Found problems: 85335
Let $S^{2}\subset \mathbb{R}^{3}$ be the unit sphere. Show that for any $n$ points on $ S^{2}$, the sum of the squares of the $\frac{n(n-1)}{2}$ distances between them is at most $n^{2}$.
The first AMC $8$ was given in $1985$ and it has been given annually since that time. Samantha turned $12$ years old the year that she took the seventh AMC $8$. In what year was Samantha born?
$\textbf{(A) }1979\qquad\textbf{(B) }1980\qquad\textbf{(C) }1981\qquad\textbf{(D) }1982\qquad \textbf{(E) }1983$
$k$ stones are put into $2012$ boxes in such a way that each box has at most $20$ stones. We are chosing some of the boxes. We are throwing some of the stones of the chosen boxes. Whatever the first arrangement of the stones inside the boxes is, if we can guarantee that there are equal stones inside the chosen boxes and the sum of them is at least $100$, then $k$ can be at least?
$ \textbf{(A)}\ 500 \qquad \textbf{(B)}\ 450 \qquad \textbf{(C)}\ 420 \qquad \textbf{(D)}\ 349 \qquad \textbf{(E)}\ 296$
Let $ABC$ be an isosceles triangle with $BC=CA$, and let $D$ be a point inside side $AB$ such that $AD< DB$. Let $P$ and $Q$ be two points inside sides $BC$ and $CA$, respectively, such that $\angle DPB = \angle DQA = 90^{\circ}$. Let the perpendicular bisector of $PQ$ meet line segment $CQ$ at $E$, and let the circumcircles of triangles $ABC$ and $CPQ$ meet again at point $F$, different from $C$.
Suppose that $P$, $E$, $F$ are collinear. Prove that $\angle ACB = 90^{\circ}$.
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that
\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
Consider an isosceles triangle $ABC$ with $AB=AC$, and a circle $\omega$ which is tangent to the sides $AB$ and $AC$ of this triangle and intersects the side $BC$ at the points $K$ and $L$. The segment $AK$ intersects the circle $\omega$ at a point $M$ (apart from $K$). Let $P$ and $Q$ be the reflections of the point $K$ in the points $B$ and $C$, respectively. Show that the circumcircle of triangle $PMQ$ is tangent to the circle $\omega$.
In an acute triangle $ABC$, the altitude from $C$ intersects $AB$ at $E$ and the altitude from $B$ intersects $AC$ at $D$. $CE$ and $BD$ intersect at a point $H$. A circle with diameter $DE$ intersects $AB$ and $AC$ at points $F,G$ respectively. $FG$ and $AH$ intersect at $K$. If $\overline{BC} = 25$, $\overline{BD} = 20$, and $\overline{BE} = 7$, the length of $AK$ is of the form $\frac{p}{q}$ , where $p, q$ are relatively prime positive integers. Find $p + q$.
Peter wants to make an unusual die having different positive integers on each of its faces. For neighbouring faces the corresponding numbers should differ by at least two. Find the minimal sum of the six numbers.
(Folklore)
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $P$, and there exist a circle $\Gamma$ tangent to the extensions of $AB,BC,AD,DC$ at $X,Y,Z,T$ respectively. Circle $\Omega$ passes through points $A,B$, and is externally tangent to circle $\Gamma$ at $S$. Prove that $SP\perp ST$.
Prove that however we choose the majority of numbers among an even number of the first consecutive natural numbers, there will be two numbers among this choosing whose sum is a prime.
How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }1 \qquad
\textbf{(C) }2 \qquad
\textbf{(D) }3 \qquad
\textbf{(E) }4 \qquad
$
Let $ f$ be a polynomial with positive integer coefficients. Prove that if $ n$ is a positive integer, then $ f(n)$ divides $ f(f(n)\plus{}1)$ if and only if $ n\equal{}1.$
What is the maximum number of integers that can be chosen from $1,2,\dots,99$ so that the chosen integers can be arranged in a circle with the property that the product of every pair of neighbouring integers is 3-digit number?
Let be three complex numbers $ z,t,u, $ whose affixes in the complex plane form a triangle $ \triangle . $
[b]a)[/b] Let be three non-complex numbers $ a,b,c $ that sum up to $ 0. $ Prove that
$$ |az+bt+cu|=|at+bu+cz|=|au+bz+ct| $$
if $ \triangle $ is equilateral.
[b]b)[/b] Show that $ \triangle $ is equilateral if
$$ |z+2t-3u|=|t+2u-3z|=|u+2z-3t| . $$
Mary's top book shelf holds five books with the following widths, in centimeters: $ 6$, $ \frac12$, $ 1$, $ 2.5$, and $ 10$. What is the average book width, in centimeters?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
Let $ABCD$ be a cyclic quadrilateral and $\omega_1, \omega_2$ the incircles of triangles $ABC$ and $BCD$. Show that the common external tangent line of $\omega_1$ and $\omega_2$, the other one than $BC$, is parallel with $AD$
Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series
$$\sum_{i=1}^{\infty} \frac{1}{a_i }$$
be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show
that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$
A nonempty word is called pronounceable if it alternates in vowels (A, E, I, O, U) and consonants (all other letters) and it has at least one vowel. How many pronounceable words can be formed using the letters P, U, M, A, C at most once each? Words of length shorter than $5$ are allowed.
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.
The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]
Alex and Bob play a game 2015 x 2015 checkered board by the following rules.Initially the board is empty: the players move in turn, Alex moves first. By a move, a player puts either red or blue token into any unoccopied square. If after a player's move there appears a row of three consecutive tokens of the same color( this row may be vertical,horizontal, or dioganal), then this player wins. If all the cells are occupied by tokens, but no such row appears, then a draw is declared.Determine whether Alex, Bob, or none of them has winning strategy.
A circle and a square overlap such that the overlapping area is $50\%$ of the area of the circle, and is $25\%$ of the area of the square, as shown in the figure. Find the ratio of the area of the square outside the circle to the area of the whole figure.
[img]https://cdn.artofproblemsolving.com/attachments/e/2/c209a95f457dbf3c46f66f82c0a45cc4b5c1c8.png[/img]
Call a number $T$ [i]persistent[/i] if the following holds: Whenever $a,b,c,d$ are real numbers different from $0$ and $1$ such that
$$a+b+c+d = T$$
and
$$\frac{1}{a}+\frac{1}{b} +\frac{1}{c}+\frac{1}{d} = T,$$
we also have
$$\frac{1}{1 - a}+\frac{1}{1-b}+\frac{1}{1-c}+\frac{1}{1-d}= T.$$
(a) If $T$ is persistent, prove that $T$ must be equal to $2$.
(b) Prove that $2$ is persistent.
Note: alternatively, we can just ask “Show that there exists a unique persistent number, and determine its value”.
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Let $ \sigma(S_n,k)$ denote the sum of the $ k$th powers of the lengths of the sides of the convex $ n$-gon $ S_n$ inscribed in a unit circle. Show that for any natural number greater than $ 2$ there exists a real number $ k_0$ between $ 1$ and $ 2$ such that $ \sigma(S_n,k_0)$ attains its maximum for the regular $ n$-gon.
[i]L. Fejes Toth[/i]