Found problems: 85335
Elmo is drawing with colored chalk on a sidewalk outside. He first marks a set $S$ of $n>1$ collinear points. Then, for every unordered pair of points $\{X,Y\}$ in $S$, Elmo draws the circle with diameter $XY$ so that each pair of circles which intersect at two distinct points are drawn in different colors. Count von Count then wishes to count the number of colors Elmo used. In terms of $n$, what is the minimum number of colors Elmo could have used?
[i]Michael Ren[/i]
Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.
Let the product $(12)(15)(16)$, each factor written in base $b$, equal $3146$ in base $b$. Let $s=12+15+16$, each term expressed in base $b$. Then $s$, in base $b$, is
$\textbf{(A)}\ 43\qquad
\textbf{(B)}\ 44\qquad
\textbf{(C)}\ 45\qquad
\textbf{(D)}\ 46\qquad
\textbf{(E)}\ 47$
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$
[i]Proposed by TuZo[/i]
Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$.
[i]Proposed by Andrew Wu[/i]
Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$
Prove that for all positive real numbers $a,b,c,d$,
$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.
Given a square with area $A$. A circle lies inside the square, such that the
circle touches all sides of the square. Another square with area $B$ lies inside the
circle, such that all its vertices lie on the circle.
Find the value of $\frac{A}{B}.$
A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.
Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*}
abcd &= 2007,\\
a &= \sqrt{55 + \sqrt{k+a}},\\
b &= \sqrt{55 - \sqrt{k+b}},\\
c &= \sqrt{55 + \sqrt{k-c}},\\
d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*}
Show that among any seven coplanar unit vectors there are at least two of them such that the magnitude of their sum is greater than $ \sqrt 3. $
[i]Ion Tecu[/i] and [i]Teodor Radu[/i]
Let the sequence $(a_n)_{n \in \mathbb{N}}$, where $\mathbb{N}$ denote the set of natural numbers, is given with $a_1=2$ and $a_{n+1}$ $=$ $a_n^2$ $-$ $a_n+1$. Find the minimum real number $L$, such that for every $k$ $\in$ $\mathbb{N}$
\begin{align*} \sum_{i=1}^k \frac{1}{a_i} < L \end{align*}
Bob chose six of the nine digits from $1$ to $9$ and wrote the list, ordered from smallest to largest, of all three-digit numbers that can be formed using the digits you chose. At Bob's list, the number $317$ appears at position $22$. What number appears at position $60$ in the list from Bob? Find all possibilities.
Acute scalene triangle $\triangle{}ABC$ has circumcenter $O$ and orthocenter $H$. Points $X$ and $Y$, distinct from $B$ and $C$, lie on the circumcircle of $\triangle{}ABC$ such that $\angle{}BXH=\angle{}CYH=90^{\circ{}}$. Show that if lines $XY$, $AH$, and $BC$ are concurrent, then $OH$ is parallel to $BC$.
Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that
\[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\]
where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).
A convex 51-gon is given. For each of its vertices and each diagonal that does not contain this vertex, we mark in red a point symmetrical to the vertex relative to the middle of the diagonal. Prove that strictly inside the polygon there are no more than 20400 red dots.
[i]Proposed by P. Kozhevnikov[/i]
Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $r_1$ is the radius of the circle passing through $A$ and $B$ touching $CD$; and similarly $r_2$ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that \[ r_1 + r_2 \geq \frac{5}{8} ( a + b) . \]
$P$ is a convex polyhedron such that:
[b](1)[/b] every vertex belongs to exactly $3$ faces.
[b](1)[/b] For every natural number $n$, there are even number of faces with $n$ vertices.
An ant walks along the edges of $P$ and forms a non-self-intersecting cycle, which divides the faces of this polyhedron into two sides, such that for every natural number $n$, the number of faces with $n$ vertices on each side are the same. (assume this is possible)
Show that the number of times the ant turns left is the same as the number of times the ant turn right.
A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$. Find the perimeter of this rectangle.
Solve in the set $R$ the equation
$$2 \cdot [x] \cdot \{x\} = x^2 - \frac32 \cdot x - \frac{11}{16}$$
where $[x]$ and $\{x\}$ represent the integer part and the fractional part of the real number $x$, respectively.
In each unit cell of a finite set of cells of an infinite checkered board, an integer is written so that the sum of the numbers in each row, as well as in each column, is divided by $2002$. Prove that every number $\alpha$ can be replaced by a certain number $\alpha'$ , divisible by $2002$ so that $|\alpha-\alpha'| <2002$ and the sum of the numbers in all rows, and in all columns will not change.
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]
Let $n \geq 3$ be an integer. Let $a_1,a_2,\dots, a_n\in[0,1]$ satisfy $a_1 + a_2 + \cdots + a_n = 2$. Prove that
$$\sqrt{1-\sqrt{a_1}}+\sqrt{1-\sqrt{a_2}}+\cdots+\sqrt{1-\sqrt{a_n}}\leq n-3+\sqrt{9-3\sqrt{6}}.$$
Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$.
(V.Yassinsky)