Found problems: 85335
Bill has the expression $1+2+3+\cdots+8.$ He replaces two different addition symbols with multiplication symbols uniformly at random. The value that he obtains on average can be expressed as a common fraction $\tfrac{m}{n}.$ Find $m+n.$
Let x,y,z be positive real numbers such that $x+y+z=1$.
For positive integer n, define $S_n = x^n+y^n+z^n$
Furthermore, let $P=S_2 S_{2005}$ and $Q=S_3 S_{2004}$.
(a) Find the smallest possible value of Q.
(b) If $x,y,z$ are pairwise distinct, determine whether P or Q is larger.
in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.
a) in what ratio does $EF$ cut the digonals?(13 points)
b) find $\frac{AF}{FD}$.(5 points)
A point $P$ lies inside an equilateral triangle $ABC$ such that $AP=15$ and $BP=8$. Find the maximum possible value of the sum of areas of triangles $ABP$ and $BCP$.
Let $F_0 = 0, F_1 = 1$ and $F_{n+1} = F_n + F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence.
Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that
$F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .
Given a semicircle of diameter $ AB $, with $ AB = 2r $, be $ CD $ a variable string, but of fixed length $ c $. Let $ E $ be the intersection point of lines $ AC $ and $ BD $, and let $ F $ be the intersection point of lines $ AD $ and $ BC $.
a) Prove that the lines $ EF $ and $ AB $ are perpendicular.
b) Determine the locus of the point $ E $.
c) Prove that $ EF $ has a constant measure, and determine it based on $ c $ and $ r $.
Show that for any positive real numbers $a$, $b$, $c$ the following inequality takes place
$$\frac{a}{\sqrt{7a^2+b^2+c^2}}+\frac{b}{\sqrt{a^2+7b^2+c^2}}+\frac{c}{\sqrt{a^2+b^2+7c^2}} \leq 1.$$
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
$2$ identical red tokens and $2$ identical black tokens are placed on distinct cells of a $5\times5$ grid. Suppose it is impossible to color some additional cells of the grid red or black such that there exists a red path between the red tokens and a black path between the black tokens. Find the number of possible arrangements of the tokens on the grid.
(A red path is a path of edge adjacent red cells, and same for a black path.)
An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position($12$ o'clock, etc.) but only every $69$ minutes. At time and one-half for overtime, the extra pay to which a $\textdollar 4.00$ per hour worker should be entitled after working a normal $8$ hour day by that slow running clock, is
$\textbf{(A) }\textdollar 2.30\qquad\textbf{(B) }\textdollar 2.60\qquad\textbf{(C) }\textdollar 2.80\qquad\textbf{(D) }\textdollar 3.00\qquad \textbf{(E) }\textdollar 3.30$
How many ordered integer pairs of $ (a,b)$ satisfying $ a^2b \plus{} ab^2 \equal{} 2009201020092010$ ?
$\textbf{(A)} 4 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$
The altitudes of an acute triangle $ABC$ with $AB<AC$ intersect at a point $H$, and $O$ is the center of the circumcircle $\Omega$. The segment $OH$ intersects the circumcircle of $BHC$ at a point $X$, different from $O$ and $H$. The circumcircle of $AOX$ intersects the smaller arc $AB$ of $\Omega$ at point $Y$. Prove that the line $XY$ bisects the segment $BC$.
[i]Proposed by A. Tereshin[/i]
For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
Suppose that each of $20$ students has made a choice of anywhere from $0$ to $6$ courses from a total of $6$ courses offered. Prove or disprove : there are $5$ students and $2$ courses such that all $5$ have chosen both courses or all $5$ have chosen neither course.
Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$?
[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 25;
draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle);
draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle);
label("$A$",dir(135),NW);
label("$B$",dir(45),NE);
label("$C$",dir(315),SE);
label("$D$",dir(225),SW);
label("$E$",dir(135-r),N);
label("$F$",dir(45-r),E);
label("$G$",dir(315-r),S);
label("$H$",dir(225-r),W);
[/asy]
$\textbf{(A) }20^\circ\qquad\textbf{(B) }30^\circ\qquad\textbf{(C) }32^\circ\qquad\textbf{(D) }35^\circ\qquad\textbf{(E) }45^\circ$
Let $BM$ be the median of triangle $ABC$. On the extension of $MB$ beyond $B$, the point $K$ is chosen so that $BK =\frac12 AC$. Prove that if $\angle AMB=60^o$, then $AK=BC$.
(Mykhailo Standenko)
A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?
Given a circle, construct a chord that is trisected by two given noncollinear radii.
Find the largest value of the real number $ \lambda$, such that as long as point $ P$ lies in the acute triangle $ ABC$ satisfying $ \angle{PAB}\equal{}\angle{PBC}\equal{}\angle{PCA}$, and rays $ AP$, $ BP$, $ CP$ intersect the circumcircle of triangles $ PBC$, $ PCA$, $ PAB$ at points $ A_1$, $ B_1$, $ C_1$ respectively, then $ S_{A_1BC}\plus{} S_{B_1CA}\plus{} S_{C_1AB} \geq \lambda S_{ABC}$.
Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.
Alice and Bob are running around a rectangular building measuring 100 by 200 meters. They start at the middle of a 200 meter side and run in the same direction, Alice running twice as fast as Bob.
After Bob runs one lap around the building, what fraction of the time were Alice and Bob on the same side of the building?
Writing $1989$ in base $b$, we obtain a three-digit number: $xyz$. It is known that the sum of the digits is the same in base $10$ and in base $b$, that is, $1 + 9 + 8 + 9 = x + y + z$. Determine $x,y,z,b.$
Clarabelle wants to travel from $(0,0)$ to $(6,2)$ in the coordinate plane. She is able to move in one-unit steps up, down, or right, must stay between $y=0$ and $y=2$ (inclusive), and is not allowed to visit the same point twice. How many paths can she take?
[i]Proposed by Connor Gordon[/i]
The sides $AB$, $BC$, $CD$ and $DA$ of the quadrilateral $ABCD$ are respectively equal to the sides $A'B'$, $B'C'$, $C'D' $ and $D'A'$ of the quadrilateral $A'B'CD$' and it is known that $AB \parallel CD$ and $B'C' \parallel D'A'$. Prove that both quadrilaterals are parallelograms.
(V Proizvolov, Moscow)