Found problems: 85335
Let $ABC$ be an acute triangle. Let $I$ be a point inside the triangle and let $D$ be a point on the line $AB$. The line through $D$ which is parallel to $AI$ intersects the line $AC$ at the point $E$, and the line through $D$ parallel to $BI$ intersects the line $BC$ in point $F$. prove that
$$\frac{EF \cdot CI}{2} \ge area (\vartriangle ABC) $$
Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal.
Proposed by [i]Sena Başaran[/i]
$\log_8 2 = 0.2525$ in base $8$ (to $4$ places of decimals). Find $\log_8 4$ in base $8$ (to $4$ places of decimals).
A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$, there are only finitely many values of $j$ such that $A_j\subseteq A_i$. Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$.
Let $ABC$ be a triangle with $AB < AC < BC$ and $\Gamma$ its circumcircle. Let $\omega_1$ be the circle with center $B$ and radius $AC$ and $\omega_2$ the circle with center $C$ and radius $AB$. The circles $\omega_1$ and $\omega_2$ intersect at point $E$ such that $A$ and $E$ are on opposite sides of the line $BC$. The circles $\Gamma$ and $\omega_1$ intersect at point $F$ and the circles $\Gamma$ and $\omega_2$ intersect at point $G$ such that the points $F$ and $G$ are on the same side as $E$ in relation to the line $BC$. With $K$ being the point such that $AK$ is a diameter of $\Gamma$, prove that $K$ is circumcenter of triangle $EFG$.
Let $ x_1, x_2, \cdots, x_n$ be positive real numbers . Prove that:
$$\sum_ {i = 1}^n x_i ^2\geq \frac {1} {n + 1} \left (\sum_ {i = 1}^n x_i \right)^2+\frac{12(\sum_ {i = 1}^n i x_i)^2}{n (n + 1) (n + 2) (3n + 1)}. $$
Inside the triangle $A_1A_2A_3$ with sides $a_1$, $a_2$, $a_3$, three points are given, which we label $P_1$, $P_2$, $P_3$ so that the product of their distances from the corresponding sides $a_1$, $a_2$, $a_3$ is as large as possible. Prove that the triangles $P_1A_2A_3$, $A_1P_2A_3$, $A_1A_2P_3$ cover the triangle.
[hide=original wording]V trojúhelníku A1A2A3 se stranami a1, a2, a3 jsou dány tři body, které označíme Pi, P2, P3 tak, aby součin jejich vzdáleností od odpovídajících stran a1, a2, a3 byl co největší. Dokažte, že trojúhelníky P1A2A3, A1P2A3, A1A2P3 pokrývají trojúhelník.[/quote]
Consider the natural implementation of computing Fibonacci numbers:
\begin{tabular}{l}
1: \textbf{FUNCTION} $\text{FIB}(n)$: \\
2:$\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1 \\
3:$\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$
\end{tabular}
When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur?
The sequence $(a_n)$ is given by $a_1=2$, $a_2=500$, $a_3=2000$ and
$$\frac{a_{n+2}+a_{n+1}}{a_{n+1}+a_{n-1}}=\frac{a_{n+1}}{a_{n-1}}\qquad\text{for }n\ge2$$Prove that all terms of this sequence are positive integers and that $a_{2000}$ is divisible by $2^{2000}$.
Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.
Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.
Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.
The diagnols $\overline{AC}$ and $\overline{BD}$ of a quaderilateral $ABCD$ meet at $O$. Let $s_1$ be the area of $\triangle{AOB}$ and $s_2$ be the area of $\triangle{OCD}$. Then show that $$\sqrt{s_1}+\sqrt{s_2} \leq \sqrt{s}$$ Also find a geometrical condition for equality to hold (By geometrical condition we mean something like parallel lines, perpendicular lines,bisecting lines etc.)
Consider a rectangle $ABCD$ with $AB = 2$ and $BC = \sqrt3$. The point $M$ lies on the side $AD$ so that $MD = 2 AM$ and the point $N$ is the midpoint of the segment $AB$. On the plane of the rectangle rises the perpendicular MP and we choose the point $Q$ on the segment $MP$ such that the measure of the angle between the planes $(MPC)$ and $(NPC)$ shall be $45^o$, and the measure of the angle between the planes $(MPC)$ and $(QNC)$ shall be $60^o$.
a) Show that the lines $DN$ and $CM$ are perpendicular.
b) Show that the point $Q$ is the midpoint of the segment $MP$.
$a$ is irrational , but $a$ and $a^3-6a$ are roots of square polynomial with integer coefficients.Find $a$
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.
Prove that the sets
$$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , $$
$$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, $$
are not homeomorphic on the Euclidean topology induced on them.
Suppose that
\[ \frac {2x}{3} \minus{} \frac {x}{6}
\]is an integer. Which of the following statements must be true about $ x$?
$ \textbf{(A)}\ \text{It is negative.} \qquad \textbf{(B)}\ \text{It is even, but not necessarily a multiple of }3\text{.}$
$ \textbf{(C)}\ \text{It is a multiple of }3\text{, but not necessarily even.}$
$ \textbf{(D)}\ \text{It is a multiple of }6\text{, but not necessarily a multiple of }12\text{.}$
$ \textbf{(E)}\ \text{It is a multiple of }12\text{.}$
An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$).
Prove that there is a circle of radius $\frac{1}{4}$ containing the path.
Illustration of an example path:
[b]4.[/b] Let $F_{\epsilon} (0<\epsilon<1)$ denote the class of non-negative piecewise continuous functions defined on $[0,\infty)$ which satisfy the following condition: $f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)$. Find the value of
$s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx$
[b](R. 5)[/b]
Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$, ${f_1}(x) = 3x$, and
${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$
Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$.
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit and the tens digit is equal to the units digit and which are squares of integers.
A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?
$\textbf{(A)} ~77\qquad\textbf{(B)} ~86\qquad\textbf{(C)} ~92\qquad\textbf{(D)} ~98\qquad\textbf{(E)} ~104\qquad$