Found problems: 85335
Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let
$$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and
$$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.
Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear.
[i]proposed by Amirhossein Zabeti[/i]
A round robin tournament is held with $2016$ participants. Each round, after seeing the results from the previous round, the tournament organizer chooses two players to play a game with each other that will result in a win for one of the players and a loss for the other. The tournament organizer wants each person to have a different total number of wins at the end of $k$ rounds. Find the minimum possible value of $k$ for which this can always be guaranteed.
[i]Proposed by Nathan Ramesh
Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$.
[i]Proposed by Lewis Chen[/i]
Alice and Bob play a game on a complete graph $G$ with $2014$ vertices. They take moves in turn with Alice beginning. At each move Alice directs one undirected edge of $G$. At each move Bob chooses a positive integer number $m,$ $1 \le m \le 1000$ and after that directs $m$ undirected edges of $G$. The game ends when all edges are directed. If there is some directed cycle in $G$ Alice wins. Determine whether Alice has a winning strategy.
Let $ABCD$ be a square of center $O$. The parallel to $AD$ through $O$ intersects $AB$ and $CD$ at $M$ and $N$ and a parallel to $AB$ intersects diagonal $AC$ at $P$. Prove that $$OP^4 + \left(\frac{MN}{2} \right)^4 = MP^2 \cdot NP^2$$
Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.
Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$.
a) Prove $|a -3b|\ge 1$.
b) Find all pairs of positive integers $(a, b)$ satisfying the equality.
Compare the numbers $P = 2^a,Q = 3, T = 2^b$, where $a=\sqrt2 , b=1+\frac{1}{\sqrt2}$
(A) $P < Q < T$, (B) $T < P < Q$, (C) $P < T < Q$, (D) $T < Q < P$, (E) $ Q < P < T$
Assume that the differential equation
$$y'''+p(x)y''+q(x)y'+r(x)y=0$$has solutions $y_1(x)$, $y_2(x)$, $y_3(x)$ on the real line such that
$$y_1(x)^2+y_2(x)^2+y_3(x)^2=1$$for all real $x$. Let
$$f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2.$$Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation
$$y'+Ap(x)y=Br(x).$$
On the coordinate plane, draw a set of points whose coordinates $(x, y)$ satisfy the equation $y=x+|y-3x-2x^2|$.
The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$, $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$.
The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.
A regular hexagon is divided by straight lines parallel to its sides into $6n^2$ equilateral triangles. On them, there are $2n$ rooks, no two of which attack each other (a rook attacks in directions parallel to the sides of the hexagon). Prove that if we color the triangles black and white such that no two adjacent triangles have the same color, there will be as many rooks on the black triangles as on the white ones.
Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$
Positive integers $a,b,c,x,y,z$ satisfy:
$a^2+b^2=c^2$, $x^2+y^2=z^2$
and
$|x-a| \leq 1$ , $|y-b| \leq 1$.
Prove that sets $\{a,b\}$ and $\{x,y\}$ are equal.
If for every real $x$, $ax^2 + bx+c \geq 0$, where $a,b,c$ are reals such that $a<b$, what is the smallest value of $\dfrac{a+b+c}{b-a}$?
$
\textbf{(A)}\ \dfrac{5}{\sqrt 3}
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ \dfrac{\sqrt 5}2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \dfrac{\sqrt 7}2
$
In a $50 \times 50$ checkered square, each cell is painted in one of $100$ given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a $1 \times 2$ rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given $100$ colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?
In a triangle, each median forms an angle with the side it is drawn to, which is less than $\alpha$. Prove that one of the angles of the triangle is greater than $180^\circ-\frac{3}{2}\alpha$.
[i]Proposed by Sergey Berlov[/i]
What is the greatest number of parts that $5$ spheres can divide the space into?
Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$.
Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a\plus{}b$ divides $ ab$.
The equation $z^6 + z^3 + 1$ has one complex root with argument $\theta$ between $90^\circ$ and $180^\circ$ in the complex plane. Determine the degree measure of $\theta$.
The numbers $1, 2, ..., 2012$ are written on a blackboard, in some order, each of them exactly once. Between every two neighboring numbers the absolute value of their difference is written and the original numbers are deleted. This process is repeated until only a number remains on the board. What is the largest number that can stay on the board?