$ \textbf{(A) } \text{esterification} \qquad\textbf{(B) } \text{hydrolysis}\qquad\textbf{(C) } \text{neutralization} \qquad\textbf{(D) } \text{saponification}\qquad$
In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.
Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops?
[i]Proposed by Harry Kim[/i]
The distinct points $X_1, X_2, X_3, X_4, X_5, X_6$ all lie on the same side of the line $AB$. The six triangles $ABX_i$ are all similar. Show that the $X_i$ lie on a circle.
Given is a number $a$ with 0 $\le \alpha \le \pi$. A sequence $c_0,c_1, c_2,...$ is defined as
$$c_0=\cos \alpha$$
$$C_{n+1}=\sqrt{\frac{1+c_n}{2}} \,\, for \,\,\, n=0,1,2,...$$
Calculate $\lim_{n\to \infty}2^{2n+1}(1-c_n)$
Prove that the number of orientations of a connected $3$-regular graph on $2n$ vertices where the number of vertices with indegree $0$ and outdegree $0$ are equal, is exactly $2^{n+1}$ $ {2n} \choose {n}$.
A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.
Ali has $100$ cards with numbers $1,2,\ldots,100$. Ali and Amin play a game together. In each step, first Ali chooses a card from the remaining cards and Amin decides to pick that card for himself or throw it away. In the case that he picks the card, he can't pick the next card chosen by Amin, and he has to throw it away. This action repeats until when there is no remaining card for Ali.
Amin wants to pick cards in a way that the sum of the number of his cards is maximized and Ali wants to choose cards in a way that the sum of the number of Amin's cards is minimized. Find the most value of $k$ such that Amin can play in a way that is sure the sum of the number of his cards will be at least equal to $k$.
Angela, Bill, and Charles each independently and randomly choose a subset of $\{ 1,2,3,4,5,6,7,8 \}$ that consists of consecutive integers (two people can select the same subset). The expected number of elements in the intersection of the three chosen sets is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$.
Show that the line $DE$ bisects the angle $\angle BFC$.
In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.
Let $ ABC $ be an obtuse triangle with $ AB=AC, M $ the symmetric point of $ A $ with respect to $ C, $ and $ P $ the intersection of the line $ AB $ with the perpendicular bisector of the segment $ \overline{AB} . $
Knowing that $ PM $ is perpendicular to $ BC, $ show that $ APM $ is equilateral.
For a sequence $x_1,x_2,\ldots,x_n$ of real numbers, we define its $\textit{price}$ as \[\max_{1\le i\le n}|x_1+\cdots +x_i|.\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price $D$. Greedy George, on the other hand, chooses $x_1$ such that $|x_1 |$ is as small as possible; among the remaining numbers, he chooses $x_2$ such that $|x_1 + x_2 |$ is as small as possible, and so on. Thus, in the $i$-th step he chooses $x_i$ among the remaining numbers so as to minimise the value of $|x_1 + x_2 + \cdots x_i |$. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price $G$.
Find the least possible constant $c$ such that for every positive integer $n$, for every collection of $n$ real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality $G\le cD$.
[i]Proposed by Georgia[/i]
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
Let $O$ be a point (in the plane) and $T$ be an infinite set of points such that $|P_1P_2| \le 2012$ for every two distinct points $P_1,P_2\in T$. Let $S(T)$ be the set of points $Q$ in the plane satisfying $|QP| \le 2013$ for at least one point $P\in T$.
Now let $L$ be the set of lines containing exactly one point of $S(T)$. Call a line $\ell_0$ passing through $O$ [i]bad[/i] if there does not exist a line $\ell\in L$ parallel to (or coinciding with) $\ell_0$.
(a) Prove that $L$ is nonempty.
(b) Prove that one can assign a line $\ell(i)$ to each positive integer $i$ so that for every bad line $\ell_0$ passing through $O$, there exists a positive integer $n$ with $\ell(n) = \ell_0$.
[i]Proposed by David Yang[/i]
Find all values of real parameter $a$ for which equation $2{\sin}^4(x)+{\cos}^4(x)=a$ has real solutions