Found problems: 85335
Wu starts out with exactly one coin. Wu flips every coin he has [i]at once[/i] after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$.
[i]Proposed by Bradley Guo[/i]
The circles $S_{1}$ and $S_{2}$ intersect at $M$ and $N$.Show that if vertices $A$ and $C$ of a rectangle $ABCD$ lie on $S_{1}$ while vertices $B$ and $D$ lie on $S_{2}$,then the intersection of the diagonals of the rectangle lies on the line $MN$.
Let $f$ be a function of the positive reals in the positive reals, such that
$$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$
(a) Find $f(1)$.
(b) Find $f(x)$.
Find all triples $(x,y,z)$ of positive integers, with $z>1$, satisfying simultaneously that \[x\text{ divides }y+1,\quad y\text{ divides }z-1,\quad z\text{ divides }x^2+1.\]
Does there exist an infinite sequence $\{x_{n}\}$ of reals satisfying the following conditions
i)$|x_{n}|\leq 0,666$ for all $n=1,2,...$
ii)$|x_{m}-x_{n}|\geq \frac{1}{n(n+1)}+\frac{1}{m(m+1)}$ for all $m\not = n$?
From a set of integers $\{1,...,100\}$, $k$ integers were deleted. Is it always possible to choose $k$ distinct integers from the remaining set such that their sum is $100$ if
[b](a) $k=9$?[/b]
[b](b) $k=8$?[/b]
Which is larger: $2^{\sqrt{2}}$; $2^{1+\frac{1}{\sqrt{2}}}$ and 3 ?
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?
Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]
Let $X$ be an $n$-element set and let $A_1,\ldots ,A_m$ be subsets of $X$ such that
i) $|A_i|=3$ for each $i=1,\ldots ,m$.
ii) $|A_i\cap A_j|\le 1$ for any two distinct indices $i,j$.
Show that there exists a subset of $X$ with at least $\lfloor\sqrt{2n}\rfloor$ elements which does not contain any of the $A_i$’s.
Call integer number $x$ as far from squares and cubes, if for every integer $k$ it is true : $|x-k^2|>10^6,|x-k^3|>10^6$.
Prove, that there are infinitely many far from squares and cubes degrees of $2$
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that
\[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]
How many ways are there to arrange the letters $A,A,A,H,H$ in a row so that the sequence $HA$ appears at least once?
[i]Author: Ray Li[/i]
In an exam, $6$ problems were proposed. Every problem was solved by exactly $1000$ students, but in no case has it happened that two students together have solved the $6$ problems. Determine the smallest number of participants that could have been in said exam.
[hide=original wording]En un examen fueron propuestos 6 problemas. Cada problema fue resuelto por exactamente 1000 estudiantes, pero en ningun caso ha ocurrido que dos estudiantes en conjunto, hayan resuelto los 6 problemas.
Determinar el menor numero de participantes que pudo haber en dicho exame[/hide]
On the numerical line, arrange a system of closed segments of length $1$ without common points (endpoints included) so that any infinite arithmetic progression with any non zero difference and any first term has a common point with a segment of the system.
On a side of the triangle, take four points $K$, $P$, $H$ and $M$, which are respectively the midpoint of this side, the foot of the bisector with the opposite angle of the triangle, the touchpoint of this side of the circle inscribed in the triangle and the foot of the corresponding altitude. Find $KH$ if $KP = a$, $KM =b$.
Let $n$ be an even positive integer. A sequence of $n$ real numbers is called complete if for every integer $m$ with $1 \leq m \leq n$ either the sum of the first $m$ terms of the sum or the sum of the last $m$ terms is integral. Determine
the minimum number of integers in a complete sequence of $n$ numbers.
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.
Let $\triangle ABC$ be a triangle such that $AB=6, BC=8,$ and $AC=10$. Let $M$ be the midpoint of $BC$. Circle $\omega$ passes through $A$ and is tangent to $BC$ at $M$. Suppose $\omega$ intersects segments $AB$ and $AC$ again at points $X$ and $Y$, respectively. If the area of $AXY$ can be expressed as $\frac{p}{q}$ where $p, q$ are relatively prime integers, compute $p+q$.
The polynomial $P (x)$ is such that the polynomials $P (P (x))$ and $P (P (P (x)))$ are strictly monotone on the whole real axis. Prove that $P (x)$ is also strictly monotone on the whole real axis.
Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
The sides of $\triangle ABC$ form an arithmetic sequence of integers. Incircle $I$ is tangent to $AB$, $BC$, and $CA$ at $D$, $E$, and $F$, respectively. Given that $DB = \tfrac32$, $FA = \tfrac12$, find the radius of $I$.
$\textbf{(A) } \dfrac12\qquad\textbf{(B) } \dfrac{\sqrt{15}}7\qquad\textbf{(C) } \dfrac{\sqrt{15}}6\qquad\textbf{(D) } \dfrac{2\sqrt{15}}{9}\qquad\textbf{(E) } \dfrac{\sqrt{15}}{4}$
Find all pairs of positive integers $(a,b)$ such that $a^2 + b^2$ divides both $a^3 + 1$ and $b^3 + 1$.
For a given positive integer $n >2$, let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that
$C_{1}\cap C_{2}, C_{2}\cap C_{3},C_{1}\cap C_{3}$ are finite. Find the maximum number of points of the sets $C_{1}\cap C_{2}\cap C_{3}$.