Found problems: 85335
Let $D{}$ be a point inside the triangle $ABC$. Let $E{}$ and $F{}$ be the projections of $D{}$ onto $AB$ and $AC$, respectively. The lines $BD$ and $CD$ intersect the circumcircle of $ABC$ the second time at $M{}$ and $N{}$, respectively. Prove that \[\frac{EF}{MN}\geqslant \frac{r}{R},\]where $r{}$ and $R{}$ are the inradius and circumradius of $ABC$, respectively.
Let $n \in \mathbb N$, $n \geq 2$.
(a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \]
(b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \]
[i]Ion Savu[/i]
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.
Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.
[i]Proposed by Iman Maghsoudi[/i]
Let $n$ be a positive integer and let $\triangle$ be the closed triangular domain with vertices at the lattice points $(0, 0), (n, 0)$ and $(0, n)$. Determine the maximal cardinality a set $S$ of lattice points in $\triangle$ may have, if the line through every pair of distinct points in $S$ is parallel to no side of $\triangle$.
For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$
Compute $\displaystyle\sum_{k=1}^\infty \dfrac{k^4}{k!}$.
The functions $f(x)-x$ and $f(x^2)-x^6$ are defined for all positive $x$ and increase. Prove that the function $f(x^3) -\frac{\sqrt3}{2} x^6$ also increases for all positive $x$.
Jar $A$ and Jar $B$ each contain $10$ beans. The number of beans in jar $A$ is doubled, and the number of beans in jar $B$ is halved. How many beans are now in jars $A$ and $B$?
$\textbf{(A) }15\qquad\textbf{(B) }20\qquad\textbf{(C) }25\qquad\textbf{(D) }30\qquad\textbf{(E) }40$
As shown in the figure, it is known that $BC= AC$ in $\vartriangle ABC$, $M$ is the midpoint of $AB$, points $D$, $E$ lie on $AB$ such that $\angle DCE= \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F $(different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$, $O_2$ respectively. Prove that $O_1O_2 \perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/8/c/62d4ecbc18458fb4f2bf88258d5024cddbc3b0.jpg[/img]
In how many ways can 1992 be expressed as the sum of one or more consecutive integers?
Liam writes the number $0$ on a board, then performs a series of turns. Each turn, he chooses a nonzero integer so that for every nonzero integer $N,$ he chooses $N$ with $3^{- |N|}$ probability. He adds his chosen integer $N$ to the last number written on the board, yielding a new number. He writes the new number on the board and uses it for the next turn. Liam repeats the process until either $8$ or $9$ is written on the board, at which point he stops. Given that Liam eventually stopped, find the probability the last number he wrote on the board was $9.$
Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^y=9$, compute $xy$.
There are four lines in the plane, no three concurrent, no two parallel, and no three forming an equilateral triangle. If one of them is parallel to the Euler line of the triangle formed by the other three lines, prove that a similar statement holds for each of the other lines.
Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?
$\textbf{(A) }90\qquad\textbf{(B) }105\qquad\textbf{(C) }120\qquad\textbf{(D) }135\qquad \textbf{(E) }150$
In the $xyz$ space, consider a right circular cylinder with radius of base 2, altitude 4 such that
\[\left\{
\begin{array}{ll}
x^2+y^2\leq 4 &\quad \\
0\leq z\leq 4 &\quad
\end{array}
\right.\]
Let $V$ be the solid formed by the points $(x,\ y,\ z)$ in the circular cylinder satisfying
\[\left\{
\begin{array}{ll}
z\leq (x-2)^2 &\quad \\
z\leq y^2 &\quad
\end{array}
\right.\]
Find the volume of the solid $V$.
Find all four-element sets of positive integers $\{w,x,y,z\}$ such that $w^x+w^y=w^z$.
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_0=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n$. Define
\[
q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j},
\]
and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n$. Evaluate $M(2020)$.
Find all polynomial functions with real coefficients for which $$(x-2)P(x+2)+(x+2)P(x-2)=2xP(x)$$ for all real $x$
In a plane are given n points $P_i \ (i = 1, 2, \ldots , n)$ and two angles $\alpha$ and $\beta$. Over each of the segments $P_iP_{i+1} \ (P_{n+1} = P_1)$ a point $Q_i$ is constructed such that for all $i$:
[b](i)[/b] upon moving from $P_i$ to $P_{i+1}, Q_i$ is seen on the same side of $P_iP_{i+1}$,
[b](ii)[/b] $\angle P_{i+1}P_iQ_i = \alpha,$
[b](iii)[/b] $\angle P_iP_{i+1}Q_i = \beta.$
Furthermore, let $g$ be a line in the same plane with the property that all the points $P_i,Q_i$ lie on the same side of $g$. Prove that
\[\sum_{i=1}^n d(P_i, g)= \sum_{i=1}^n d(Q_i, g).\]
where $d(M,g)$ denotes the distance from point $M$ to line $g.$
Let $n \geq 3$ and $k \geq 2$ be integers, and form the forward differences of the members of the sequence
$1,n,n^2,...n^{k-1}$
and successive forward differences thereof, as illustrated on the right for case $(n,k) = (3,5)$. Prove that all entries of the resulting triangles of positive integers are distinct from one another.
Diagram:
http://www.cms.math.ca/Competitions/IMTS/imts5.html
For any natural number $n$, consider a $1\times n$ rectangular board made up of $n$ unit squares. This is covered by $3$ types of tiles : $1\times 1$ red tile, $1\times 1$ green tile and $1\times 2$ domino. (For example, we can have $5$ types of tiling when $n=2$ : red-red ; red-green ; green-red ; green-green ; and blue.) Let $t_n$ denote the number of ways of covering $1\times n$ rectangular board by these $3$ types of tiles. Prove that, $t_n$ divides $t_{2n+1}$.
Solve equation $(3a-bc)(3b-ac)(3c-ab)=1000$ in integers. (V.Brayman)
Find the least positive integer $k$ so that $k + 25973$ is a palindrome (a number which reads the same forward and backwards).
In the acute triangle $ABC$, the midpoint of side $BC$ is called $M$. Point $X$ lies on the angle bisector of $\angle AMB$ such that $\angle BXM = 90^o$. Point $Y$ lies on the angle bisector of $\angle AMC$ such that $\angle CYM = 90^o$. Line segments $AM$ and $XY$ intersect in point $Z$. Prove that $Z$ is the midpoint of $XY$ .
[asy]
import geometry;
unitsize (1.2 cm);
pair A, B, C, M, X, Y, Z;
A = (0,0);
B = (4,1.5);
C = (0.5,3);
M = (B + C)/2;
X = extension(M, incenter(A,B,M), B, B + rotate(90)*(incenter(A,B,M) - M));
Y = extension(M, incenter(A,C,M), C, C + rotate(90)*(incenter(A,C,M) - M));
Z = extension(A,M,X,Y);
draw(A--B--C--cycle);
draw(A--M);
draw(M--interp(M,X,2));
draw(M--interp(M,Y,2));
draw(B--X, dotted);
draw(C--Y, dotted);
draw(X--Y);
dot("$A$", A, SW);
dot("$B$", B, E);
dot("$C$", C, N);
dot("$M$", M, NE);
dot("$X$", X, NW);
dot("$Y$", Y, NE);
dot("$Z$", Z, S);
[/asy]
Is there a set $S\subset\mathbb{R}^3$ of $2006$ points such that not all its points are coplanar, no three of the points are collinear, and for any $A,B\in S$ we can find points $C,D\in S$ for which $AB||CD$?