Found problems: 85335
The names of $8$ people are written on slips of paper and placed in a hat. Each of the $8$ people then randomly draw a piece of paper (without replacement). Then, the people are formed into groups satisfying the following requirements:
(i) Each person is in the same group as the person who drew his piece of paper.
(ii)There are as many groups as possible while still satisfying condition (i).
On average, how many groups will there be? (There might be “groups” of only one person.)
Let D be an arbitrary point inside $\Delta ABC$. Let $\Gamma$ be the circumcircle of $\Delta BCD$. The external angle bisector of $\angle ABC$ meets $\Gamma$ again at $E$. The external angle bisector of $\angle ACB$ meets $\Gamma$ again at $F$. The line $EF$ meets the extension of $AB$ and $AC$ at $P$ and $Q$ respectively. Prove that the circumcircles of $\Delta BFP$ and $\Delta CEQ$ always pass through the same fixed point regardless of the position of $D$. (Assume all the labelled points are distinct.)
The sequence $ (a_n)$ satisfies $ a_1 \equal{} 1$ and $ \displaystyle 5^{(a_{n\plus{}1}\minus{}a_n)} \minus{} 1 \equal{} \frac{1}{n\plus{}\frac{2}{3}}$ for $ n \geq 1$. Let $ k$ be the least integer greater than $ 1$ for which $ a_k$ is an integer. Find $ k$.
Let $a, b, c$ be positive real numbers. Prove that
$$\frac{a^3}{a^2 + bc}+\frac{b^3}{b^2 + ca}+\frac{c^3}{c^2 + ab} \ge \frac{(a^2 + b^2 + c^2)(ab + bc + ca)}{a^3 + b^3 + c^3 + 3abc}$$
Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds?
Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]
The triangle $ABC$ satisfies $AB=10$ and has angles $\angle{A}=75^{\circ}$, $\angle{B}=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the area of triangle $EOD$ can be written as $\tfrac{a\sqrt{b}}{c}$ for square-free $b$ and coprime $a,c$. Find the value of $a+b+c$.
Let $ABC$ be a triangle and $H$ be its orthocenter. If it is given that $B$ is $(0,0)$, $C$ is $(1,2)$ and $H$ is $(5,0)$, find $A$.
Let $k$ be a fixed even positive integer, $N$ is the product of $k$ distinct primes $p_1,...,p_k$, $a,b$ are two positive integers, $a,b\leq N$. Denote
$S_1=\{d|$ $d|N, a\leq d\leq b, d$ has even number of prime factors$\}$,
$S_2=\{d|$ $d|N, a\leq d\leq b, d$ has odd number of prime factors$\}$,
Prove: $|S_1|-|S_2|\leq C^{\frac{k}{2}}_k$
Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $$
\varepsilon_{i+1}=(\delta_i-a_{i+1})(\delta_i-b_{i+1}), \quad \delta_{i+1}=\delta_i+(-1)^{\delta_i}\varepsilon_{i+1} \quad (0 \leq i \leq n-1).
$$. Let $w(a,b)=\varepsilon_0+\varepsilon_1+\varepsilon_2+\dots +\varepsilon_n$ . Find $f(n)=\sum\limits_{a,b \in {B_n}} {w(a,b)} $.
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The sequence of real numbers $\{a_n\}$, $n \in \mathbb{N}$ satisfies the following condition: $a_{n+1}=a_n(a_n+2)$ for any $n \in \mathbb{N}$. Find all possible values for $a_{2004}$.
Let $C$ be a circle with two diameters intersecting at an angle of $30$ degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius $1$. Find the largest possible radius of $C$.
In a circle with radius $R$, there is inscribed a quadrilateral with perpendicular diagonals. From the intersection point of the diagonals, there are perpendiculars drawn to the sides of the quadrilateral.
(a) Prove that the feet of these perpendiculars $P_1,P_2,P_3,P_4$ are vertices of the quadrilateral that is inscribed and circumscribed.
(b) Prove the inequalities $2r_1\le\sqrt2 R_1\le R$ where $R_1$ and $r_1$ are radii respectively of the circumcircle and inscircle to the quadrilateral $P_1P_2P_3P_4$. When does equality hold?
[i]H. Lesov[/i]
Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$?
[i]Proposed by Pavel Ciurea[/i]
Calculate $ \int \frac {dx}{x^{2008}(1 \minus{} x)}$
Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$?
[i]J. Szucs[/i]
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.
Consider an $m \times n$ grid of unit squares. Let $R$ be the total number of rectangles of any size, and let $S$ be the total number of squares of any size. Assume that the sides of the rectangles and squares are parallel to the sides of the $m \times n$ grid. If $\frac{R}{S} =\frac{759}{50}$ , then determine $mn$.
Calculate $1^2-2^2+3^2-4^2+...-2018^2+2019^2$.
In a coordinate system there are drawn the graphs of the functions $y=ax+b$ and $y=bx+a, (a\neq b)$. Their intersection is marked with red and their intersections with the $Oy$ axis are marked with blue. Everything is erased except the marked points. Using only a ruler and a compass, find the origin of the coordinate system.
Let $n$, $m$ and $k$ be positive integers satisfying $(n-1)n(n+1)=m^k.$ Prove that $k=1.$
If $ a$ and $ b$ are nonnegative real numbers with $ a^2\plus{}b^2\equal{}4$, show that: $ \frac{ab}{a\plus{}b\plus{}2} \le \sqrt{2}\minus{}1$ and determine when equality occurs.
Define a function $P(n)$ from the set of positive integers to itself, where $P(1)=1$ and if an integer $n > 1$ has prime factorization $$n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$$
then $$P(n) = a_1^{p_1}a_2^{p_2} \dots a_k^{p_k}.$$
Prove that $P(P(n)) \le n$ for all positive integers $n.$
[i]Proposed by Evan Chang (squareman), USA[/i]
The function $ f$ has the property that for each real number $ x$ in its domain, $ 1/x$ is also in its domain and
\[ f(x) \plus{} f\left(\frac {1}{x}\right) \equal{} x.
\]What is the largest set of real numbers that can be in the domain of $ f$?
$ \textbf{(A) } \{ x | x\ne 0\} \qquad \textbf{(B) } \{ x | x < 0\} \qquad \textbf{(C) }\{ x | x > 0\}\\
\textbf{(D) } \{ x | x\ne \minus{} 1 \text{ and } x\ne 0 \text{ and } x\ne 1\} \qquad \textbf{(E) } \{ \minus{} 1,1\}$
How many of the fi rst $1000$ positive integers can be written as the sum of fi nitely many distinct numbers from the sequence $3^0$, $3^1$, $3^2$ ,$...$?