Found problems: 85335
Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and
\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\]
For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?
In triangle $ABC$, the side $BC = a$ and the radius $r$ of the circle tangent to the side BC and the extensions of $AB$ and $AC$ ($A$-excircle) are known. It is also known that inside the triangle there is a point $M$ such that $$BC - AM = CA - BM = AB - CM$$ Find the radius of the circle inscribed in the triangle $BMC$.
Let $a_0, a_1, a_2, ...$ be a sequence such that: $a_0 = 2$; each $a_n = 2$ or $3; a_n =$the number of $3$s between the $n$th and $n+1$th $2$ in the sequence. So the sequence starts: $233233323332332 ...$ . Show that we can find $\alpha$ such that $a_n = 2$ iff $n = [\alpha m]$ for some integer $m \geq 0$.
Let $ a\plus{}ar_1\plus{}ar_1^2\plus{}ar_1^3\plus{}\cdots$ and $ a\plus{}ar_2\plus{}ar_2^2\plus{}ar_2^3\plus{}\cdots$ be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is $ r_1$, and the sum of the second series is $ r_2$. What is $ r_1\plus{}r_2$?
$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ \frac{1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac{1\plus{}\sqrt{5}}{2}\qquad \textbf{(E)}\ 2$
For every integer $ a>1$ an infinite list of integers is constructed $ L(a)$, as follows:
[list] $ a$ is the first number in the list $ L(a)$.[/list]
[list] Given a number $ b$ in $ L(a)$, the next number in the list is $ b\plus{}c$, where $ c$ is the largest integer that divides $ b$ and is smaller than $ b$.[/list]
Find all the integers $ a>1$ such that $ 2002$ is in the list $ L(a)$.
Let $\vartriangle ABC$ be a right triangle with $AB = 4, BC = 5$, and hypotenuse $AC$. Let I be the incenter of $\vartriangle ABC$ and $E$ be the excenter of $\vartriangle ABC$ opposite $A$ (the center of the circle tangent to $BC$ and the extensions of segments $AB$ and $AC$). Suppose the circle with diameter $IE$ intersects line $AB$ beyond $B$ at $D$. If $BD =\sqrt{a}- b$, where a and b are positive integers. Find $a + b$.
$P$ is any point of the tetrahedron $ABCD$ except $D$. Show that at least one of the three distances $DA$, $DB$, $DC$ exceeds at least one of the distances $PA$, $PB$ and $PC$.
In $xyz$ space, let $l$ be the segment joining two points $(1,\ 0,\ 1)$ and $(1,\ 0,\ 2),$ and $A$ be the figure obtained by revolving $l$ around the $z$ axis. Find the volume of the solid obtained by revolving $A$ around the $x$ axis.
Note you may not use double integral.
A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color.
Marius Ghergu
Prove that $(a^2 + b^2 + c^2)^ 2 = 2 (a^4 + b^4 + c^4)$ if $a + b + c = 0$.
An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?
Consider a regular heptagon ( polygon of $7$ equal sides and angles) $ABCDEFG$ as in the figure below:-
$(a).$ Prove $\frac{1}{\sin\frac{\pi}{7}}=\frac{1}{\sin\frac{2\pi}{7}}+\frac{1}{\sin\frac{3\pi}{7}}$
$(b).$ Using $(a)$ or otherwise, show that $\frac{1}{AG}=\frac{1}{AF}+\frac{1}{AE}$
[asy]
draw(dir(360/7)..dir(2*360/7),blue);
draw(dir(2*360/7)..dir(3*360/7),blue);
draw(dir(3*360/7)..dir(4*360/7),blue);
draw(dir(4*360/7)..dir(5*360/7),blue);
draw(dir(5*360/7)..dir(6*360/7),blue);
draw(dir(6*360/7)..dir(7*360/7),blue);
draw(dir(7*360/7)..dir(360/7),blue);
draw(dir(2*360/7)..dir(4*360/7),blue);
draw(dir(4*360/7)..dir(1*360/7),blue);
label("$A$",dir(4*360/7),W);
label("$B$",dir(5*360/7),S);
label("$C$",dir(6*360/7),S);
label("$D$",dir(7*360/7),E);
label("$E$",dir(1*360/7),E);
label("$F$",dir(2*360/7),N);
label("$G$",dir(3*360/7),W);
[/asy]
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)$, $ \left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)$, $ \ldots$, $ \left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)$ of $ k$-tuples of finite sets satisfying the following two properties:
(i) for every $ i$ and every $ j \neq j^{\prime}$, $ A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset$, and
(ii) for every $ i\neq i^{\prime}$ there exist $ j\neq j^{\prime}$ for which $ A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset$. Prove that
\[ \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1.
\]
There are any number of stamps with the values $134$, $135$, $...$, $142$ and $143$ cents available. Find the largest integer value (in cents) that cannot be represented by these stamps.
(G. Woeginger, TU Eindhoven, The Netherlands)
[hide=original wording]Es stehen beliebig viele Briefmarken mit den Werten 134, 135. . .., 142 und 143 Cent zur Verfügung. Man bestimme den größten ganzzahligen Wert (in Cent), der nicht durch diese Briefmarken dargestellt werden kann.[/hide]
The sides of a right triangle are $a, a+d,$ and $a+2d$, with $a$ and $d$ both positive. The ratio of $a$ to $d$ is:
$ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 1:4 \qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 3:1\qquad\textbf{(E)}\ 3:4 $
Consider a square $8 \times 8$ board with its $64$ cells initially white. Determine the minimum number of colors needed to color the cells (each one with only one color) in such a way that if four cells on the board can be covered by an $L$-shaped tile as shown in the figure, then the four cells are of different colors.
[asy]
size(3cm);
draw((1,0)--(1,1)--(2,1)--(2,0)--(1,0)--(0,0)--(0,1)--(0,2)--(1,2)--(1,1)--(0,1)--(1,1)--(2,1)--(3,1)--(3,0)--(2,0));
[/asy]
[b]Note:[/b] The $L$-shaped tile can be rotated or flipped.
Let $ O $ be circumcenter of triangle $ABC$. For a point $P$ on segmet $BC$, the circle passing through $ P, B $ and tangent to line $AB $ and the circle passing through $ P, C $ and tangent to line $AC $ meet at point $ Q ( \ne P ) $. Let $ D, E $ be foot of perpendicular from $Q$ to $ AB, AC$. ($D \ne B, E \ne C $) Two lines $DE $ and $ BC $ meet at point $R$. Prove that $ O, P, Q $ are collinear if and only if $ A, R, Q $ are collinear.
$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$).
Prove that $l_1 || l_2$.
From the vertex $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus.
[i]Proposed by Tommy Walker Mackay, United Kingdom[/i]
Prove the inequality $$\frac{a + b}{a^2 -ab + b^2} \le \frac{4}{ |a + b|}$$ for all real numbers $a$ and $b$ with $a + b\ne 0$. When does equality hold?
Find the value of the expression
$$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$
assuming $f(x) =\frac{x^2}{1 + x^2}$ .